wmmmm PRACTICAL I PHYSICS TOK STUDf:N|:s ui SCIENCE AND ENGINEERING MEASUREMENTS AND PROPBRT!!;^ ^ '^ MAjTf-K v\KT II BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Henrg W. Sage ■^'^ n<*» »"' IW**"" (jb»»V Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924031363108 A OOUESE OF PEACTICAL PHYSICS FOR STUDENTS OF SCIENCE AND ENGINEERING ERVIN S.'' FERRY Professor of Physics in Purdue University PART I FUNDAMENTAL MEASUREMENTS AND PROPERTIES OE MATTER PART II HEAT PRINTED EXPRESSLY FOR THE STUDENTS OP PURDUE UNIVERSITY PUBLISHED BY THE AUTHOR 1903 T A COURSE OF PRACTICAL PHYSICS For Students of Science and Engineering, Bu ERVIN S. FERRY. PART I. Fundamenbal Measuiemenbs and Piopeities of Mabter PART 11. Heab. Parts I and II bound in one volume, price $2.25. PART III. Wave Mobion, Sound and Lighb. In pieparabion. PART IV. Eleobilcal Measuiemenbs. In pieparabion. Copyright. 1903, by Ervin S. Ferry. All rights reserved. TRESS OP BURT-TERRT-WILSON CO. LflPHYETTE, IHD.. u. 5. R. PREFACE. The aim of the present work .is to furnish the student with a laboratory manual of physical processes and measurements in which the explanation of the theory and the description of the method of manipulation of each experiment is so complete as to preclude the necessity of consulting either another book or a labor- atory instructor. The work is so designed that it can be begun at the commencement of the second college year. In the selection of experiments the plan has been to include only such methods as are strictly scientific, that have as practical a bent as possible, and that can be depended upon to give good results in the hands of the average student. This enlists the energy of the student and saves that of the instructor. Although the majority of the experiments are quantitative measurements, some few methods and principles which experience has found to give difficulty to the average college student have been illustrated by qualitative exercises. It has been assumed that the experiment is rare that is so important as to justify a student performing it before he understands the theory involved and the derivation of the formulae required. Consequently the theory of each experiment is given in detail and the required formula de- veloped ' at length. Since, in general, a student can appreciate most completely the physical significance of the various steps in an analytical discussion when couched in terms with which he has been for some time acquainted, it has appeared desirable to use as elementary mathematical methods as possible even though brevity be thereby sometimes sacrificed. It is hoped that the numerous illustrations, all of which have been made especially for this book, will assist in making the text clear and interesting. At the end of many of the experiments is placed a set of questions designed to test the"studtnt's understand- ing of the principles involved in the preceding experiment. A collection of diagrams and engravings is placed at the end of the book so that the student can illustrate his laboratory reports without the labor of making drawings. A few pieces of apparatus, experimental methods, and proofs have been given that may possess some novelty, although the fixed purpose has been to use the standard classical forms except in such cases where a trial of not less than a year by a class of one hundred or more students has demonstrated the superiority of the proposed innovation. In conclusion I desire to acknowledge my indebtedness to Professor Jacob Westlund, of the department of mathematics in Purdue University, for many scholarly suggestions throughout the progress of the work ; and also to Mr. Lloyd E. King, form- erly instructor of physics in Purdue University, who has been so good as to make many suggestions the incorporation of which into the text has added materially to its accuracy and clearness. I am also indebted to Mr. F. L. Shinn of the department of physical chemistry in the University of Wisconsin for the collection and verification of the data on vapor densities contained in thf tables. ERVIN S. FERRY. Purdue University, LaFayette, Ind. September 15, 1903. CONTENTS PART I FUNDAMENTAL MEASUREMENTS AND PROPERTIES OF MATTER Chapter I. GENERAL NOTIONS REGARDING PHYSICAL MEASUREMENTS. Page. Relative accuracy required 1 Discussion of Elrors 2 Rules for Computation 4 Plotting of Curves , 6 Chapter II. LENGTH. Instruments for the Measurement of Length S I. — Measurement of the Diameter of a rod by means of a Microm- eter Gauge and a Vernier Caliper 14 II. — Graduation of a glass scale by means of a Dividing Engine. 15 III. — Verification of a barometer scale by means of a Cathetometer , i 16 The Adjustment of a Cathetometer 17 rV. — Radius of Curvature and Sensitiveness of a spirit level .... 19 V. — Determination of the Thickness of a thin plate by means of a Spherometer and an Optical Lever 23 The Adjustment of a Telescope and Scale 25 VI. — Determination of the Radius of Curvature of a. Spherical surface by means of the Spherometer and the Optical Lever i 26 Vn. — Correction for Eccentricity in the mounting of a divided circle ; . . 29 Chapter III. WEIGHING OE THE COMPAEISON OF MASSES. The Balance 32 VIII. — Weighing by the Method of Vibrations 34 Reduction to the weight in vacuum 37 IX. — Weighing by the Method of Gauss 38 X. — Calibration of a set of Standard Masses 40 Chapter IV. TIME. Instruments for the Measurement of Time 42 The Method of Passages 44 The Method of Coincidences 45 XI. —Study of Falling Bodies 46 Chapter V. VELOCITY AND ACCELERATION. XII. — Determination of the Change of Speed of a flywheel during a revolution ', ; 49 XIII. — Determination of the Speed of a projectile by the Ballistic Pendulum 50 XIV. — ^The Acceleration of Gravity by means of a Pendulum 52 Chapter VI. MEASUREMENT OF AREA AND VOLUME. XV. — Determination of the Area of a plane figure with a Planimeter I 50 XVI. — Calibration of a Burette 63 XVII. — Determination of the Volume of a solid by Immersion 64 XVIII. — Determination of the Volume of a solid with a, Pyknometer 66 XIX. — IDetermination of the Volume of a solid body by the Volumenometer 67 Chapter VII. DENSITY AND SPECIFIC GRAVITY. Definitions 70 XX. — Determination of the Density and Specific Gravity of a, liquid with a Pylaiometer 70 XXI. — Determination of the Specific Gravity of a liquid vpith the Mohr-^Vestphal balance, 72 XXII. — Calibration of an Areometer of variable immersion 74 XXIII. — ^Determination of the Density ar,d Specific Gravity of a solid with a Pyknometer 78 XXTV. — Determination of the Density and Specific Gravity of a solid by Immersion 79 XXV. — Determination of the Relative Densities of gases with Bunsen's Elffusiometer 81 Chapter VIII. FRICTION. XXVI. — Determination of the Coefficient of Friction between two plane surfaces 84 XXVII. —The Friction of a belt on a pulley 86 XXVIII. — Determination of the Coefficient of Friction between a lubricated journal and its bearings 88 Chapter IX. MOMENT OF INERTIA. Definitions 91 Transformation Formulae for the computation of the Moment of Inertia of a body about a given axis in terms of the Moment of Inertia about other axes ... 92 Moment of Inertia of a solid cylinder about its geomet- rical axis 93 Moment of Inertia of a cylindrical ring about its geometrical axis 94 Moment of Inertia of a system consisting of two cylin- ders, about an axis parallel to the geometrical axes of the cylinders 94 Moment of Inertia of a cylinder about an axis passing through its center normal to its length 95 XXIX. — Determination of the Moment of Inertia of an irregular body 97 Chapter X. ELASTICITY. Definitions 99 XXX. — Determination of the Elastic Limit, , Tenacity and Brittle- ness of a wire 100 XXXI. — Young's Modulus by stretching of a wire 101 XXXII. — Study of Flexure of rectangular rods 104 XXXIII. —Young's Modulus by bending of a rod 108 XXXIV. — Determination of Simple Rigidity (Vibration Method) ... .112 XXXV. — Determination of Simple Rigidity (Statical Method) 116 XXXVI. — Determination of the Modulus of Elastic Resilience of a rod 119 Chapter XI. VISCOSITY. XXXVII.— study of Damped Vibrations 121 XXXVIII. — Determination of the absolute Coefficient of Viscosity of a liquid. (Poiseuille's Method.) '. 129 XXXIX. — Determination of the Specific Viscosity of liquids. (Coulomb's Method.) 134 PART II. HEAT. Chapter XII. THERMOMETRY. Errors of the Mercury-in-(jilas& Thermometer 137 The Bi'fkmann Differential Thermometer 140 XL. — C'alibration of a Mercury -in-Ulass Thermometer 141 The Determination of Unknown Temperatures 147 XLl. — Calibration of a, Platinum Resistfuuce Thermometer 148 Chapter XIII. expansion of SOLIDS AND LIQUIDS. Fundamental Equations 153 Reduction of Barometric Readings 155 XLH. — Deleriiiination of the Coeffieient of Linear Expansion of Solids 157 ^Llll. — Determination of the Coellicient of Absolute Expansion of a Liquid by the Method of Balancing Columns 162 XL1\'. — Determination of the Coefficient of Cubical Expansion of Glass 164 Chapter XIV. PKOPKRTIES OF GASES AND SOLUTIONS. The Fundamental Law of Gases , 167 Application to Solutions. Van't Hoff's Law 169 External Work done during Expansion under Constant Pressure 171 External Work done during Vaporization and Solution. .171 The EtHeieney of a perfectly Reversible Thermodynamic Engine 171 XLA'. — Detei-mination of tlie Coefficient of Expansion of a Gas by means of an Air Thermometer 170 XLVI. — Determination of the maximum Vapor Pressure of a Liquid at temperatures below 100° C, by the Statical Method 180 XLVII. ■ — Determination of the maximum Vapor Pressure of a Liquid at various temperatures by the Dynamical Method 184 XLVin. — ^Determination of the Density of an Unsaturated Vapor by Victor Meyer's Method 186 The Relation between Density and Molecular Weight.. 189 Chapter XV. HYGROMETRY. Definitions and General Equations 191 XLIX. . — Determination of the Relative Humidity of the Air with Daniell's Dew Point Hygrometer 192 L. — Determination of the Relative Humidity of the Air with the Wet and Dry Bulb Hygrometer 194 Chapter XVI. CALORIMETRY. Thermal Unit. Thermal Water Equivalent 196 The Correction for Radiation 197 LI. — Determination of the Emissivity and Absorbing Power of flifterent Surfaces 204 LIL —Determination of the Specific Heat of a. Liquid by the Method of Cooling 20S LIII. LV. LVI. Lvn LVIII. LIX. LX. LXI. LXII. LXIII.. LXIV. LXV. LXVI. LXVIl. LXVm. —Determination of the Specific Heat of a Solid by the Method of Mixtures 212 — Determination of the Specific Heat of a Solid by the Method of Stationary Temperature 217 —Determination of Specific Heat with Joly's Steam Calorimeter '. ; 219 Chapter XVII. CHANGE OF STATE. —The Flash Test, Fire Test and Cold Test of an Oil 223 -Determination of the Boiling Point of a Solution 22(5 —Determination of the Freezing Point and Melting Point of a Solution 228 -Determination of the Heat Equivalent of Fusion of Ice. . . .230 -Determination of the Heat Equivalent of Vaporization of Water 233 Chapter XVIII. THERMOCHEMICAL MEASUREMENTS. The Gram Molecule and Gram Equivalent 236 Thermochemical Notation 237 — Determination of the Molecular Heat of Solution of a Salt . 238 — Determination of the Heat of Dilution of a Solution 240 — Determination of the Heat of Neutralization of an Acid and Base 241 — Determination of the Heat Value of a Solid or Liquid with the Combustion Bomb Calorimeter 242 — ^Determination of the Heating Value of a, Gas with Jun- ker's. Calorimeter , 248 — ^Determination of the Molecular Weight of a Substance by the Boiling Point Method 2,i2 — ^Determination of the Molecular Weight of a Substance by the Freezing Point Method 258 — ^Determination of the Degree of Dissociation of a Sub- stance in Solution by the Freezing Point Method 263 Chapter XIX. LXIX. HEAT CONDUCTION. Definitions and Fundamental Equations 265 -Determination of the Relative Thermal Conductivity of two Substances 267 ClIAPTKR XX. THERMODYNAMICS. LXX. — Determination of the Mechanical Equivalent of Heat by Puluj's Method 270 TABLES. 1. — Conversion Factors 27i 2. — Mensuration 27(1 3.— The Greek Alphabet 276 4. — Density and Specifie Gravity of Solids and Liquids 277 5. — Specifie Gravity of Gases and Vapors 278 6. — Specific Gravity of Water at Diflferent Temperatures 279 7. — Specifie Gravity of Aqueous Solutions of Alcohol 279 8. — Specific Gravity of Aqueous Solutions at 15° C 280 9. — Reduction of Arbitrary Areometer Scales 280 10. — Elastic Constants of Solids 281 11. — Coefficients of Friction 281 12. — Viscosity of Liquids 282 13. — Corrections for the Influence of Gravity on the Height of the Barometer 283 14. — Boiling Point of Water under different Barometric Pressures 284 15. — Pressure of Saturated Aqueous "\'apor , 285 16. — Pressure of Saturated Mercury Vapor 285 17. — The Wet and Dry Bulb Hygrometer 286 18. — Coefficient of Linear Expansion of Solids 287 19. — Coefficient of Cubical Expansion of Liquids 287 20. — Heat "S'alues of various Fuels 287 21. — Thermal Emissivity of Different Surfaces 288 22. — Coefficient of Absolute Thermal Conductivity 288 23.— Specific' Heat of Solids and Liquids 289 24. — Specific Heat of Aqueous Solutions at 18° C 289 2-1. — Melting Points and Heat Equivalents of Fusion 290 26. — Boiling Points and Heat Equivalents of Vaporization 290 27. — Cryohydrie Temperatures of various Salts 290 28.^Heats of Solution and Dilution of Sulphuric, Nitric and Acetic Acid 291 29. — Heat of Dilution of various Salts 291 .30.— Heat of Solution of Chepiical Substances 292 31. — ^Heat of Neutralization of dilute Acids and Bases 293 32. — The Atomic Weight and "^'^alencies of the principal Elements 294 33. — ^Degree of Dissociation of various Substances in one per cent. Aqueous Solution 295 Index 296 Illustrations 299 Chapter I. GENERAL NOTIONS REGARDING PHYSICAL MEASUREMENTS. Physical measurement consists in the comparison of the mag- nitude of a quantity with the magnitude of the unit of the same kind as the quantity to be measured. For instance, the distance between two points is given in terms of the length of a certain bar which is taken arbitrarily as the unit of length. Similarly the mass of a body is determined in comparison with a material stan- dard of mass constructed in accordance with an arbitrary defini- tion of the unit of mass. In the same way, an interval of time is measured in terms of a unit arbitrarily defined. These are ex- amples of direct measurements, but much the larger number of quantities must be measured indirectly. For instance. Young's Modulus of Elasticity of a certain ma- terial cannot be measured directly, but can be determined by ap- plying a force of F units to a wire made of it, of length L and radius r so as to cause an elongation e. Then from the laws of elasticity, Young's Modulus of Elasticity M = — ^— . The meas- urement of the different quantities entering into the value of Af are not made with equal ease, and an inspection of the equation shows that errors made in measuring the various quantities affect the re- sult in different degree. This leads to the consideration of the relative degree of accuracy required in the different component measurements enter- ing into the required result. In the above example, it is obvious that the same percentage error in the value of P, L ov e would affect the result equally, but that the same percentage error in the value of r would affect the result much more from the fact that 2 FuNDAMBNTAI, MeASURBMUNTS r appears in the equation to the second power. However, the elongation e, is a very small quantity compared with P or h, and in order that it may be measured with a percentage error no greater than that of P or L, much greater care must be exercised in its accurate measurement. In addition, the fact that the two quantities in which small errors cause the greatest effect in the result are multiplied together, renders the result more affected by their errors than would be the case if they appeared in the formula in any other manner. As an additional example, con- sider the measurement of the specific heat of a substance by the method of mixtures. The specific heat of a substance can be determined indirectly by heating a specimen of the substance of mass M to z. temperature T and then plunging it into a mass m of water at a temperature t contained in a vessel of water equiva- lent B, and observing the common temperature attained by the mixture. In this case the specific heat C, of the substance is given by the equation (m + A') (e — <) ^~ M. K'i—^) It is obvious that in this equation an error in affects the final result more than an equal percentage error in the value of either of the other temperatures, on account of the fact that, dif- ferent from the other temperatures, Q occurs in both numerator and denominator, but with opposite signs. From these examples, it is apparent that before beginning an indirect measurement that requires the determination of sev- eral component quantities, the law connecting the quantities must be examined, and the degree of accuracy deterrriined that is essen- tial in each of the component measurements. The methods and apparatus best suited to the various meas- urements can then be selected. Discussion of Errors. — Any physical measurement is sub- ject to many errors, of which some are determinate and can be eliminated by proper selection of apparatus and experimental method, or whose value can be computed. But after all precautions have been taken, there will still be left some indeterminate sources Errors 3 of error which give rise to what are called residual errors. Thus, after all known determinate errors have been corrected for, the terms in a -series of observations of the same quantity will not in general be identical, but will vary one from another. It thus be- comes necessary to determine from the series of observations the value that has the greatest probability of approximating most nearly to the true value. When all the observations are taken with equal care under the same conditions, the arithmetical mean of all the observations gives the value that has the greatest pro- bability of approximating most closely to the true value. This arithmetical mean is not, in general, the true result, but is the best approximation attainable. The difference between any single observation and the mean of the series is called the deviation of that observation from the mean. The average deviation (or the deviation measure) of the single deviations from the mean, is de- fined as the mean of the magnitudes of the deviations taken with- out regard to sign. By fractional deviation (or fractional pre- cision) of an . observation is meant the ratio of the deviation of the observation from the mean, to the observation itself. Per- centage deviation is one hundred times the fractional deviation. In the case of an extensive series of observations of the same quantity made by the same method with equal care it has been shown experimentally that (o) there is an equal number of posi- tive and negative deviations of the same magnitude, (6) small deviations are more numerous than large ones (c) large devia- tions are few in number. In fact, in an extensive series, a devia- tion having a magnitude four times the mean deviation will occur, on the average, but once in a thousand observations. In a series of observations of the same quantity presumably taken with equal care, sometimes one term will appear that is widely divergent from all the others. From the laws of devi- ations given in the preceding paragraph, this would suggest a mistake due to mental confusion in making or recording the read- ing, or due to some accidental disturbance occuring at the time the reading was taken. These laws of deviations also suggest a rule for the rejection of observations differing widely from the others. Since in a series of carefully made observations it is 4 FUNDAMBNTAI. MEASUREMENTS improbable that there would be more than one deviation in a thousand greater than four times the average deviation of the other terms of the series, one can with propriety assume that if an observation occurs in a short series whose deviation is greater than this amount, that this observation contains a mistake and should be neglected. Computation.- — Computation should be so accurate that errors thereby introduced are certainly less than the errors of observation. The length to which computations should be car- ried depends upon the magnitude of the deviations of the observa- tions, and can be conveniently expressed in terms of the number of significant figures that should be retained in the final result. A significant figure is any digit other than zero. The number of significant figures is independent of the position of the decimal point. The decimal point is simply a mathematical sign to indi- cate the position of the units place. Thus the following numbers all certain two significant figures — 2100.0, 21.0, 0.00021. The following useful rules are taken from Holman's Precision of Measurements where their proof is derived. 1. In computing deviations, retain two significant figures. For example d:=o.o25, ^^31.0. 2. In the single observations, retain enough significant figures to include the place in which the second significant figure of the deviation occurs. For example, Observations. Deviations from mean. Adjusted reading. 46.308 —0.09s 46.308 46.503 o.io 46.50 46.382 • — 0.021 46.382 46.423 0.020 46.423 46.402 0.0016 46.402 232.018 232.015 Mean = 46.4036 Adjusted mean = 46.403 3. When two or more observations are to be multiplied together, or divided one by another, find by inspection the one for which the percentage deviation is the greatest. If the percent- age deviation of this term is between ten per cent, and one per cent, use four significant figures in the multiplicand, multiplier and product, or in division, dividend and quotient respectively. Simi- Computation 5 larly if the percentage deviation of this term is between one per cent and one-tenth of one per cent., use five significant figures throughout. For example, suppose the observations given in the preceding paragraph are to be multiphed by w . The percentage deviation of the first term of this series is 0.20 per cent., so that in multiplying this term by ^ we should use five figures through- out, t. e. 46.308 X 3-1416 = 14548 When logarithms are used, retain as many places in the niantissae as there are significant figures in the data. Above ex- ample becomes log 46.308=1.66566 log 3.1416 =0.49714 log 145.48=2.16280 Many measurements are of such a nature that it is impossi- ble to obtain a number of observations of the same quantity, and for this reason it is impossible to determine the deviation required by the above rule. In this event, the product or quotient should contain the same number of significant figures as the quantity entering into the product or quotient that contains the smallest number of significant figures. In the case of an indirect measurement that involves the de- termination of the ratio of two quantities, it must be noted that it is inaccurate and usually unallowable to use the mean of a series of readings for the numerator of the fraction, and to divide this by the mean of the series of measurements made of the quan- tity entering into the denominator. As an example consider the formula already adduced for Young's Modulus by stretching M = — n— • T 1- P. Ordinarily, L, r and •"■ may be considered to be constant, and then a series of observations is taken of the elongation e produced by various forces P . And since no term of the series of observa- tions oi e or F will in general be the true value of that quantity, it follows that in general the series of ratios obtained by dividing the applied force, F, by the corresponding elongation produced, Cj will not be constant. If these ratios are not constant it is clear that the mean value of M will be given by Fundamental Measurements L 1 (F' F" , F'" Fn irr'' n \ e' ^ e" ^ e'" e„ / and the true mean value of M will not be expressed by M = L /F' + F" +F'" F„ ( F' + F" + F'" F„ \ \ e' A- e" A- e"' e„ ) irr^ \ e' + e" -\- e"' e„ Pi,oTTiNG OP Curves. — When a series of simultaneous obser- vations of two variable quantities has been made, the relation be- tween the two variables is usually rendered more apparent from a graphical representation or "plot" of the observations on co- ordinate axes than from an inspection of the series. This plot is usually made on cross-section paper ruled into centimeters or inches divided into tenths. Jn making a plot, the following rules should be observed. 1. The size of the plot should vary directly with the accu- racy of the observations and inversely with the accuracy of plot- ting. The error oi plotting depends upon the error made in locat- ing the points representing the data on the cross-section paper, and upon the size of the plot. The fractional deviation of the plotting equals the ratio of the average deviation measure of locat- ing the points, to the length of the ordinates of the points. Usually it is not feasible to locate a point on cross-section paper with a smaller average deviation then half a millimeter, so that if the cross-section paper is 20cm long, the fractional deviation of the plotting will not be less than ' = 0.0025 or one quarter of one per cent. Consequently, since errors of plotting affect the third place of significant figures, it is useless and mis- leading to carry plotting beyond the third place of significant figures. If the data includes more than three places of significant figures, the figures beyond the third must be neglected in making the plot. 2. Choose the scale of abscissas so that the largest value reaches across the horizontal space assigned to the plot. Choose the scale of ordinates so that the largest value extends through the vertical distance assigned to the curve. Always choose scales so that each space of the cross-section paper represents a con- venient number of units, or a convenient fractional part of a unit. Plotting of Curves 7 In general the plot should be made on such a scale that the fractional precision of plotting the two co-ordinates will be equal, and also that the fractional precision of plotting equals the frac- tional precision of the data. Thus if X and Y represents one pair of simultaneous observations of the two variables, and x and y the distances proportional to fhe magnitude of X and Y measured off along the axes of abscissas and ordinates of the plot; and if DX, DY, Dx, Dy represents the precision measure of the data X and Y, and the precision measure of plotting the lines x and y respectively, then, the above statement becomes Dx Dy D X X ^^ and ^y -DF X y ~ y X D X y Dy ~ DX " Y ~ DY whence, — ^ and —y- give the number of units of plot per unit of data, so that these ratios give the scale of the plot for the axis of abscissas and ordinates respectively. Dx and Dy are equal; and if centimeter cross-section paper is used for the plot will be equal to about 0.05, while if inch paper is used Dx and Dy will be equal to about 0.025. The data furnishes the value of DX and DY. In an experiment in which one of the variables is determined with many times the precision of the other, it may not be advis- able to plot the two co-ordinates with equal fractional precision. In such a 'tftse, if the co-ordinates were plotted with equal precis- ion, the curve drawn through the points would make such a small angle with one of the co-ordinate axes that in order to keep the size of the plot within reasonable limits, the scale must be made unduly small. When this occurs, it is better to use different scales for plotting the two variables, even though some of the precision of one element of the data is thereby sacrificed in plotting. 5. Draw two heavy lines to represent the rectangular co- ordinate axes of the plot. Along the co-ordinate axes put the scale and names of the quantities plotted. 8 Fundamb;ntal Measurements 4. The points plotted should he plainly indicated by either a puncture in the center of a small circle, or a small cross inter- secting at the point plotted and having its arms parallel to the co- ordinate axes. 5. Draw a fine smooth curve or straight line that best fits all the points plotted. In general this line will not pass through all of the points plotted. This line should be drawn so that there are about as many points on one side of it as on the other side. The points should not be located mainly on one side of the line at any region throughout its entire length. 6. Give each curve a title. This title will usually call atten- tion to the two variable quantities treated and the condition under which particular values were observed. Chapter II. LENGTH. Instruments for the Measurement of Length. — The principle of the Micrometer Screw is of wide application in the accurate measurement of short lengths. A carefully made screw of a pitch of either one millimeter or one-half millimeter turns in an accurately fitting nut. Attached to one end of the screw is a divided head by means of which is measured the fraction of the distance between two threads through which the end of the screw advances when the screw is rotated a fraction of a turn. The principle of 'the micrometer screw is employed in the Micrometer Screw Gauge, the Spherometer, the Dividing En- gine, fhe Filar Micrometer Microscope. Instruments for the Measurement of Length Fig. 1 The micrometer Screw Gauge (Fig i), consists of an accu- rately made screw ^A B Mwi ■■i'^ 'F'"'"""W---^g!^Xjx>H!a which can be ad- vanced toward or away from the stop A. The whole number of millimeters d i s- tance between A and B is indicated by the millimeter divisions on the shank C uncovered by the sleeve D, while the frac- tion of a millimeter is given by the graduated circle on the edge of the sleeve D. If the pitch of the screw is half a millimeter and if the head is divided into fifty equal spaces, one division on the shank will be uncovered by the sleeve for every two complete turns of the screw, and each space on the divided head corre- sponds to an advance of the screw of o.oi mm. The Spherometer, (Fig. 2) is a micrometer screw with a very large divided head passing ver- tically through a nut mounted at the center of an equilateral tripod. If the pitch of the screw is one half millimeter, the head will be usually divided into 500 equal spaces so that a difference of length of 0.00 1 mm. can be read directly. The precision of reading the spherometer is greater than the precision of setting. This instrument is especially useful in measuring the radius of curvature of spherical surfaces — whence its nam' In the Dividing Engine (Fig. 3) a long micrometer screw with a large divided head A is mounted horizontally in a massivo base between a pair of tracks in such a way that it has no longitu- dinal movement, but when rotated causes a nut to advance par allel to the tracks. Attached to the nut B, is a carriage C which Fig. 2 lo Fundamental Measurements slides along the tracks with the advance of the nut. Fastened to the base are one or two microscopes M, with cross hairs in the eye pieces, which can be focussed upon an object resting upon the sliding carriage. In making the measurement of the distance be- tween two points, the carriage is slid along until one point is under the cross hairs of a microscope and then the micrometer screw is turned until the other point comes under the cross hair. Fig. 3. The difference between the reading of the micrometer screw when . one point was under the cross hair, and the reading when the other point was under the cross hair gives the distance between the two points. The dividing engine receives its name from the fact that it is most often used to rule divided scales. Fastened to the base is a system of levers by which a stylus or tracing point 5" can be drawn across the sliding carriage in the direction normal to its motion. By this means a line can be drawn upon an. object fastened to the top of the carriage, the carriage advanced by a definite amount, another line drawn parallel to the first, and so on until a scale is constructed. The mechanism carrying the stylus is often arranged with cams D which permit lines to be drawn of unequal length, so that in ruling a scale every fifth and tenth line may be drawn longer than the others. The Filar Micrometer Microscope is a microscope having in the focus of the eyepiece a reticule supporting a cross hair which can be moved across the field of vision by means of a micrometer Instruments for the Measurement oE Length ii screw. Attached to the inside of the tube of the microscope, in the focal plane in which moves the cross hair, there is usually fixed a serrated edge whose teeth serve as a scale to indicate the .whole number of turns made by the micrometer screw. The distance on the microscope stage corresponding to one turn of the microm- eter must be determined by means of a standard scale. A filar micrometer microscope is illustrated at M Fig 3. The Eyepiece Micrometer is a very finely divided scale ruled on thin glass that can be placed in the focal plane of a microscope. After being put into position the eyepiece micrometer is stan- dardized by means of a standard scale placed on the microscope stage. Vernier's scale is a device employed for the estimation of fractions of the smallest divisions of a scale. It consists of a short auxiliary scale capable of sliding along the edge of the prin- cipal scale. The precision attainable with the vernier is about three times that attainable with the unaided eye. On the ordinary or direct vernier, a length equal to (n — i) divisions of the principal scale is divided into n equal spaces, so that the difference in length between a vernier division an^ a scale division is i-^m of a scale division. Suppose now that the zero line of the vernier be placed in coincidence with the zero line of the principal scale ; then the first line of the vernier will be distant l-^» scale divisions from the first line of the scale, or in general, the r th line of the vernier will be distant ?•-=-« scale divisions from the rth line of the scale. Conversely, when the r th line of the vernier is coincident with any line of the scale, then the fraction of a scale division between the vernier zero and the next preceding line on the principal scale is r -=- n of a scale division. Obviously a vernier can be constructed for either a linear or circular scale and n can have whatever value is f 1 f most convenient. In the case of a centimeter ' Ji'i'i'i 'ill' I I'l 1 ' 1 1 I II scale divided into millimeters, the vernier would usually be nine millimeters long and divided into ten equal spaces. (Fig 4.) This would per- mit readings to be made in tenths of millimeters. In the case of a centimeter scale divided into half millimeters, Fig. 5. 12 FUNDAMENTAI, MjJASUREMENTS llie vernier would usually be 24 scale divisions in length and be divided into 25 equal spaces. (Fig. 5.) This would permit readings to be made directly in fiftieths of millimeters. In the case of a scale divided into inches and fiftieths of an inch, the vernier could conveniently be made 19 scale •divisions long and be divided into 20 equal spaces. (Fig. 6.) This would permit readings to be made directly in thousandths of inches. In the case of a circular scale divided into de- grees and thirds of degrees,the vernier is often made of a length corresponding to 59 scale divisions and is divided into 60 equal spaces. (Fig. 7.) This arrangement permits readings directly to thirds of minutes, i. e. to 20 seconds. The figure in the mar- gin shows such a vernier and also illustrates the manner in which verniers are often numbered so that readings can be read off directly without computation. In this particular case, as each vernier division corresponds to one third of a minute it is natural to number the fif- teenth division 5, the thirtieth division 10, etc., minutes. As an exercise the student is advised to verify the readings of the vernier scales illustrated in the text. The reading of vernier scale in Fig. 4 is 8.6, in Fig 5 1 1. 10 in Fig. 6 29.111 in Fig. J 145° 48', and 11° 14' 40". Piw. 9. Instruments for the Measurement op Length 13 The Vernier Caliper, (Fig. 9), consists of a finely divided steel scale C with a fixed jaw A at one end, and a jaw B provided with a vernier scale D that can slide along the length of the scale. The Cathetometer, Fig 10, is an instrument for measuring ver- tical distances consisting of a rigid vertical column, accurately graduated throughout its length, supported upon a heavy tripod furnished with spirit levels and leveling screws. A car- riage movable up and down the column supports a tele- scope on which is a spirit level. The carriage can be clamped at any point along the length of the column and its position read by means of the scale on the column and a vernier V attached to the car- riage. With the column ver- tical and the telescope hori- zontal, the vertical distance between two points is deter- mined by setting the cross hairs of the telescope first upon one point, taking the reading of the position of the carriage on the column, then elevating or lowering the car- riage until the cross hairs come opposite the second point and observing the new reading of the position of the carriage. The difference be- tween those two readings is it'Mi. 10. the distance required. For greater precision, the telescope is often provided with a screw micrometer by means of which a cross hair can be moved across the focal plane. 14 FUNDAMENTAI, MEASUREMENTS I. MEASUREMENT OF THE DIAMETER OF A ROD BY MEANS OF A MICROMETER GAUGE AND A VERNIER CALIPER. Object and Theory of Experiment. — The object of this experiment is to measure the diameter of the same object with two instruments and to compare the precision of measurement obtained by the two methods. The theory of the two instruments has been given in the preceding paragraphs. Manipulation and Computation. — Before using any measuring instrument having a divided scale, the position of the zero point must usually be noted. It is unusual and unneces- sary for the zero point to be located at the position marked. In using the micrometer gauge, grasp the milled head of the screw loosely with the finger tips and turn the screw until it just slips between the fingers. To find the zero point, bring the end of the screw into contact with the stop and observe the reading on the sleeve of the screw head. Take the average of five such readings as the zero point. Then, placing the object to be meas- ured between the stop and end of the screw, take ten readings. Find the mean of these ten observations, corrected for the zero point ; find the deviation of each observation from the mean, and the average fractional precision of the series of observations. In using the vernier caliper, first ascertain the correction for the zero point by closing the jaws and noting the reading. The average of five readings thus obtained is taken as the cor- rection for the zero. Place the same object previously meas- ured between the jaws of the instrument, close them gently, clamp the jaws by means of the set-srew, and take the reading. Re- peat this operation ten times and determine the average frac- tional precision of the series of readings. Compare the precision of the two methods of measurement. Measurement of Length 15 II. GRADUATION OF A GLASS SCALE BY MEANS OF A DIVIDING ENGINE. Object and Theory of Experiment. — The object of this exercise is to construct a millimeter scale by etching fine lines on a strip of glass. The theory of the dividing engine has been given briefly in a preceding paragraph. The details of the use of the instrument will be best understood from an actual study of it. Manipulation. — The pitch of the screw of the dividing engine is first determined by observing the number of turns of the screw necessary to move the carriage through a known dis- tance. Fasten a standard scale on the carriage in a direction par- allel to the screw. Focus the microscope on one of the divi- sions of the scale, observe the reading of the head of the screw, rotate the screw until another division of the scale comes exact- ly under the cross hairs of the microscope and again observe the reading of the head of the screw. The distance through which the carriage has advanced divided by the number of turns of the screw is the pitch of the screw. Since a screw turning in a nut always has a certain amount of backlash, before taking the first reading the screw must be turned sufficiently to take up any backlash these may be, and then the direction of rotation of the screw must not be changed throughout the series of readings. Clean the glass strip with caustic potash, coat it evenly with parafifin or beeswax, and fasten it to the carriage parallel to the screw. Adjust the ruling mechanism to rule lines along the middle of the strip of glass of the required length, and regulate the cogged wheels of the ruling mechanism so that every fifth and tenth line will be longer than the others. Adjust the stops about the head of the screw A so that when the head is l6 FUNDAMENTAI, MEASUREMENTS turned through the distance between the two stops, the carriage will advance exactly one millimeter. The proper distance be- tween the stops is deterrnined from the pitch of the screw already found. Now the dividing engine is set ready for ruling. Draw the stylus or ruling point S across the glass strip, rotate the screw head through the distance between the stops, and make another mark. This operation is repeated until the required length of scale is divided. Remove the glass strip from the dividing engine, repair any abrasions in the wax surface with a hot iron, and number the divisions with a fine pointed pen. Cover the ruled surface with hydrofluoric acid by means of a large feather. When sufficient- ly etched, the plate is rinsed in running water, the wax scraped of?, and finally cleaned with turpentine. III. VERIFICATION OF A BAROMETER SCALE BY MEANS OF A CATHETOMETER. Object and Theory oe Experiment. — In the ordinary form of Fortin's barometer, the lower end of the tube dips into a reservoir filled with mercury which can be raised or lowered by a screw. By this means, before taking an observation, the sur- face of the mercury in the reservoir is always brought to the level of the point of an ivory pin extending downward from the cover of the reservoir. The barometric height is the length of the mercury column from this point to the horizontal tangent to the meniscus at the upper end of the column. A brass scale attached to the metal tube enclosing the barometer tube is adjusted so that its divisions indicate distances measured from the point of the ivory pin. Not infrequently this scale is inac- curately adjusted by the maker, or becomes out of adjust- ment through handling. The object of this experiment is to measure the barometric height by a cathetometer, to compare this height with the barometric reading, and if they differ, to adjust the scale so as to give correct readings. Measurement oif Length 17 Manipulation and Computation. — Set the cathetometer on a stand of adjustable height about a meter distant from the barometer. The adjustments now to be made on the catheto- meter are four in number. First, parallax. Set the intersection of the cross hairs of the telescope on the image of a dot, and draw the eye piece in and out until there is no relative motion of the dot and the cross hairs when the eye is moved from one side to the other in front of the eye piece. Second. The intersection of the cross hairs must be in the line of collimation of the telescope. With the eye at the eye piece, rotate the telescope about its axis." If the intersection of the cross hairs describes a circle about the image of the dot, this adjustment must be made. When the intersection of the cross hairs and the image of the dot are farthest apart, reduce the sep- aration half way by moving the carriage supporting the tele- scope vertically by means of the fine adjustment screw B Fig. 10 and then bring the intersection of the cross hairs into coincidence with the image of the dot by means of the screw of the eye piece micrometer. Repeat this operation until the intersection of the cross hairs and the image of the dot remains in coincidence throughout a revolution of the telescope about its axis. Third. The spirit level resting on the telescope must be paral- lel to its axis. Bring the bubble to the middle of the level tube by tilting the telescope. Take the telescope out of its Wyes and replace it reversed end for end. If the bubble is now not in the middle of the level tube, bring it half way back by tilting the tele- scope by means of screw D, and the remaining distance by means of the adjusting screws at the ends of the spirit level. Repeat this operation until the bubble will remain in the middle of the tube when the telescope is pointing |n either direction. The student is not to change the adjusting screws at the ends of the spirit level without permission of an instructor.^ Fourth. The cathetometer column must be vertical and the telescope perpendicular to it. If there are spirit levels fastened to the base of the instrument they are used only for the coarse adjustment. Loosen the clamp C so that the column can be rotated about its axis. While the column is being rotated, note 1 8 Fundamental Measurements the position of the bubble in the telescope level. If it does not remain at the middle of the tube, clamp the column at the posi- tion where the bubble is most displaced, and bring the bubble half way back by tilting the telescope and the remaining distance by means of the tripod leveling screws. Repeat this operation until the bubble remains in the middle of the tube throughout a complete revolution of the column. The cathetometer is now ready for use. Raise the tele- scope until the horizontal cross hair in the eye piece is tangent to the meniscus at the upper end of the barometer column. Take the cathetometer reading by means of the scale and vernier. Lower the telescope until the horizontal cross hair coincides with the level of the mercury in the reservoir and again take the cathetometer reading. The difference between these two read- ings is the barometric height. Make five determinations. Now, by means of the screw T, Fig. II, at the bottom of the barometer, bring the surface of the mercury in the reservoir to the level of the ivory point, P. Read the barometric height by means of the scale and vernier, V, at- tached to the case. Attached to the sliding vernier, there is a similar piece of metal directly back of the barome- ter tube. These two slides move together. In order to avoid parallax, when taking a reading the vernier is moved up and down until the position is found where the lower edge of the vernier, the upper surface of the men- iscus and the lower edge of the rear slide are in line. Take the mean of five readings. If the value of the barometric height obtained by reading the ba- rometric scale differs from that ob- FiG. 11. tained by means of the cathetometer, Measurement of Length 19 the barometer scale is out of adjustment. Find the error of ad- justment of the barometer scale by taking the difference be- tween the mean of five determinations of the barometric height by means of the barometer scale and with the cathetometer. IV. RADIUS OF CURVATURE AND SENSITIVENESS OF A SPIRIT LEVEL. Object and Theory of Experiment. — In many measure- ments in which a spirit level is used in connection with other phy- sical apparatus, it is necessary that the sensitiveness of the level be of the same order as that of the other apparatus. An example is the case of the telescope and level of an engineer's transit. When used in leveling or in measuring vertical angles, the least "Verti- cal motion of the telescope should make itself evident both by a displacement of the level-bubble and of the cross hair. It is there- fore important that the magnifying power of the telescope and the sensitiveness of the spirit level should be of the same' order of magnitude. The suitability of a level for a particular use includes the determination of the uniformity of the run of the bubble in the vial and the sensitiveness of the spirit level. The sensitiveness of a spirit level may be defined as the distance the bubble moves for an inclination of the level of one minute. Since the sensitive- ness is directly proportional to the radius of curvature of the vial, Fig. 12, CO Fundamental Measurements it is often designated by the radius of curvature. The object of this experiment is to make a test of a spirit level. In the laboratory a spirit level is usually tested by means of a Level Trier consisting of a base plate upon which rests a T shaped casting supported by two projecting steel points B and F Fig. 12, at the end of the arms of the T, and a micrometer screw M, at the foot of the T. The pitch of the micrometer screw must Fig. 13. be measured and also the perpendicular distance from the micrometer screw to the line connecting the points E and F. The level to be tested L is placed on the T and the position of the bubble in the vial is noted by means of a scale engraved upon the glass or by a scale, S, attached to the level trier. In case it is inconvenient to separate the level from a piece of ap- paratus of which it forms a part, the entire apparatus-, e. g. a tele- scope or theodolite, may be mounted in the grooves ABC or DBF. After the level tube is in place, the micrometer reading is noted. The level trier is now tilted through a small angk by turn- ing the micrometer screw, and readings are again taken of the micrometer screw and the position of the bubble. In Fig. (13), let AB, AC represent the two positions of the level trier inclined at an angle 6 to one another : let the upper face of the spirit level be represented by the circular arcs (BG) and (CF) of radius GJ=FH. Now since a free liquid surface is always perpendicular to the direction of the force acting upon it (in this case the attraction of gravitation), the bubble is always at that point of the arc of the vial at which a plane perpendicular to the force is tangent to the arc. Whence, if in the diagram, GJ and BH are vertical lines, and if in the first position of the spirit level the bubble was at G, then in the second Measurement of Length 21 position it would be at E, having been displaced through a dis- tance FB. Prom the figure, 9 = ^ / radians, and also G = 5^ radians. FH When B is small, the arc {BD) may be considered equal to BC. For purposes of abbreviation, denote BA by x, BC by y, {HF) by d, and the radius of the vial FH by R. 36o° Since one radian ^^ =57.296 =3437.75', we can transform radians into minutes by multiplying by 3437.75. It follows from the definition of the sensitiveness of a level that „ . . d x d , , . Sensitiveness = -^ = „.-„.„.■ — • (i) 9 3437.75 y %l d. Again, since 9 = ^— = -77- ' we have the radius of curvature R = — (2) ?/ Whence it is seen that the sensitiveness of a spirit level is directly proportional to its radius of curvature. Manipui-ation and Computation. — Measure the perpen- dicular distance from the end of the micrometer screw to the line connecting the two pivotal points from an impression of these three points obtained by pressing them on a piece of paper. The pitch of the micrometer scfew is obtained in tHe following man- ner : after placing the spirit level on the trier adjust the microme- ter screw until one end of the bubble is directly under a scale division near the middle of the vial ; then insert under the microme- ter screw a small piece of plate glass whose thickness has been al- ready measured with a spherometer or micrometer gauge, and again adjust the micrometer screw until the bubble rests at the same point as before. The number of turns of the micrometer screw necessary to make this adjustment divided by the thickness of the glass plate is the pitch of the screw. Again, adjust the micrometer screw until one end of the bubble is directly under a scale division near one end of the vial. 22 Fundamental Measurements Observe the micrometer screw reading and the scale readings at both ends of the bubble ; rotate the micrometer screw through lo spaces and take readings as before. Continue this operation until the bubble has been moved to the other end of its run, and then return step by step in the same manner. Repeat this series of read- ings three times. A series of such readings may be conveniently tabulated in the following form. Number of Observation Micrometer Eeaaing Level Scale Readings. Displacements. Length of Bubble Left Enfl Bight End Lett End Eight End 1 5 1.3 10.2 8.9 2 15 6.1 14.9 4.8 4.7 8.8 3 25 11.1 19.8 5.0 4.9 8.7 4 35 ■ 16.2 25.1 5.1 5.3 8.9 5 45 21.1 30.2 4.9 5.1 9.1 4 35 16.1 25.1 5.0 5.1 9.0 3 25 11.1 19.9 5.0 5.2 8.8 2 15 6.2 15.0 4.9 4.9 8.8 1 5 1.3 10.2 4.9 4.8 8.9 Me an 4.95 5.00 8 88 The values in columns 5 and 6, show the uniformity of the run of the bubble, or the variation in sensitiveness when the bubble is at different positions in the vial. The average sensi- tiveness and the radius of curvature of the vial is obtained by substituting for d in eqs. (i) and (2) the mean displacement ob- tained from columns 5 and 6. The student must be careful to measure the three quantities X, y and d in the same units, that is, all three must be measured in inches, or all in centimeters, etc. Care must also be taken to keep the entire vial at the same temperature. It must not be touched by the fingers or breathed upon, as when unequally heated, the bubble tends to move toward the point of highest temperature. Test Questions and Problems. I. In the case of a spirit level attached to the telescope of Measurement op Length 23 Fig. 14. an engineer's transit or theodolite, the ra- dius of curvature of the spirit level can be calculated from an observed target move- ment /, at a given distance Lj produced by a deflection of the telescope such that the bubble is displaced by an amount d. In this case prove that the radius of curva- ture of the level is V. DETERMINATION OF THE THICKNESS OF A THIN PLATE BY MEANS OF A SPHEROMETER AND AN OPTICAL LEVER. Object and Theory oe Experiment. — The object of this experiment is to measure the thickness of a thin microscope cover glass by two methods and to compare the precision of measurement obtained by the two methods. The spherometer has already been described. The theory of the optical lever will now be deduced. The optical lever to be used in this experiment consists of a piece of sheet brass 3 cms. long and i cm. wide mounted upon four pointed legs, one at each end and the other two midway between the end legs and in a line normal to the line joining them. Fastened on the upper side of the optical lever, with its reflecting surface in the plane of the two middle legs is a small glass mirror. The length of the four legs may be such that when the optical lever rests upon a piece of plate glass all four legs are in contact with the glass, or the end legs may be slightly shortened so that the optical lever can be tilted forward and backward about the ends of the middle legs. From the difference in the angle through which the optical lever can be tilted when the middle legs rest directly upon a large plane surface, and the angle through which it can be tilted when a thin plate is interposed 24 Fundamental Measurements between the middle legs and the plane surface, the thickness of the thin plate can be determined. Fig, 15. Let mna be the optical lever with its mirror approximate- ly normal to the base mn, T a telescope and O'O" a vertical scale distant about one meter from the optical lever. First as- sume that the ends of the feet of the lever are all in one plane. Imagine the thin plate x placed under the middle feet of the op- tical lever. When the lever is tilted forward an observer at the telescope sees the point of the scale at 0' reflected in the mir- ror, and when the mirror is tilted backward the reflected image of the scale at 0" comes opposite the cross hair of the telescope. Meantime the optical lever has been tilted through the angle 6 , consequently the angle between the normals to the mirror in its two positions is also Q. And since the angle of reflect- ion equals the angle of incidence, the angle between o'a and o"a' o' o" equals 2 0. When is very small, = 2 6 radians, and also = e, consequently 2mm' oa Let the thickness of the thin plate be denoted by h, the dis- tance mn between the two end feet be denoted by 2I, the dis- tance oa between the scale and mirror be denoted by L and Measurement of Length 25 the scale reading o'o" be denoted by S. Again, since c is mid- way between tn and n, it follows that the distance wm' is twice the thickness h of the thin plate. Making these substitutions the thickness of the plate h = -r-7 (approx.) (1) This is for the case of an optical lever having the lower ends of all four feet in one plane. But if the end feet are shortened so that the lever is capable of being tilted enough to produce a deflection S' when placed upon a plane surface, then the above value of the total deflection 5 must be diminished by the de- flection S' . For this case, the thickness of the thin plate h = ^^^^ (approx.) (2) Manipulation and Computation. — In using the spherome- ter a series of five observations of the zero point must first be made by placing the instrument on a piece of plate glass and noting the readings on the two scales when the point of the screw is just in contact with the glass plate. Now raise the screw, place under it the thin plate whose thickness is to be measured, lower the screw until it is just in contact with the thin plate and note the reading on the two scales. Repeat five times. The difference between the mean of these five readings and the mean zero reading is the thickness of the plate. In using the optical lever, the telescope and scale must first be adjusted ; that is, the telescope, scale and mirror of the optical lever must be placed in such relative positions that on look- ing through the telescope toward the mirror a reflected image of the scale will be visible. In making this adjustment, place the vertical scale facing the mirror distant about a meter from it, then standing behind the scale and looking at the mirror, move the eye about until a reflected image of the scale is seen in the mirror. If the telescope is now placed where the eye is sit- uated, it is obvious that on looking through the telescope toward the mirror the same image will be visible, as soon as the telescope is properly focused. If the position in which this would require the telescope to be placed is inconveniently high 26 Fundamental Measurements or inconveniently low, the mirror may be tilted with respect to the base of the optical lever until the image of the scale is visi- ble in a more convenient position. Next adjust the eyepiece of the telescope, by sliding it back and forth, until the cross hairs are not only distinctly visible but also do not move with refer- ence to the image of the scale when the eye is moved. When the telescope and scale are adjusted, with the optical lever on a piece of plate glass, observe the scale reading in the telescope when the optical lever is tilted forward and when it is tilted back. The difference between these two readings is S'. Now place under the middle legs of the optical lever, the thin plate whose thickness is to be measured and take two similar readings. The difference between these readings is 5. The distance L, may be measured with a meter stick or steel tape. The distance /■ is best obtained from the measurement of the distance between prick-marks made by pressing the feet of the optical lever on a sheet of paper. Make five determinations of the thickness of the thin plate by the optical lever method, and compare the precision of the two methods. Test Questions and Problems. I. Discuss the approximations involved in eq. (2_), showing which assumptions can be easily fulfilled and under what condi- tions the optical lever method is most accurate. VI. DETERMINATION OF THE RADIUS OF CURVATURE OF A SPHERICAL SURFACE. Object and Theory oE Experiment. — There are three prin- cipal methods for determining the radius of curvaure of a spherical surface. They are by rneans of (a) the spherometer, (&) the optical lever, (c) the reflection of light. The last method is ap- plicable only to highly polished surfaces and its consideration will be delayed until the subject of light is taken up. The object of Measurement oe Length 27 this experiment is to determine the radius of curvature of a spherical surface by means of the spherometer and by means of the optical lever, and to compare the precision of the two methods. The theory of the two methods will now be considered. (a) By means of the sphcroinctcr. The curvature of the sur- face is determined from the dimensions of the spherometer and the distance through which the point of the screw must be moved from the plane of the ends of the three legs in order that all four points may be brought into contact with the spherical surface. L,et ABC Fig. 16 be the positions of the three fixed feet, and F the position of the point of the screw, when all four are in one plane. Let the dis- tance between ^ny two legs of the tripod be denoted by /, and the distance between the points of the screw and any one of the feet be denoted by d. Then if R is the required radius, and h is the height of the point of the screw above the plane of the ends of the three feet when all four are in contact with the spherical surface, then from Fig. 17. h^ + d' Fig. 17. 7? = ^ (R-hr + d^ = 2 h (1) But since / can be measured more accurately than can cf, the value of d in terms of I is usually substituted in this equation. Fig. 16. il 4 GB w /37^ -1" = /3\^ Thus from the diagram. cP = GF^ + But GF = Whence I d = —^ V 3 (2) (3) Substituting value of d from eq. (2) in eq. (i) (b) By means of the optical lever. Fig. 18 and Fig. 19 rep' resent two views of the optical lever resting on the curved sur 28 Fundamental Measurements face, at right angles to one another. Let the end points of the lever touch the spherical surface at F and D, and the middle points Fig. i8. Fig. 19. at B and H. Let R represent the required radius of curvature of the spherical surface, 2I, the distance between the end points, and 2b, the distance between the two middle points. From Fig. 18, . R^'z^iR—ACy+CD'- 2R-AC=2R {AB+BC)=^C''+CD'' Similarly from Fig 19 2 R-AB^AB'+BH' But since AC is small compared with CD, and AB is small com- pared with BH; and CD and BH an approximately equal respec- tievly to I and bj the above equations may be written, 2i? {AB+BC)=P (4) and 2R-AB=2R-AB=b'' (5) since AB in Fig. 18, obviously equals AB in Fig. 19. From the theory of the optical lever, it has been shown (eq. 2 Ex- periment V) that BC--= (S—S') I 4 L (6) Substituting for AB and BC their values obtained from eqs. (5) and (6) eq. (4) becomes 2 R I (S-S') ,, Whence h' + R (7) Manipulation and Computation, (a) When sphero- meter is used, observe the zero point of the spherometer by plac- ing the instrument on a sheet of plate glass and noting the read- Measurement op Length 29 ing on the two scales when the point of the screw is just in con- tact with the glass plate. Then press all four points of the sphero- meter on a piece of paper or tinfoil and measure / by means of a pair of sharp pointed dividers and a finely divided scale. Now rotate the screw until when placed upon the spherical surface whose curvature is to be determined, the point of the screw and the points of the three legs will all be in contact with the curved surface. Again take a reading of the two scales of the instrument. The difference between this reading and the zero reading is h. Substitute these values of / and h in eq. 3, and solve for R. Obtain the mean of five values of R determined in this way from five sets of observations. (b) When the optical lever is used. Press the end points of the lever on a piece of paper or tinfoil and measure 2/ by means of a pair of sharp pointed dividers and a finely divided scale. In the same way. measure 2h. Place lever on the curved surface, adjust a telescope and scale, measure L, and observe the differences of scale readings 5* and S' exactly as described in the preceding experiment. Make five distinct sets of observations, substitute value in eq. (7) and take mean result for the value of R. Compare the precision of the observations taken in the two methods. The spherometer method is especially useful for finding the radius of curvature of a surface of considerable extent, while the optical lever method is available for surfaces of limited extent and small curvature. VII CORRECTION FOR ECCENTRICITY IN THE MOUNT- ING OF A DIVIDED CIRCLE. Object and Theory of Experiment. — Angles are often measured by means of a divided circle and an index or vernier attached to an arm capable of rotation about an axis passing through the center of the circle. This method is subject to a source 3° Fundamental Measurements of error due to the mechanical difficulty of mounting the arm carrying the vernier so that its axis of rotation accurately coin- cides with the normal axis of the divided circle. The object of this experiment is to construct a correction curve for a divided circle having an eccentrically mounted vernier. Let C be the center of the divided circle, A and B the zero points of two verniers car- ried on an arm capable of rotation about the point D. If the line AB passes through D and D coincides wih C, there is no eccen- tricity in the mounting, and correct angu- lar readings are obtained by means of a single vernier. But in the general case ^ where neither of these conditions is fulfilled, correct angular readings can only be ob- tained from simultaneous readings of the two verniers A and B. Let A° and 5° be the observed readings. Draw A.^ B^ through C parallel to AB. If there were no eccentricity in the mounting, that is if the points D and C coincide and if AB passes through D, the readings would be .4°i and 5°i. In other words /4°i and B°-^ are the true readings corresponding to the observed readings A° and B° . Draw through C the Hues BB and AF. Since A^B^ is parallel to AB and AC equals CB, the angle BCA^=BCB^=ABB—BAC=ACA^. Therefore XCA^=y2(XCE+XCA) or A\=y2 {B°+A°). If the divisions on the divided circle be numbered as shown in the figure, B°^=B° — i8o°. Consequently the corrected reading of the vernier A is A\=y2 {B°+B°—i8o°) (1) This is the corrected observation of the vernier giving the numer- ically smaller reading. In precisely the same manner, since B°.^=^'y (B°-{-F°) and since B°^^i8o°-\-A° we obtain for the corrected reading of the vernier B, B\=y2 {A°+B°+i8o°) (2) Weighing 31 This is the corrected observation of the vernier giving the numer- ically larger reading. In this manner, by means of two verniers, is obtained the reading of either vernier corrected for eccentricity of mounting. MANIPU1.AT10N AND Computation. — Starting with one vernier at the zero point of the circle take observations of both verniers. Then move the vernier ten degrees and again read both verniers. Repeat at intervals of ten degrees until the entire cir- cumference is traversed. From these data- compute the corrections of the observed vernier readings. On cross-section paper lay off observed vernier readings on axis of abscissas and the corresponding corrections on the axis of ordinates. The curve drawn through the points thus obtained is the correction curve required. From the form of this curve de- cide whether the axes through C and D are coincident, and whether the line AB passes through D. Test Questions and Problems. I . Derive the equation analogous to eq. ( i ) that applies to a graduated circle having the scale divisions numbered from the zero point in both directions to 180" . Chapter III. WEIGHING, OR THE COMPARISON OF MASSES. The mass of a body is the quantity of matter it contains in comparison with the quantity contained in some particular body arbitrarily assumed to contain the unit quantity of matter. By definition, the unit of mass is taken as the mass of one cubic centimeter of water at the temperature of its maximum density. This unit is called the gram. In order to realize a practical standard of mass Borda constructed a cylinder of platinum which was intended to represent the mass of one thousand grams. This kilogram is deposited in the Archives of Paris, 32 Fundamental Measurements and although its mass slightly differs from the definition of a kilogram, it constitutes the prototype standard in terms of which all scientific measurements of mass are made. The weight of a body is the force with which it tends to move toward the earth. Since the weight of a body equals the product of its mass and its acceleration, it follows that so long as the acceleration remains constant, the masses of any bodies are proportional to their weights. Consequently masses of bodies can be compared by means of a comparison of their weights. This method of comparing masses is called weigfhing. Fig. 21. The Balance. In accurate weighing, a balance is used having arms of equal length. The beam is capable of rotation at its middle point upon a knife edge which rests upon an agate plate. Suspended from knife edges at the ends of the beam are .two scale pans in which are placed the masses to be xompared. Fastened to the beam is a long pointer which swings in front of a graduated scale when the beam vibrates. Weighing 33 The unknown masses are compared with standard masses popularly through improperly called "weights." Ordinarily, standard masses of one gram and upwards are made of brass, and fractional masses are made of platinum or aluminum. As metallic bodies are liable to gradually become lighter by abra- sion through use and cleansing, standard masses are some- times made of quartz. As very small masses are inconvenient to use, they are usually not made of less than ten milligrams (o.oio grms.) mass. Instead of using smaller ones, a ten milli- gram mass is made of aluminum wire, which can be placed at different points along the beam of the balance by means of a rod extending out through the sides of the balance case. This bent wire is called a rider. The beam of the balance from the point directly above the central knife edge to the point directly above the knife edge supporting one of the scale pans, is divid- ed into ten equal parts. The division directly above the central knife edge is marked zero, and the others are numbered in order I, 2, 3, etc., to the end of the arm. If now, the lo milligram rider is placed on the beam at the point marked i, it produces the same moment as would a mass of one milligram placed in the scale pan : if placed at the point marked 6, the lo milligram rider produces the same moment as would a mass of six milli- grams if placed in the scale pan. Naturally a rider of any other mass can be used if the beam be divided accordingly. Usually each of the numbered divisions of the balance beam is subdivid- ed into tenths, so that by means of the lo milligram rider, differ- ences of mass not greater than o.i milligram (o.oooi gms.) can be directly measured : and by estimating tenths of the small sub- divisions of the beam, differences of mass of o.oi milligrams (o.ooooi gms.) can be measured. In order to protect the knife edges of the balance when not in use, and while the masses in the pans are being changed, a balance is provided with an arrestment by which the weight of the beam and of the pans can be lifted off the thin knife edges. This arrestment is operated by means of a milled head protrud- ing from the front of the base of the balance case. 34 FuNDAMENTAIv MEASUREMENTS Precautions in the use of a Balance. 1. Raise the arrestment whenever masses are to be changed. 2. Never touch the standard masses with the fingers — use forceps. 3. All substances that are liable to injure the pans must be weighed in appropriate vessels. 4. Never stop the swings of a balance so suddenly as to cause a jerk. 5. When the weighing is finished, raise the arrestment, put the standard masses in their proper places in the box, dust off the pans and the floor of the case with a camel's hair brush, and close the case. 4. Never stop the swings of a balance so suddenly as to cause a jerk. 5. When the weighing is finished raise the arrestment, put the standard masses in their proper places in the box, dust off the pans and the floor of the case with a camel's hair brush, and close the case. VIII. WEIGHING BY THE METHOD OF VIBRATIONS. Object and Theory of Experiment. — The object of this experiment is to determine the mass of a body in terms of the masses of bodies taken as standards of mass. The zero point of a balance is the point of the ivory scale at which the pointer comes to rest when the balance is not loaded. As a rule the zero point is not exactly at the middle of the scale. In the case of a balance with arms of equal length, when the object to be weighed is on one pan and the standard masses on the other are so adjusted that the pointer rests at the zero point, then the mass of the object equals the mass of the standards. It would be extremely tedious and in fact it is quite unnec- essary to wait until the pointer actually comes to rest in order WEIGHING 35 to determine the zero point of a balance. It is also unnecessary to continue the adjustment of the standards until the masses in the two pans are equal. By means of the method of vibrations, the point on the ivory scale at which the pointer would finally come to rest can be determined from the observation of a few oscillations of the pointer. By the sensibility of a balance for any given load is meant the difference in the scale reading pro- duced by the addition of one milligram to one pan, when each pan has acting upon it a weight equal to the given load. The sensibility of a balance is different for different loads. Weigh- ing by the method of vibrations comprises three operations : first, the zero point is determined; second, the object to be weighed is placed on one pan and standards of mass placed upon the other until the point of rest for the loaded balance is within two or three scale divisions of the zero point ; third, a small overweight is added to one pan and the new point of rest de- termined. From the second and third operations the sensibility is determined. From the sensibility can be computed the amount of mass that must be added to, or subtracted from, that upon the mass pan in order to bring the point of rest of the loaded balance to the zero point. This gives the relative mass of the body. Manipulation and Computation. — The details of the ma- nipulation and computation will be clearly understood from the consideration of an example. With the balance pans empty and case closed, release the arrestment so as to cause the point- er to swing back and forth through a small arc. It will be noticed the vibration of the pointer is like a pendulum in that the period of vibration is constant while the amplitude of the swings constantly diminishes. Note the turning points of the pointer for a number of successive oscillations. Guard against parallax in observing the pointer. In some balances the mid- dle point of the scale is marked zero and the end points — lo and -(- 10 respectively. The inconvenience of the positive and negative sign can be obviated by mentally calling the left hand division o, the middle division lo, and the right hand division 20. For the determination of the zero point the following turn- ing points of the pointer were observed. 36 Fundamental Measurements Oscillations to the left. Oscillations to the right. (2) 13-4 (4) .13- (I) 7.6 (3) 7.2 (5) 6.8 Average =7.2 Average =13.2 Since the friction causing the gradual diminution of the ampli- tude of vibration acts for equal lengths of time on each com- plete vibration the mean of any two successive swings to the left equals the length of the intermediate swing to the right. Or in general, if the first and last swing observed are on the same side, the mean of the left swings equals the mean of the right swings. Consequently the zero point of the balance, that is, the point where the pointer would finally come to rest is the mean be- tween the average. left-hand swings and the average right-hand swings. In the above example the zero point is 10.2 scale divi- sions. While the arrestment is elevated' so as to lift the beam off the knife edge, place the object on one pan and add standard masses to the other pan, until on lowering the arrestment, the pointer does not swing off the scale. Right-handed persons will find it most convenient to place the object on the left pan so that the mass pan is in front of the hand that makes the ad- justment of the standards. In the same manner as when the zero point was determined, observe a series of turning points, and from these turning points deduce the point of rest. Suppose that with the object in the left-hand pan, a load of 24.166 gms. in the mass pan gives the point of rest at 7.4 scale divisions. Since this point of rest is to the left of the zero point, the mass in the right hand pan is in excess by the amount necessarv to move the pointer from 10.2 to 7.4. of 2.8 sc^le divisions. In order to determine the mass that will produce this deflection of 2.8 scale divisions, note the change produced in the point of rest by a small change of mass. Suppose that when, by means of the rider, the mass in the right side of the balance is dimin- ished by 2 milligrams (0.002 gm.), the new point of rest deter- mined as before, is found to be 11.6 scale* divisions. Then since 2 mg. produces a change of 11. 6 — 7.4=4.2 scale divisions, Weighing 37 it follows that one milligram produces a deflection of 2.1 scale divisions. This number, 2.1, is the sensibility of the balance for the particular load on the object pan. Since i mg. produces a deflection of 2.1 scale divisions, the nun.'ber of mg that must be taken from the mass pan in order to bring the point of rest from 7.4 scale divisions to the zero point of the balance, 10.2 equals (10.2 — 7.4)-^2. 1=1.3 "ig. Consequently the apparent mass of the object in air is 24.166 — 0.0013^24.1647 gms. In or- der to find the true mass of the object, this apparent mass must be corrected for the buoyant force of the air acting on the object and on the standards of mass. Reduction to the weight in vacuum. A body immersed in air or any other fluid loses an amount of weight equal to the weight of the air or other fluid displaced. Consequently bodies of equal weight in air but of unequal vol- ume, are not of equal mass. Masses of bodies are strictly pro- portional to their weights only when the weighings are either performed in vacuum, or the weights reduced to what they would be if the weighings had been actually made in vacuum. This reduction is easily effected. Let / represent the weight of the body in air; let P repre- sent the weight of the body in vacuum ; and let D, d, p represent respectively the density of the body, of the standard masses and of the air. The volume of the body V=F-^D ; the volume of the standard masses v^f^^d. The loss of weight of the body by the buoyancy of the air =p T^and the loss of the standard masses = p v. T'herefore since in air, the standard masses balanced the body, F — p V=f — pv F f .-. F =f ^ F [9 p p^ p^ p' p' Now since the density of air, which is about 0.0012 gms. per c. c, is a very small quantity compared with D or d, the terms 38 FUNDAMENTAI, MEASUREMENTS in the above series containing p= and higher powers may be neg- lected without affecting the result. Therefore the weight re- duced to vacuum is In the case of brass standards of mass, 1^=8.4 gms. per c. c. And since p=D.ooi2 gms. per c. c. the above equation becomes F=f [2 + 0.0012 (-1 -0.12)] approx. Test Questions and Problems. I. Explain the effect of the following quantities on the sen- sibility of a balance, — (a) length of beam, (b) mass of beam, (c) distance between the central knife edge and the center of gravity of the beam. IX. WEIGHING BY THE METHOD OF GAUSS. Object and Theory op Experiment. — There are four prin- cipal sources of error to be considered in making an accurate weighing. The are (i), a difference in density between the bodies on the two scale pans: (2), errors in the standards of mass: (3), inequality of the lengths of the arms of the balance; (4), inequality of the masses of the scale pans. The method of correcting for the first error has been considered in the preceding section : the method of determining the second is described in the succeeding experiment entitled Calibration of a Set of Standards of Mass; while the method of eliminating the third and fourth sources of error is the object of the method of Gauss now to be considered. In this method the object is weighed twice by the method of vibrations, first in one pan and then in the other. Let R and L represent the lengths of the right and left arms of the balance respectively, P and Q the weights of the right and left scale pans respectively. Let an object of true weight P have an appar- Weighing 39 parent weight of /' when placed on the right scale pan, and an ap- parent weight of f when placed on the left scale pan. Then from the principle of moments, {F+P)R={f+Q)L. iF+Q)L=if'+P)R. In the case where the balance is in equilibrium when the beam is horizontal, PR= QL; therefore for this case, the above equa- tions become FR=fL. (1) FL=f'R. (2) Multiplying eq. (i) by eq. (2). F = / ^ (3) That is, the true weight of the body is the geometrical mean of the two apparent weights obtained by first weighing the body in one scale pan and then in the other. Manipulation and Computation. — Proceeding exactly as in the preceding experiment find the zero point of the balance. If this is found to be not more than two scale divisions from the center of the scale, then the requirement that the beam must be horizontal when in equilibrium, is fulfilled with sufficient closeness to permit the employment of equation (3). Using the method of vibrations, weigh the body first in one pan and then in the other, thus obtaining / and f. A more convenient form of eq. (3) can be obtained by an ap- proximation so close as to be nearly always applicable, as fol- lows: F=Vff = i (f+f) - i (//- ^f')\ In the case of any balance in ordinary adjustment, / and f will not differ by more than one thousandth of the magnitude of either f or /'. Consequently the quantity yi Wf— Vf'f is quite negligible compared with the quantity V2 (f -\- f). The value of the true weight then becomes approximately equal to the arith- metical mean of the two apparent weights obtained by first weighing the body on one scale pan -and then on the other. That is, f+f F = — =— (approx.) 40 FuNDAMENTAi, Measureme;nts Test Questions and Problems. I. Dividing eq. (i) by eq. (2) obtain the ratio of the lengths of the arms of the balance, and show that R f —f -77 = 1 + -^y^(approx.) ^4) CALIBRATION OF A SET OF STANDARD MASSES. Object and Theory of Experiment. — Only in a most care- fully adjusted set of standard masses will the masses of the vari- ous members of the set be equal to ttieir marked value, or even be strictly proportional to their supposed value. It is conse- quently necessary in any accurate work, to compare their masses one with another ; and where absolute weighings are to be made, the masses in the set must be compared with an ultimate standard. The object of this experiment is to compare the masses of the various members of a set and to construct a table of correc- tions. The theory of the method is to compare, by means of a sen- sitive balance, masses or groups of masses supposed to be equal, forming frofn these results as many separate equations as there have been weighings performed, and from these equations deduc- ing the ratio of the masses to some convenient unit. In this experiment the unit of comparison will be the mass of one of the standards in the set being calibrated. Ordinarily there must be made as many separate weighings as there are standard masses to be compared. If the arms of the balance are of equal length, the weighings may be made by the ordinary method of vibra- tions ; but if the arms are not of equal length, the method of double weighing must be employed. Consider a set comprising the following brass standard masses, — three i gram, one 2 gram, one 5 gram, two 10 gram, one 20 gram, and one 50 gram. In addition there will ordinari- ly be an equal number of fractional grams, made of aluminum or Weighing 41 platinum, of- masses one hundredth of the corresponding brass standards. Distinguish the brass standards by the symbols i', i", 1'", 2' 5', 10', 10", 20', 50'. Let the usual case be considered where the arms of the balance are of unequal length. Perform a double weighing with i' in one pan of the balance, and i" in the other. Suppose that with i" in the left pan and i' in the right scale pan, a certain small weight / must be added to the right scale pan in order to bring the balance to its zero point : and that when the masses are reversed, a certain small weight /' must be added to the left scale pan in order to bring the balance to its zero point. Then from the principle of double weighing. In the same manner is obtained 2'=i"+i"'+%(/t-|-;j')- etc.^etc. For purposes of abbreviation, represent the quantities )4 (/+/')> y^ig+g'), y2ih+h'), etc., by A, B, C, D, etc. Obviously the quantities A, B, C, etc., may be either additive or substractive. Then we have the following equations : i"=i'+^. i"'=i"+B. 2'=l"+l"'+C. 5'z=i'+i"+i"'+2'+D. io'= i'+ 1"+ 1"'+2'+5'+£. io"=io'+P'. 20'=I0'+I0"+G. So'= i'+ 1"+ 1"'+2'+5'+ io'+ 10' '+20'+H. From this series of equations, the value of all the masses in the set can be determined if any one of them is known. For exam- ple suppose the value of i' is definitely known in comparison with an ultimate standard. Then, comparing all the other standard masses with i', we have I'=l'. i"=i'+^. i"'=i'+^+5. 42 Fundamental Measurements 2'=2{l')+2A+B + C. 5'=S{i')+4A+2B+C+D. iQ'=io{i')+8A+4B+2C+D+B. io"^io{i')+8A+4B+2C+D+B+F. 2o'=2o{i') + i6A+8B+4C+2D+2B+F+G. So'^So(i')+4oA+2oB+ioC+sD+4B+2F+G+H This is the table of corrections required. If the absolute value of no standard mass is known, then the above table gives the value of various masses in terms of i'. In precisely the same manner the masses of the fractional grams and of the rider are compared with i', except that in the case of these small masses it is unnecessary to use the method of double weighing even though the lengths of the two arms of the balance are not exactly equal. Chapter IV. TIME. . Instruments for the Measurement oe Time. — In nearly all apparatus used for the measurement of time, the principle is em- ployed that the period of a body vibrating with harmonic mo- tion is constant. Examples of vibrating bodies often used for measuring time are, the pendulum, the "balance wheel" (which vibrates with circular harmonic motion), and the tuning fork. Attached to the vibrating body is usually a mechanism called "clock-work" for maintaining and counting the vibrations. The vibrating body may be kept in motion by imparting to it a slight impulse every time it passes through its position of equili- brium. The most common laboratory instrument for the measure- ment of time is the clock having a pendulum which makes one beat or half of a complete vibration every second. The error of a clock is determined from the transit of a star. In the watch Measurement op Time 43 and the chronometer a spring and balance wheel are used. These instruments are employed where accuracy must be sacrificed to portability. They are sometimes provided with a starting and stopping device by which the interval between two events can be easily ascertained. The chronograph is a recording instru- ment consisting of a cylinder carrying a sheet of paper which is revolved at a uniform rate by means of clockwork. Resting upon the paper are pens A and B (Fig. 22) attached to arma- tures actuated by electromagnets fastened to a carriage which slides on a track parallel to the axis of the cylinder. When the clockwork is running, the pen traces on the paper a helical line _J||||M™""'"l!IIIMi««iJJhlllJ.lJflfM^^ 1 1. Fig. 22. until an electric current is sent through the electromagnet. A dis- displacement of the line is then produced. A clock pendulum can be so connected to the circuit containing the electromagnet that a notch will be made in the line every second. By introducing a telegraph key into the circuit an observer can produce, in ad- dition, a series of notches in the trace on the cylinder corres- ponding to the occurrence of a series of events. The interval of time between these events is found by comparing the distance between these notches with the distance corresponding to one sec- ond. With this form of the chronograph it is possible to obtain a precision of measurement of o.oi of a second. A precision of about o.ooi of a second is possible by the use of the tuning-fork chronograph in which the record is made on a drum covered with smoked paper, by means of a stylus at- 44 FUNDAMENTAI, MEASUREMENTS tached to a prong of an electrically driven tuning fork of known period. In this case the trace is a wavy line, the time corresponding to the distance between any two crests being known. Tlie time of the beginning and end of an event is marked by minute holes in the blackened surface of the paper made by the passage of electric sparks from the stylus to the metal cylinder when a second electric circuit is completed. Measui^ement oe Time. — The measurement of a short inter- val of time between two separate events would usually be made with a stop watch or a chronograph. But for determining the period of a regularly recurring event there are several methods of procedure, the choice between which depends upon the mag- nitude of the period and the accuracy required. The most obvious method consists simply in noting by means of a clock or stop- watch the interval of time between two recurrences of the event and dividing this interval by the number of recurrences. For in- stance, the period of vibration of a body suspended by a long wire might be found by observing the number of seconds occu- pied by, say, twenty passages of a mark upon the body past a fixed point. This interval divided by twenty gives approximate- ly the time of oscillation or half vibration period of the body. Naturally, a closer approximation would be obtained by taking the mean of one hundred oscillations. It should be noted that the period of a vibrating body cannot be accurately determined from observations of the instant of turning, as at that moment the motion of the body is insensible for an appreciable length of time. The Method of Passages is a scheme by which the result ob- tained by taking the mean of a large number of oscillations is obtained from the actual observation of a much smaller number. To fix the ideas take the same example above considered. Note the time when a mark on the body passes a fixed point, and again when it passes the same point, say, twenty oscillations later. Suppose this gives an approximate time of oscillation of 15.4 seconds. It is evident that if this number be correct, the looth oscillation will occur 1540 seconds after the first. Without actually observing all the intermediate passages, we can return to the observation a few seconds before the expiration of the Measurement op Time 45 1540 seconds and note time of the passage that occurs nearest to this instant. Suppose it to be 1535 seconds, then the time of oscillation is 15.35 seconds. The length of interval to be se- lected depends upon the fractional precision of the determined approximate time of oscillation. It must never be so great that there will be any uncertainty as to the number of oscillations that have occurred during the interval. The Method of Coincidences is a most accurate method for the comparison of two nearly equal periods of vibration. Sup- pose the period of oscillation of a simple pendulum is to be com- pared with that of a clock pendulum beating seconds. If the pe- riod of the simple pendulum is slightly greater than that of the clock pendulum, a moment will occur when each is at its lowest point ; after that the clock will gain on the simple pendulum until after a certain interval it has gained a whole oscillation, when again each is at its lowest point. Let n be the interval between two such coincidences. Then while the seconds pendulum has made n oscillations, the simple pendulum has made n — i oscilla- tions. Whence the time of oscillation of the simple pendulum is n-^(n — i) seconds. Similarly if the period of the simple pendulum is less than that of the clock peijciulum, between two co- incidences the clock will have made n and the simple pendulum n-\-i oscillations. That is, the time of oscillation of the simple pendulum is, in this case, w-=-(n+i) sec- onds. One method of determining the instant of coincidence employs an electric circuit containing the two pendulums, a battery, and a telegraph sounder all in series as shown in Fig. 23. When the two pendu- lums are in coincidence , they will pass through the mercury contacts A and B at the same instant. At this instant the sound- er will remain silent. The interval between two successive clicks is n in the above ex- pressions for the time of oscillation. Fig. 23. 46 FuNDAMENTAI< MEASUREMENTS XI. STUDY OF FALLING BODIES. Object and Theory oe Experiment. — If a body be allowed to fall freely under the action of gravity, it would be expected that thq.time of falling would be some function of the distance passed through. That is ■< oi t^ where s, represents the distance passed through, t, the time of fall, and j3 a constant. If no other variable factor enters, then it may be assumed provisionally that the law of falling bodies is expressible by the equation S ^^ iC t f \\ where k and /8 are constants whose values are yet to be de- termined. The object of this experiment is to ascertain whether the relation between the distance a body falls and the time occu- pied in falling this distance can be expressed by an equation of the above form, and if it can be so expressed, to determine the numerical value of the two constants. If the distance, s^, s^, s^, passed over by a freely falling body in the times t^, t^, t^, etc., be measured, then we will have etc. Si h^ S, i/ 01, _L.-=1L_ . Jl!_ _ li- . etc S2 f P S3 ^ P Putting these equations into the logarithmic form, log Si — log s, = p\ (log- <, — log t^) log- S2 — log S3 = /3| (log ?2 — log ts) etc, in which P\ denotes the value of )3 derived from the equation expressing the ratio of s^ to s^. Measurement of Time 47 If on solving the above series of equations, it is found that i^i = ^1 = etc., within the limits of accuracy of the experiment, then the assumed equation (i) is verified, and the mean of the values of /8^=j8f = etc., is the value of P in the equation original- ly assumed.The numerical value of k can nowr be obtained from the original equation on the substitution in it of the value of /8 already determinated, together with observed values of s and tj obtained from any single experiment. The equation obtained by substituting in ( i ) the numerical values of k and ft is called an empirical or experimental formula, the verbal statement of which constitutes the fundamental law of falling bodies. ManipuIvATion and Computation. — In the apparatus . used in this experiment the falling body is a metal sphere A (Kig. 24) which is pressed against the end of a longitudinally split brass tube BB' by means of the wooden rod C and spiral spring B. The rod C fits into a hole in the falling body with sufficient fric- tion to support the body, and yet with not too much to be easily withdrawn by a jerk on the cord X. P and G are two flat metal springs which press against the supporting rod C. At D on the supporting rod C is a metal sleeve which metallically connects these two springs when the rod is withdrawn from thehole in the falling body. H is an electric bat- tery, K the electromagnet of a chronograph, L a small hinged trap door which is kept in metallic contract with M by means of the weight N until the impact of the falling body. These various parts of the apparatus are electrically connected by wires as shown in diagram. When the falling body is pressed against the end of the split tube BB', the electric current follows the route HB'ABKH. Fig. 24. 48 FUNDAMENTAI, MEASUREMENTS The electromagnet of the chronograph is now actuated by the current and the stylus is displaced. When the string X is pulled, the rod C is withdrawn from the hole in the falling body, and the latter drops, breaking the electric circuit at BB'. This breaking of the electric circuit allows the stylus of the chron- ograph to return to its normal position, thus producing a sud- den displacement of the line being drawn on the cylinder at the moment the falling body starts to fall. But when the metallic sleeve D reaches FG, the electric circuit is again complete, the current following the route HGDfLMKH. This causes the stylus. to be again drawn to one side, making a notch in the rec- ord Hne. On the impact of the falling body on the trap door, the electric circuit is broken at LM. and a third sudden displace- ment in the chronograph record line is produced. The time of fall of the body is then equal to the time occupied Ly the chrono- graph in moving through an arc equal to the distance between the first and third displacements of the record line. By this means, knowing the rate of the chronograph, the time occupied by the body in its fall is accurately determined. The length of the fall is the distance from the bottom of the body, to the top of the trap door when the body is supported against BB' and the trap door is closed. Make three determinations of the time of fall for each of three different lengths of fall. These distances should be ap- proximately 1.5, 2 and 3 meters. From these observations ob- tain the values of k and j8, substitute these values in the original equation and finally, give a clear statement of the law of falling bodies expressed analytically by the resulting empirical formula. Test Questions and Problems. I. Plot a curve having times of fall for abscissas and cor- responding lengths of fall for ordinates. From this curve find graphically the velocity of the body at various instants of time and from these values deduce the magnitude of the acceleration of the body's motion. Vei-) where 8 is the mass of unit volume of the liquid at the tempera- ture of the experiment. If the masses are taken in grams, and if the liquid employed is water at ordinary room temperature, then since one gram of the water occupies approximately one cubic centimeter, the vol- ume of the specimen is i>z= (m-\-M' — M") cubic centimeters, (approx.) (2) Manipulation and Computation. — Make all weighings by the method of vibration. When filled with liquids, as in the sec- ond and fourth weighings, be certain that no air bubbles are present, that the outside of pyknometer is dry, and that the stop- per is in place. In order to avoid changes in volume due to changes of temperature, avoid touching the filled bottle with the bare hand. XIX. DETERMINATION OF THE VOLUME OF A SOLID BODY BY THE VOLUMENOMETER. Object and Theory of Experiment. — The volume of a body which alters its condition when immersed in a liquid can not be determined by the preceding methods, The object of 68 Fundamental Measurements this experiment is to determine the volume of a solid body which is affected by ordinary liquids. In the use of the volumenometer or stereometer, application is made of Boyle's law that when the temperature remains con- stant, the product of the pressure and the volume of a perfect gas is constant. A convenient form of volumenometer con- sists (Fig.. 39) of a vertical glass tube AB fitted at the upper end with a ground glass stopper S and at the lower end with a flexible rubber tube attached to a second glass tube C capable of movement in the vertical direction in. front of a scale ruled on mirror glass. In the tube AB are two points made of colored enamel, and also a glass capsule to hold the specimen. The rubber tube and the lower portions of the two glass tubes are filled with mercury. The volume of the space above the point A, and the volume of the space be- tween A and B, must first be determined. Denote the former volume by Vj and the latter volume by V^. With the stopper removed, adjust the height of C until the mercury in AB just touches the point A. When this is the case, the point and its image seen in the mercury surface come just together. Clamp the tube C, take the reading of the level of the mercury: then replace the stopper 5 and lower C until the meniscus in AB is at the point B. Again observe the height of the mercury in C. Call the difference between these two readings h. Then if the barometric height be H, from Boyle's law VH=iV+V^) (H—h) or Vh Fig. 39. Vi = {H-h) (1) Area and Vor,UME 69 Now bring the meniscus back to A, remove the stopper S, introduce into the capsule a body of l^ (/i A 6)2 — >^ (,i A 6)^ — etc. But since A 6 is an indefinitely small quantity, the second and higher powers may be neglected. That is, log f — log /■ = |U, A e. Extending this reasoning to the entire arc of the pulley covered by the belt, we have, log.iT' _ log F ~ 11.% whence, the coefficient of static friction between the belt and pul- ley is given by log J?" — log J' M=-^ ^-^ (I) ill which the angle 9 is measured in radians. In precisely the same manner is obtained the coefficient of kinetic friction log F" — log F " = 9 (2) where P and F" are the tensions of the belt where it leaves the pulley, when the belt is slipping. Manipulation and Computation. — Stretch the belt over a pulley that can be rotated by means of a crank. To one end of the belt attach a 10 lb. weight and to the other end a vertically hanging spring balance whose lower end is fastened to the floor. Now turn the crank so as to carry the belt away from the spring balance until the belt is just on the point of slipping. The spring balance reading is now /'"; F = \0 lb^s. weight ami 9 = 180° = ir 88 Properties oe Matter radians. Consequently a value of /x can be computed. Repeat, mak- ing F successively equal to 20, 30, etc., pounds weight, until the limit of the spring balance is reached. The mean of the values of jn thus obtained is the coefficient of static friction between the belt and the pulley. Determine this coefficient for both the flesh side and hair cide of the belt. When the pulley is rotated until the belt slips and then the speed of rotation is kept constant, the spring balance reading is F", while as before, F equals the weight acting on the other end of the belt, and equals u radians. From these values, b is computed. XXVIII. DETERMINATION OF THE COEFFICIENT OF FRIC- TION BETWEEN A LUBRICATED JOURNAL AND ITS BEARINGS. Object and Theory oe Experiment. — The object of this experiment is to compare the lubricating properties of diflferent oils from their relative effect in reducing the friction between a journal and its bearings. The Thurston Oil Testing Machine to Fig. 47. Fig. 48. be used in this experiment consists of a heavy pendulum having a bearing at one end through which passes a horizontal shaft ca- pable of rotation. The bearings can be caused to exert any given Friction 89 pressure on the journal by means of a heavy coiled spring and ad- justing screw forming part of the pendulum. When the shaft is rotated the pendulum will be deflected through an angle deter- mined by the moment of the tangential effort at the circumference of the journal and the moment of the weight of the pendulum. Let W represent the weight of the pendulum ; T, the tension of the spring; P, the mean normal pressure between journal and bearings ; R, the effective arm of the pendulum ; F, the tangential effort at the circumference of the journal^numerically equal to the force of friction; r, the radius of the journal; /, the length of the journal and b, the coefificient of kinetic friction between the journal and its bearings. If the pendulum is in equilibrium when deflected through an angle 6 ,the moment of the forces FF at the circumference of the journal equals the moment of the weight W. That is, 2 / e X X ^ But IV = —7- = —T = ~~r • Consequently llie ?/■ s Fr Angular acceleration — —7- = ~Zf — g (fromcq. ])' Therefore the quantity we have called rotational inertia, or the moment of inertia has for its value, the product of tlie mass of the particle and the square of the distance of this mass from the axis of rotation. If there are a number of particles of masses m^, m-^, m^ etc., at distances r^, r^,, r^, etc. from the axis of rotation, then the moment of inertia of the system is I^nm^ r.^ -\- OTj r^ -\- m^ r^ + etc = 2 mr"^. (3) The moment of inertia of a body of simple geometrical form can be computed, but the moment of inertia of an irregularly shaped body can only be determined experimentally. The experimental determination is usually made by a comparison with a body whose moment of inertia can be computed. A few simple cases will now be considered. Transformation Formulae for the Computation of the Mo- ment OF Inertia of a Body about a given A.xis in terms of the Moment of Inertia about other Axes. Relation between moments of inertia about perpendicular axes. -Denote the moment of inertia of any plane lamina about a normal axis passing through any point c by /, and the moment of inertia about two perpendicular axes XX' and YY' ly- ing in the plane of the lamina and pass- ing through c by /i and /j respectively. Let p be the position of any particle of mass m distant x and y from the axes Fig. 50- YY' and XX' respectively. Then the Moment of Inertia 93 moments of inertia of ni about the three axes are m{cp)^, m{bp)^ and m{ap)-. Consequently the moments of inertia of the lamina about these three axes are respectively, I = I. Ill (x'+i/') , /i = S m if and I^ = 1, m x^. Whence ' I^h+h- (*) Or, in other words, the moment of inertia of any plane lamina 'about an axis normal to it at any point is equal to the sum of its moments of inertia about any two perpendicular axes lying in the plane of the lamina and passing through the given point. Relation between moments of inertia about parallel axes. — Let D be the position of any particle of the body of mass m. Let the two parallel axes intersect the plane of the diagram at A and C. Draw the line DB perpendicular to the line AC. Call the dis- Pjg -J tance AD^r, DC^j\, DB=^h, BC^l and AC^p. Then the moment of inertia of the particle about A is mr'^-=^m \_hP-\-{,p — IY\ Since p, the fixed distance between the two parallel axes, does not change whatever the position of the particle being considered, we have for the moment of inertia of the body about A, X vn ^ = 1. m Ui^ + i') + j»' S '" — 2 JO S ml. If the axis through c pass through the center of mass of the body, then from a well known proposition in elementary mechanics 2 m /=o. Therefore in this special case 2 mi-' — S mr^^ -i- p'' 2 m. (5) or, expressed in words, the moment of inertia of a body about a given axis is equal to the moment of interia of the body about the parallel axis through its center of mass, plus the product of the mass of the body and the square of the distance between the two axes. Moment of Inertia of a Soud Cylinder about its Geo- metrical Axis. — Let the cylinder have a length /, radius r, and density p. Consider the circular cross section of the cylinder to be made up of n indefinitely thin rings of thickness A r. The masses of the concentric rings, starting from the center, are, /p7rAr^ I p TT it2 A rf— A r'l ? p tt [(3 A r)^— (2 A r)='], etc. 94 Properties op Matter Consequently the moment of inertia of the entire cylinder about its geometrical axis is Z = / P TT [A r'(A r') + 3 A r^ (2 A r)= + 5 A r^ (3 A r/ \- -\ , - + (2 11— V\ A y' (n A ry. /■= ZpttA )•• [IX P +3X2^ +5X 32H 1- (2 n - 1) n^,] Substituting for the infinite series its summation obtained by the method of differences, we have I = -^ [3 «* A r* + 4 (li A r)' A r — (« A /■) A r^ ] But 11 A ?• = r, so that 7 r= --^ [3 r" + 4 r^ A r — »• A r'] Taking- as Unit Ar=o 7 = '-^(3,-^)=(Zp..^)f = ^ (6) where M is the mass of the cylinder. Moment of Inertia ot a Cylindricai, Ring about its= Geo- METRiCAL AxES.^Let r and r' be the external and internal radii re- spectively of the ring, I the length p the density ; and let M' and M" be the masses of cylinders of radii r and r' of the same length and density as the ring. Then the moment of inertia of the ring is _ M' r^ M" r" 2 ~ 2 But M' = I p IT r^ and M" = Z p x r'^ therefore I p IT I = -^ (r* — r"). But the mass of the shell is .1/ := I p ir (r^ — r'^), cc nsequently I =^ 0' + r'^). (7) Moment oe Inertia op a System Consisting of Two Cylinders, about an Axis paraleei. to the Geometrical Axes OP THE Cylinders. — Let the cylinders have masses M-^ and M^ and radii r^ and r^ ; and let /i and Ir, be their distances respectively from the axis of revolu- MoMGNT OF Inertia 95 tion of the system. The value of the moment of inertia of each cylin- der about its own geometrical axis has been already determined.De- note these values by I^ and Jj. Denote the moments of inertia of the cylinders about the parallel axis through by I\ and I\. that Fig. 52. From the transformation formula (5) page (93) it is obvious and I\ = I, + M, k- I\ = I, + M, h' ■ Substituting for /^ and /j their values obtained from eq. 6, we have for the moment of inertia of the system consisting of the two cylinders, about an axis through parallel to the geomet- rical axes of the cylinders 1= l\+I\ = M, (^+ h')-VM, (^+ 7,^) (8) Moment op Inertia op a Cyi^inder about an Axis passing Through its Center normal to its Length. — The moment of inertia of a circular lamina about an axis passing through its cen- ter normal to the plane of the lamina has been shown to be (eq. 6) / = i/^ mr'^ Since its moment of inertia is the same about any diameter, we have, from eq. (4), the moment of inertia of a circular lamina about a diameter, 7j = ^ / =: >^ w r= (9) Consider a right cylinder of length I and radius r, to be di- vided into an infinite number of thin laminae parallel to the base. If the distance of one of these laminae from the axis AA' be de- noted by p, and the thickness of the lamina by A p then from eq. 5, the moment of inertia of the given lamina about a diameter of the cylinder is given by r ■= y^ m r^ -\- m p"^ Fig. 53. 96 Properties- of Matter If the material of which the cylinder is composed has a den- sity p, then the mass of the lamina considered ?»= p nr^ Ap Whence P IT r' ^ 2^ \-^M Consequently the moment of inertia of the entire cylinder about the axis AA' passing through a diameter of the base is I A = ^^^ ^Ap + PT ,' l(Apy Ap+(2Apy A-^(3Ap)' Ap-\ -. (n A pY a pi Summing the series by the method of diiiferences, we obtain Ia =^-^ y. Ap + pnr r' (ApY0- + 2^ + 3^ + n^) P T r 2 A ^ + /) TT r2 iAp)\ g ) " 4 pTT r' 2 (^ ''^ A p^ -\- 2 ft' A p'' ■ A p -\- n Ap Ap'' 4 / Z fr A p" -4- 6 n' A p' ■ A p -\- n Av Ap^\ ■SAp+pTr'{ ^ ^ — ^ ^) Now 'Z Ap — V Ap— 1. Making- this substition and proceeding to the limit Ap = o. p TT » M, p TT r' P ( r' P I Ia = -^— + 3 =='"^'"MT + TJ C r' P I That is Ia=MI-^~+~^\ (10) In the case where the moment of inertia is taken about an axis passing through the center of the cylinder in a direction normal to its length, the cylinder may be considered to be com- posed of two parts each of length J4 L and mass M. Making this substitution in the above equation we obtain Ic=M'\^+j^\ (10) where M' = 2 M is the mass of the entire cylinder. Moment of Inertia 97 XXIX. DETERMINATION OF THE- MOMENT OF INERTIA OF AN IRREGULAR BODY. Object and Theory op Experiment. — The moment of iner- tia of a body or system of bodies can be computed only when ii has a very simple figure or configuration. But in any case where the body can be set into vibration about the axis about which the moment of inertia is required, it is a very simple matter to determine experimentally the moment of inertia of the body. The object of this experiment is to determine the moment of inertia of a body of such an irregular shape that it cannot be computed. From the equation derived to find the simple rigidity of a body by the vibration method, (eq. 4 p. 114) it follows that if a body is suspended so that it can "vibrate torsionally, its moment of inertia equals a constant times the square of its time of mak- ing a single vibration. That is, I oc t\ Now if some additional mass of known moment of inertia /' be added to the body whose moment of inertia is required, we will have I+r oc t^^ where t.^, is the new time of vibration of the system. Conse- quently _L_ ^' Whence / can be determined if the moment of inertia of the added mass be known, together with the time of vibration of the body, and the time of vibration of the body plus the added mass. Manipulation and Computation. — The first time this ex- periment is performed it is advantageous to determine the mo- 7 98 Propertiijs oie Matter ment of inertia of a body by experiment that can also be determined by computa- tion in order that the student can check the accuracy of his work. A convenient shape for the body whose moment of inertia is to be determined is that of a disc suspend- ed axially by means of a thin wire from a fixed support. Focus a reading telescope on a vertical mark on the edge of the disc. Being careful to avoid any swinging, set the disc in small torsional vibrations. By Fi<5- 54- means of a stop watch, read accurately the time of the o, loth, 2otli, etc., passages of the mark past the cross hairs of the telescope. If two observers are available, one looking through the telescope will indicate the passages by means of sharp taps on a table with a pencil, while the other ob- server will note the time. These readings give the value of t, i. e.. the time of half a complete vibration of the vibrating body. Now place on the disc a known moment of inertia. This can be in the form of a massive ring, or a pair of cylinders with their axes vertical. In either case the mass added to the body must be placed symmetrically with respect to the supporting wire. Find the new time of swing f^, as before. Errors in measuring t and *i, will have least influence on the result if /' is so large as to make (t{^ — t^) nearly equal to t^. All distances required, are to be measured with vernier caH- pers. Compare the magnitude of the moment of inertia of the disc as found by experiment with that found by computation. Test Questions and Problems. I. Why is it necessary to attach the mass of known mo- ment of inertia symmetrically about the axis of the wire ? Er,) log Zi,'" — log W" = e^ (log 7)3 - log D,) (5) log Z,'" ^ log Z,'" = 4 (log K3 - log D,) etc: The mean of the above values of ^ is the value to be given to the exi3onent of D in the original equation. The equation obtained by substituting for », p. 7. f their values thus experimentally determined is called an empirical for- mula. The statement of the facts expressed by this formula con- stitutes the Law of Bending. The values obtained for these four constants will very nearly be, a =:i,/8=:3,y= — i, « = — 3. The laws of bending would then be expressed analytically by the equation ^=.4^ (6) The value of the constant k is determined by substitutnig the values of /, F, L, B and D from any one of the preceding experi- ments. If a similar series of measurements be made upon rods of some diflferent material, the values of <*, ^, t. t, will be found to be the same as previously determined, but the value of k will be different. It is thus evident that A is a constant depending up- on the material of the bar and not upon its dimensions, whereas the other constants depend only upon the dimensions of the bar. In case the various values obtained experimentally for a, or for ^. 7, "- or k were found to be not equal one to another, within the limits of experimental error, it would be concluded that the equation previously adopted involving these quantities as con- stants is in error. Manipulation and Computation. — The rods to be experi- mented upon should be about 50 cms. long and their transverse di- mensions must be so selected that the same bars can be formed into two series, one in which the bars have constant depth and variable width, and another in which they have constant width and vari- able depth. The variable length is secured by adjusting the dis- tance between the knife edges. After a bar is placed on the sup- porting knife edges, a weight pan is suspended from the bar half way between the knife edges and sufficient weight placed in it to El the moment of this force about e equals E BejA J>)3 Ax Similarly the momj^nt about « of the force of reaction in a layer at a distance of 2 A D from the neutral axis equals E Be (A D)' ^, Ax no Properties of Matter Therefore the total moment of the forces of reaction in all the layers above the neutral axis will be ^ (P+2- + 3^+---«^)= ^ 1 6 J = :|A?j^2.(w A D)'+ SAD (« A Z»)^ + (A B)^ (n A D)] (U But n AD equals half the depth of the rod. Therefore, in the limit when A Z) =: o, the total moment of the forces of reaction in all the layers above the neutral axis is E Be£>^ 24 A.T But the moment of all the forces of reaction below the neutral axis will be the same as the moment of those above. Consequently the moment of the couple tending to right the rod wiL be 12Aar Since the bar is in equilibrium, this moment must equal the mo- ment of F about e, that is, E Be D^ ^ ,,, — p5-r = Fx. (3) 12 Ax At a and b draw aj and bk tangents to the curved surface of the rod. Since aj and bk are approximately equal to x, and since the angle between them is equal to 6, we shall have the distance jk= X 6 approximately. Let the entire fall of the end of the rod i n be called /'. /' may be regarded as made up of a number of elements similar to jk, which will be denoted by /j, /j, /g,. . In Therefore, if Ix represents the fall of the end of the rod due to the bending of an element at a distance x from the end, xe = 1:^ (4) Let the length of the rod be imagined to be made up of a very large number n of sections of length A-*- If now eqs. (3) and (4) be applied to the first section, we will have E Be D^ ^ ., 12AX =-^^^ (^' and e A X = h (6) Similarly for the second section from the end, E Be D^ Elasticity hi rom the end, 2F^x (7) 12 A a; and 2 e A a: = /^ (8) and so on throughout the entire length of the rod. Combining (5) and '(6) 127^(Aa-)^ '' ~ E B D' ^ Combining (6) and (7) ""^ EBD-" Therefore T2 /^CA jf")^ _ \2 F (A xY r w (« + 1)(2 ». + 1) 1 ^ EBD^ V 6 J 2/^ r 1 V = -EBD^ \_2 {n ^ xY + 2 ^ X (» A xY + (A ;r)2 {n A x)j (9; But w A cc =Z^, the length of the rod. Therefore in the limit when A ;r = o, eq. (9) becntnes The result just given is for a rod fixed at one end and load- ed at the other, and shows the fall of the free end of a rod of length L, breadth B, thickness or depth D, made of a material of Young's Modulus B, when a weight F acts downward on the free end. In. the case of a rod supported at both ends and loaded at its middle point with a force F (the distance between the supports be- ing L), the rod may be regarded as equivalent to a similar one fixedatthe middle point acted upon by vertical forces of j4F at each end. Or, in other words, the rod is equivalent to two rods of the same material and cross-section but of half the length of the original rod, fixed at one end and loaded at the other with a force of yiF. Making these substitutions in (10) we have F L^ _ F TY „ 112 Properties of Matter Manipulation and Computation. — Measure B and £> at a number of points along the rod by means of a micrometer gauge or vernier caliper. Measure L, the distance between the two knife edges with a meter stick. Place the rod on the knife edges and suspend from the middle point a weight pan containing sufficient load to bring the rod into good contact with the knife edges. The flexure / of the rod produced by an ad- ditional load F may be measured either by means of a micro- scope fitted with an eye piece micrometer, or by means of a micrometer screw placed above the center of the rod and moving in a nut fastened to a rigid sup- port. In case the micrometer screw. is used, the moment when the screw comes into contact with the rod can be determined either by means of a telephone in a battery circuit including the rod and micrometer screw, or by observing the image of some fixed object in a small mirror one end of which rests upon the rod while the other end rests upon an adjacent fixed support. Add, say, lOO grams to the weight pan and observe the flex- ure : add lOO grams more and observe the flexure ; add lOO grams more and observe the flexure ; then reverse the process taking off lOO grams at a time and observing the flexure produced by each change of load. The readings of flexure for a certain load should be the same the second time that they were the first time. In case they are not, repeat the observations. In order to be certain not to load the rod beyond its elastic limit, the student will find the maximum load that may be applied to the particular rod being used, on inquiry of an instructor. Give the final result in kilograms per square centimeter and also in pounds per square inch. XXXIV. DETERMINATION OF SIMPLE RIGIDITY. (vibration method.) Object and Theory oE Experiment. — The Simple Rigidity or Slide Modulus of a material is defined as the ratio of the force developed on unit area of cross section of a rod by a shearing strain, simple e =■ Elasticity 113 to tlie torsional deformation producing it. That is, the rigidity, sheat ing stress ** shearing strain The object of this experiment is to determine this constant for a material in the form of a thin wire. Consider a cylindrical rod or wire of length /, and radius r with one end fixed and the other end twisted through an angle . This will cause an element of the surface as AB to be displaced to AB'. From the diagram we have BB' BB' , r ■J— and = , consequently 9 = —j- . That is at every point of the cylindrical rod distant / from the axis and / from the fixed end, there is a (p r shearing strain equal to —j — Whence from defini- tion, the shearing stress developed at a point distant -g :t^r from the axis and at a distance / from the fixed end equals ~ii r ■ ■* I Imagine any hxpss section of the rod to be divided into a large number, n, of concentric rings of infinitesimal width A r. The area of these separate annular portions of the cross section of the rod will be, beginning from the center, ttA;-^, 3 ttA?*^, 5 TrA?-^, etc. And the moments of the forces developed on these various annular areas by the distortion, will be lid) A ]■ a (b A >■ — J— (t a r'^) A r, ^ — (Stt a r^) 2 A r, etc, Whence, the total torque action across the entire cross section of the rod when it is twisted through an angle <^ is r = '^— ' [IX 1^ + 3 X 2^ + 5 X 32 + h (2 «—!)«'] Substituting for the series its summation obtained algebraically by the method of differences. Fig. T = /J. TT 61 „ U TT A (7* ^ , „ T = ^-^-g-^ (3 «' + 4 »' - n) [3 n* A r' + 4 (m A r)^ A »- - (w A r) A r^] 114 Properties of Matter But since n is the number of concentric rings of width A r between the center and circumference of the rod, n ^r =r, so that U, d> IT T = ^-^j- [3 (■' +Ar'h.r-ris. j-^] Taking as limit A r- = o which is the moment of the couple that must be applied at the free end of a rod, having the other end fixed, when the rod has a length I and radius r, and is made of a material of rigidity fx., in order to maintain a torsion <^. Consequently if a massive body suspended from the lower end of a wire, be turned about the wire as an axis through an angle of radians from its position of zero torsion, the moment of the stresses exerted by the lower end of the wire on the massive body will have the magnitude T=^' (1) This moment will tend to twist the wire toward the position where the wire has zero torsion with an angular acceleration given by the equation ""= T-=\-2Tr)'>' '^) where a is the angular acceleration, T the moment of the result- ant stress and / the moment of inertia of the suspended body. In equation (2) the quantities within the parenthesis are all constants ; therefore the angular acceleration varies as the angular displacement. From which it follows that the motion of each part of the wire is harmonic. The time of a single simple harmonic vibration is t^^^i^ = .{-^^Y^ (3) Whence, the Simple Rigidity of the material composing the wire is Manipulation and Computation. — Suspend from the lower end of the wire a massive body of such a shape that its moment of inertia can easily be computed, for instance, a solid iron cylinder El radians, the moment of the couple developed by the twist is (eq. I, p. 114.) If the twist of the cylindrical wire or rod be ;8 degrees, its value in radians can be determined as follows. Since an angle in radians is measured by the arc subtended divided by the radius 2 TT r IT 3(0°= radians, or, 1°= i-ttt; radians and p° = ^r,„ radians = Combining eqs. (2) and (3), and transforming the angle measured in radians to degrees, we have 360 m g a d I '"" h^T'r^ (4) ii8 Properties of Matter ManipuIvATion and Computation. — Measure the diameter of the rod or wire with the greatest care in at least ten places by means of a micrometer gauge. Measure h with a meter stick or steel tape. Find three values of a and j8 by looping the threads on dif- ferent pins and by placing different loads in the weight pans. The loads in the two weight pans must be equal. If the threads are looped directly over the pins, the value of d must be determined for each observation. This distance d will, however, remain con- stant and a single measurement of it will suffice, if instead of loop- ing the supporting threads themselves over the pins, a horizontal loop of thread about 20 cms. long fastened to each of the support- ing threads serves to connect them to the pins on the edge of the disc. These horizontal threads should be of equal length and looped over diametrically opposite pins. With this arrangement d equals the diameter of the disc. Give the final result in kilograms per square centimeter and also in pounds per square inch. Test Questions and Problems. I. For heavy rods the apparatus shown in annexed figure is most convenient.One end of the rod is fastened to a rigid support, and the oth- er end is fastened to a large divided head that can rotate with very little fric- tion. The torque is pro- duced by means of a weight and cord passing over the edge if the divided head. Deduce the equation for finding the simple rig- idity of a substance with this apparatus and show what errors are inherent in this method that do not occur in the preceding method. Elasticity 119 XXXVI. DETERMINATION OF THE MODULUS OF ELASTIC RESILIENCE OF A ROD. Object and Theory 01? ExPERiMENT.-^Resilience of a body is the energy it possesses due to a strain developed in it. The ultimate resilience or modulus of resilience is the strain energy of the body when strained up to the elastic limit. Corresponding to the dif- erent types of strain are different types of resilience, as tensile re- silience, flexural resilience, torsional resilience, etc. The resilience of a material is usually given either in terms of unit mass or unit volume. The object of this experiment is to determine the flexural resilience of a bar or rod. The rod is to rest on two knife edges and be distorted by a force applied at the middle point. Let L^length of rod between supports in cms. ^^cross section in sq. cms. p = density. m=mass of rod between supports = p AL gms. F^load in dynes necessary to strain rod to its elastic limit. /=:displacement in cms. of middle point produced by F. Since until the elastic limit is reached, the distortion varies with the force applied, the average force acting while the distor- tion is increasing from zero to I is yi F. Therefore, the strain en- ergy stored up in the specimen, that is, the modulus of flexural re- silience of the rod is i?=%F/ergs. (i) The modulus of flexural resilience per unit of volume is R = -- =^-j ergs per c. c. (-') and the modulus of flexural resiliance per unit of mass is „ li Fl ^^ = Tn = 2pAL ^'^^ P^*" ^""'' *^^ or, if force is measured in grams' weight {F') instead of dynes, PI ' ,A R" = 7y — jy gram-centimeters per gram. (4) 120 Properties ot Matter Since the dimensions in eq. 4 reduce to a length, it follows that when force is reckoned in gravitational units, the modulus of elas- tic resilience per unit mass has the dimensions of length. For this reason tables of values of modulii of resilience are often given in terms of centimeters or feet. MANIPU1.AT10N AND Computation. — The apparatus consists of the rod to be examined with its ends resting upon knife edges, and a microscope fitted with an eyepiece micrometer to measure the deflection of the rod. All of the apparatus must be placed upon a rigid support free from vibration. Measure the length and cross section and determine the mass of the bar between the knife edges. Focus the microscope upon a fine cross engraved upon the the center of one of the vertical faces of the bar, or upon the point of a needle fastened rigidly to the middle of the bar. Carefully add weights to the pan suspended from the middle of the bar tak- ing a reading of the deflection after each addition. During the progress of the experiment carefully plot weights and deflections on cross section paper — the weights as abscissas and deflections as ordinates. As would be expected from Hooke's law, the line connecting these points is straight from the point of zero load up to the point representing the elastic limit, and from there it bends toward the axis of abscissas. Thus, from the curve one readily obtains the value of the load necessary to strain the bar to its elastic limit, and the deflection produced by this load. All the data is now at hand for determining the value of the modulus of flexural resilience per unit volume, or per unit mass. Test Questions and Problems. 1. Find he modulus of flexural resilience of the rod graphic- ally from the curve plotted, without the use of any equation. 2. Would the modulus of flexural resilience be greater or less than the modulus of tensile resilience? Give fully reasons for the difference. 3. If a wire of length L, made of a material of density p and Young's Modulus B be elongated by an amount e when stretched up to the elastic limit, show that the modulus of tensile resilience per gram, when forces are measured in gravitational units is Viscosity 121 E e' B, =: -pz pj centimeters. 4. Using the same nomenclature as in previous question, equate the work required to stretch the wire a distance e with the work required to raise it against gravity to a height h, and show that h = Ti rr- centimeters. 2e g L' 5. From an inspection of the equations given in the preced- ing two problems deduce a definition of modulus of tensile resil- ience. 6. By what factor must the Modulus of Resilience expressed in gms. per c. c. be multiplied in order to give the modulus in cen- timeters length? 7. By combining the equation of Young's Modulus by stretching, with the value of the tensile resilience per unit volume, show that where S represents the tensile stress at the elastic limit and B represents Young's Modulus of the material. Chapter XL VISCOSITY. XXXVII. STUDY OF DAMPED VIBRATIONS. Object and Theory 0? Experiment. — When a body original- ly vibrating with simple harmonic motion encounters friction or any retarding force, the period, amplitude and form of the wave is altered. A motion that may be considered to be produced by the effect of friction or any retarding force upon a simple harmonic motion is called a damped vibration. The object of this experi- ment is to determine from a series of observations made on a 122 ■ Properties of Matter damped vibration, the damping constant ; and also to construct two curves, one representing the damped vibration actually observed, and another representing the simple harmonic motion from which the latter may be considered to be derived. The theory of this ex- periment is of such great importance in the consideration of many phenomena in Elasticity, Electricity, Heat, Sound and Light, that it should be performed and thoroughly understood' by every stu- dent specializing in Physics, Electrical or Civil Engineering. Al- thought the theory now to be developed cannot be completely fol- lowed by a student who has worked through less than one sem- ester of calculus, still, such a student can obtain much good by accepting the final equations and making the required observations and calculations. In the case of a simple harmonic motion we have I' = A' sin ypr V (■") where V is the displacement of the body from the position of equilibrium at time f, A', is the amplitude of vibration and V the period. Our object is now to determine the corresponding equation when there is a retarding force acting on the body producing a damped vibration. Whenever such a retarding force produces a damped vibration, the eifect or magnitude of this force is propor- tional to the velocity of the vibrating body. In the case of simple harmonic motion, if the displacement of the particle "from the center of its path be denoted by /, the phase reckoned from the same point be denoted by 6 the amplitude of vibration by r, the maximum acceleration pj(5_ 52. of the particle by a', then the acceleration at any time t is given by dU , . , I — — a sin 6 = — a = — a — when the displacement is small. But when there is damping, since the retarding force is proportional to the velocity, the acceleration is diminished by an amount proportional to the velocity. There- fore, in the case of damped vibrations, the above equation becomes Viscosity 123 dU I , dl •"- df - — "' r — " dt where k is a constant depending upon the retarding force. d'l dl I ,-)< rfP + ^^-Tt + «V = ° '"^ This being a linear equation with constant coefficients, we will pro- ceed to solve it by making l^e'"' where w is a constant to be de- termined, and e is the base of the natural or Naperian logarithms. „. , mi dl ,nt dH «' since I = e , -rr —me , -rrr — ^n e , dt dV whence tlie preceding equation becomes )»" + k m + — — o ^ ^ Ik'' a' k^ a' When -3- is greater than — ^ . the motion is non oscillatory; 1,1 d' ^jjgn _ll_ is less than — > tlie roots of the above equation are 4 r itniginary, so that above equation m:iy be put into the form m = — ^ ± fi i (3) where II =-\j— k^ r 4 and i has its usual mathematical significationi/— 1. We have now two values of m : denoting them by m^ and m^, they are '"^ , • ^ ^ )«! — — -J — f" i" * ^"" "'2 ~ 2 '*'■ Hence if Wi and m^ satisfy equation (3) ("Ms an integral of equation (2) ; and if Wi and Wj are distinct roots of equation (3) the solution of (2) will be of the form Substituting the values of m^ and m^ I — Ci e e + C, e e ,^ it fji.it _y^kt _ ^ 1 124 Properties of Matter Prom trigonometry e '*' = cos n t -\- i siii m t. Consequently ? = e [Ci cos M < + Ci J sin n t-\- C^ cos /n < — Cj «' sni ju <] - e ' [(cos /* t) (Ci + C2) + (i sin ^ /) (Ci — 6%) 2 (^4 cos lit -\- B sin yu. Since this equation is true whatever be the time t, it will hold for the time t^ ; and if time and displacement be reckoned from the position of equilibrium, then when t^o, l=^o and consequently A=o. Whence kt_ I = Be ^ sinut (4) In order to make this equation comparable with eq. ( i ) , the value of fji. must be determined in terms of the period, of vibration T of the damped vibration. Differentiating eq. (4) with respect to t, we obtain the value of the velocity at any time t, kt * —TT = B e (m cos ju. r — -5- siii /i t) In periodic motion the velocity equals zero at the end of a half oscillation, at the end of three halves, five halves, etc. Denoting the time of a half oscillation by t„ then when t^^to, the above equation must equal zero. By inspection of tne above equation it is evident that if the velocity is zero when t = to. it will also be zero when TT 2 TT 3 TT t= to + — , t = to+ — , t =to + . etc. /* M ^ Since the period of a vibration is defined as the time occupied by a complete vibration, that is, a double oscillation, the period in this case is r = ^ (5) Substituting this value of n in eq. (4) we have for the equation of a damped vibration Viscosity 125 _ kt I —. B e ^ sin T If there had been no retarding force, k would equal zero, that is, g — H *< = I J and if there had been no retarding force the period T would be equal to that denoted by T in eq. (i). Consequently A'=^B. The above equation now assumes its final form This is the equation for a damped vibration. It is the form equa- tion (i) assumes when the body moving with simple harmonic motion is acted upon by a retarding force proportional to the speed. The ratio of the length of any two succeeding swings in the same direction will now be deduced. Since the period of the damped vibration T= — . the time between two consecutive os- cillations in the same direction is ~ij,. Consequently, representing by /j and l^ the amplitude of any two consecutive oscillations kt r, li = A' e sin -y i =:A'e ' ' " ^ sin?^^ = ^'« M j kt h-A'e ' '*'sin^< = ^'e 1^ )e sin -^ < k V Therefore, -y- = ^ ^ = ^ (a constant) '2 Substituting the value of ja from eq. (5) and putting into log- arithmic form, the above equation becomes k T nat. log. 0- = —7y- ^- X (a constant) Ic I02* & or log. -r = — ^ T = \ log « (7) That is, the logarithm of the ratio of the amplitude of any two suc- ceeding oscillations in the same direction equals a constant quan- tity times the period of the vibration. The natural or Naperian logarithm of this constant ratio is 126 Properties of Matter usually represented by X and is called the logarithmic decrement of the vibration. Since the logarithmic decrement includes all the terms effecting the damping and as it is easily determined experi- mentally, damped vibrations are most frequently discussed in terras of the logarithmic decrement. The relation between the period of a simple harmonic vibra- tion, (T') and the period of the damped vibration (T) can be readily obtained. In the case of a simple harmonic motion 7^'= 2 ■"• -v— , ' a 2 T and in the case of a damped vibration T'= — (eq 5 ) whence I 1 1 / a \ I / a a k^\ from definition of /*. Whence (8) It will be noticed that in the final equations (6), (7), (8), the damping factor appears as yih. It may make the equation appear in a simpler form if this factor be denoted by a single symbol. De- noting it by Y we can write the equations that will be hereafter used as follows: 2t ,1,. V = A' sin -7p v. (1 ) This is the equation of a simple harmonic motion, in which /' rep- resents the displacement from the position of equilibrium attained 2 IT in time t' ; A' the amplitude; T, the period: siid -fr the phase expressed in radians. For the case of damped vibrations, I = A' e sin -^ t (^') ^^^ or \ = 2.3026 log- t is to construct the calibra- tion curve of a thermometer, or the curve of corrections by means of which the error in any reading due to either the ir- regularity of the bore or the location of the fixed points can be corrected. The experiment consists of two parts. First, the length of a short thread of mercury is measured at different parts of the tube, and from these lengths is determined the position of points throughout the whole length of the tube that separate equal volumes. Second, the position of the fixed points is determined by placing the thermometer in the vapor of boiling water and also in melting ice. Manipulation and Computation. — The short thread brok- en off from the column of mercury can be of any convenient length not to exceed the length of ten degrees of scale. The separation of this short thread requires some manipulation. In 142 Thermometry blowing the bulb on a thermometer tube, usually a slight con- striction is left where the bulb and tube join. If such a ther- mometer is inverted and then is given a sudden jar, the thread will separate at this point. If thei'e be no constriction at this point, the thread may be separated by laying the thermometer on a table and striking the upper end of the tube with a little mallet consisting of a rubber stopper on the end of a short rod. If this is not carefully done, however, cracks may be produced inside the stem near the bulb. If the bore has an enlargement at the upper end, let the entire column of mercury run into this en- largement and remain there throughout the remainder of the experiment. Then warm the bulb slightly until a thread of mercury of the proper length runs into the tube, and separate this from the mercury in the bulb. This is the thread that will be used throughout the experiment. In case the capillary has no enlargement at the upper end in which the mercury column can be stored throughout the subsequent experiment, it may be necessary to use two short threads to calibrate the two ends of the tube. When this is the case cool the bulb with a freezing mixture of ice and salt until all the mercury has run into the bulb except the short length that is to be broken off. Separate this short thread and let it run to the farther end of the tube. In order to make measurements in the lower end of the tube, this part of the thermometer must be freed of mercury and an- other short thread separated as before. The discussion of the remainder of the experiment may be facilitated by considering a concrete example. In the thermo- meter used, there was an enlargement at the upper end so that one short thread could be used throughout the entire length of the scale. The thermometer was placed horizontal on a table, the short thread was set with its lower end near the point marked zero, and the positions of the two ends were carefullv read to tenths of a degree. Then the thread was moved along about half of its length and the positions of its end points again read. The thread was again moved along about half of its length, and so on until the point marked lOO was reached. The readings ob- tained are given in the following table : C.\I,IBR.\TION Ol? ThRRMOMETEUS 143 Lower end Upper end ThiCiid length Lower end of Upper end of Thread k'n(,'th of thrend at of thr^-ad at in sj.do divs. thread at thread at in scale tli ^ s —3.00 5- SO 8.50 48.00 56.23 8.23 2.00 10.50 8.50 53.60 61.74 8.14 6.60 15.10 8.50 59.50 67.55 8.05 12.40 20.90 8.50 65.60 73.54 7.94 17.30 25. SO 8.50 71.50 79.35 7.85 22.70 31.17 8.47 76.30 84.06 7.76 27.40 35.84 8.44 83.40 91.03 7.63 32.20 40.60 8.40 88.70 96.28 7.58 37.50 45.85 8.35 94.00 101.50 7.50 42.10 50.40 8.30 99.40 106.83 7.43 The quantities in the first and third cokimns when plotted in a curve give Fig. 67 which shows graphically the length of the "short thread at various points of the capillary. From this curve 10 20 30 40 50 60 70 80 90 100 Lengths of the capillary having equal volume at various positions of the scale. Fig. 67. we \vill con:pute the points along the scale between which the volumes of the capillary are equal. From these values the cor- rections to be applied at the different points along the scale to correct for the inequalities of the bore can be determined. For example, from the curve note that if the short thread be placed at the zero point of the thermometer, it would be 8.5 scale divisions long. Lay off this distance from the zero point along the axis of abscissas and the point marked A , on the plot is reached. From the curve it is seen that if the short thread had its lower end at /I, it would be 8.5 scale divisions long, or, it v/ould extend from A to B. Continuing this process, the points A, B, C, D, B, etc., are obtained, the distance between any two fucceeding points marking off equal volumes of the capillary. Construct a tabic of four columns, in which the first column gives the distance in scale divisions of the points A, B, C, D, etc. from the starting point. In the second column put the length of the 144 Thermometry short thread between each succeeding pair of points just de- termined. From these values, determine the average length of the thread between the equal volume points. In the' third column, put the position which the equal volume points would have had, if the capillary had been uniform. In the fourth column put the difference between the quantities in column three and column one. These values in column four give the correc- tions to be applied to readings made at diflfertnt parts of the scale, to take account of the inequalities of the bore of the tube. Construct a curve by plotting as abscissas the quantities in column one, and as ordinates, the corresponding values from column four. Points on scale be- Length of thread Position of equal vol- Corrections tween wliicli volunaes between equal volume ume points it bore for points in of bore are equal spaces. had been uniform. first column. 8.50 8.50 8.15 —0.35 17.00 8.50 16.30 , —0.70 25.50 8.50 ,24.45 —1.05 33.96 8.46 32.60 —1.36 42.35 8.39 40.75 —1.60 50.65 8.30 48.90 —1.75 58.83 8.18 57.05 —1.78 66.89 8.06 65.20 —1.69 74.81 7.92 73.35 —1.46 82.61 7.80 81.50 —1.11 90.27 7.66 89.65 —0,62 97.83 7.56 97.80 —0.03 Av. leng-th of thread, 8.15 The correction curve for the bore of the thermometer being considered is given hy A B C Fig. 68. This curve gives the -I-1-0 Calibration curve of tlierraometer No. 3165. Fig. 68. Calibration of Thermometers 145 corrections that must be applied to readings at different points along the scale on account of irregularities in the bore. The correction for the displacement of the fixed points will now be considered. By definition, the lower fixed point (0° C. or 32" F.) is the temperature of melting ice. The upper fixed point (100° C. or 212° F.) is defined as the temperature of the steam produced by boiling water at sea level and latitude 45" under a barometric pressure of 76 centimeters of mercury when the barometer is at the temperature 0° C. Observe the barometric height, noting the temperature of the barometer b}- means of the thermometer attached 'to the instru- ment. Ascertain from the laboratory instructor the latitude and altitude of the laboratory. From these data compute, in the man- ner explained on page 155 the corrected barometric pressure H reduced to standard conditions. Suspand the thermometer in the vapor of boiling water. It must not be immersed in the boiling water or be so near the surface that the bulb will be spattered by drops of water, be- cause the temperature of boiling water is not constant but is in- fluenced by the nature of the surface composing the vessel and by the presence of sHght quantities of dissolved impuri- ties. The temperature of the vapor, however, depends only upon the pressure. Regnault's hypsometer is very satisfactory Fig. 69. Fig. 70. ^^j. (.j^jg purpose. It con- s'ists of a reservoir R Fig. 6g in which the water is boiled, sur- mounted by a tube in which the thermometer is suspended. After passing through this tube the steam passes through the jacket / and escapes into the air at B. For precise work Guillaume's hyp- someter, is employed. This consists of the boiler A Fig. 70, 10 146 Thermometry surmounted by the jacketed tube B in which the thermometer is suspended. In this instrument, the steam instead of escaping into the air is condensed by a current of cold water circulating in the condenser Cj and then trickles back into the boiler. Both forms of hypsometer have a water manometer M which serves to measure any difference of pressure between the steam inside and the air outside. If the manometer indicates a pressure of d millimeters of water, or d-^i^.S millimeters of mercury, then the total pressure on the surface of the boiling water is H -{- (d -f- 13.6). Call the observed boiling point T'. Remove the thermometer from the hypsometer, allow it to cool in the air to about 40" C, and then immerse in a vessel filled with snow or shaved ice which contains enough water to fill the interstices. This gives the depressed zero point. By reference to Table 14 obtain the temperature of the vapor of water boiling at a pressure oi H -\- ((i-^13.6). Call this temperature T". Manifestly (T" — T') is the error of the upper fixed point of the thermometer. In the example above considered T" — T' = 99 . 8 — 99 . 4 = o. 4" C, and the error of the freezing point was — 0.3° C. That is, the correction to be appHed to the observed boiling point is — 0.4° C, and the correction to be applied to the zero point is +0-3" C. If now on the same sheet of coordinate paper containing the cor- rection curve for irregularities of bore, the freezing point cor- rection be entered along the axis of ordinates opposite the zero of abscissas, and the boiling point correction be entered opposite the observed boiling point, and these points be connected by a straight line, as shown by D E F Fig. 68, this line gives the cor- rections for all intermediate points of the scale resulting from the displacement of the fixed points. By adding the ordinates of the correction curve for the irregularities of bore, A B C, to the corresponding ordi- nates of this correction curve for displacement of the fixed points, ■D B P, the new curve H G F is obtained. This is the Calibration Curve of the thermometer. In case the calibration is done with two mercury threads Calibration of Thbrmometbrs 147 instead of one, the calibration should extend from each end to a distance past the middle of the tube. The curve analogous to Fig. 67 obtained from these data will not be a continuous line. Along the region where data were taken with both mercury threads, one branch of the curve will be above the other. In this region find the ratio of the ordinates of the two curves for a given position on the thermometer scale. This ratio must be the same for all points on the thermometer scale. By multiplying any ordinate of one curve by this constant ratio, the correspond- ing ordinate of the other curve will be obtained. Proceeding in this manner, a continuous curve is obtained just as though all of the calibration had been performed with a single mercury thread. The Determination op Unknown Temperatures. When thermometric measurements are taken under such conditions that the depressed zero point remains constant, the measurement of an unknown temperature requires simply that the observed reading be corrected for stem exposure, and then this result corrected for nonuniformity of bore and for displac- ment of the fixed points of the thermometer by means of the cali- bration curve. During a series of measurements of temperatures extending over a considerable range of either temperature or time, the con- stancy of the depressed zero point cannot be assumed. In this case take the thermometric reading so soon as the indication has become stationary and then obtain the value of the depressed zero by plunging the thermometer into a bath of melting ice. After correcting the observed thermometric reading for stem exposure, nonuniformity of bore and for displacement of the fixed points of the thermometer, subtract from this result the value of the depressed zero. 148 Thermometry XU. CALliBRATION OF A PLATINUM RESISTANCE THERMOMETER. Object and Theory oe Experiment. — The mercury in glass thermometer is unavailable for the measurement of tem- peratures much below — 30° C, or above -)- 300° C. Although the gas thermometer can be used for any temperature for which a suitable material to construct the bulb can be found, it is such a large awkward instrument, and the difficulties of the manipula- tion are so considerable that it is suitable only for standardizing more convenient types of thermometer. Since the electrical re- sistance of most metals varies continuously with the temperature according to definite laws, and since the accurate measurement of resistance is attended with no considerable difficulty, thermo- meters depending upon this property are in common use for measuring high and low temperatures. Platinum is the material usually employed from the fact that its resistance at any given temperature does not change with time, and also from the fact that its temperature coefficient of resistance is large. Platinum is also especially convenient from the fact that the law connect- ing the temperature and resistance of a wire made of this ma- terial is expressible by a simple formula. It has been shown by experiment that if /?„ represents the resistance of a piece of pure platinum at 0° C.^ then the resistance Rg at t° C, is expressible by the equation R, =R,[i + at + bt^] (I) where Rq, a and b are constant quantities. The object of this ex- periment is to determine the value of these constants for a given platinum resistance thermometer. If the resistance of the wire at three different temperatures be known, three equations of the fonn of eq. (i) are obtained, and from these equations the values of the three constants can be calculated. The freezing point and boiling point of water are Calibration of Thermometers 149 two convenient temperatures for the experiment. The remaining temperature required by the experiment can be the boiling tem- perature of any convenient substance, c. g. sulphur which boils at 444.5° C. But from measurements of the resistance of wires at very low temperatures, Dewar and Fleming have shown that at the absolute zero of temperature, it is highly probable that the resistance of all pure metals is zero. Assuming this relation, the resistance of the thermometer wire need be measured at but two temperatures. This simplified process gives the values of the constants in eq. (i) with sufficient accuracy for most purposes. Representing the resistances cf the thermometer wire at the tem- peratures t' , t" and absolute zero by the symbols R', R" and R'", we shall have R' ^R^[i+at' + b (ty] (2) R" =R^[i+at"+b it"r] (3) R'" = R^ [1 — 273 a + (273)' b] = o (4) From these three equations the values of the three constants a, b and R„ can be obtained. Knowing these constants together with the resistance of the thermometer wire when at any un- known ternjurature, this temperature can be calculated from eq. (I). The Wheatstone bridge method of measuring resistance will be employed in this experiment. In this method two conductors ABC and ADC, Fig. 71 are joined in parallel to the ter- Battery Battery — 4° — \^ BL -e Galv. _KL K Fig. 72. minals of a battery. It is obvious that corresponding to any point B on the conductor ABC, there is a point D on the conductor ADC which is at the same potential. If two such points be con- nected by a conductor B D, no current will flow along this bridge wire. Represent the potential at the points A, B, C and D by the symbols V^ , F3 ,etc. Let the resistance of the arms A B, B C, ISO Thermometry A D and D C be denoted by R-^, R^ R^ and R^ respectively. Let the current intensity in these arms be denoted by i^, i^, ig and h respectively. In the case considered, since F^ = Fj, , Ji = 4 and h = h- Then by Ohm's law, , _Va- Vb i _,■ _ Vb - Vc Va —Vb . . Vb— Vc whence (6) (7) R, ■ '4 -'3- ^^ Va— Vb Vb — Vc (5 ■^1 Ri Va - Vb ^ Vb — Vc ■Rz ^i Dividing eq. (5) by (6) we have Rl A*4 Thus if three of the resistances are known, the remaining un- known resistance can be determined. This arrangement can be modified so that only one known resistance is required. Suppose the conductor ^ D C is a long uniform wire. Let /g and /j be the lengths of this wire from A to D and from D to C respectively. Then since the resistance of a uniform wire is proportional to its length, it follows that when no current passes along the bridge B D, | = t- ^«) Knowing one of the resistances and the lengths l^ and l^, the re- maining resistance is determined. Manipulation and Computation. — The resistance ther- mometer consists of a coil of fine platinum wire wound on a mica frame enclosed in a wrought iron capsule. In order to eliminate Fig. 7z. Caijbration 01? Thermometers 151 errors due to a change in the temperature of the leads extending from the coil to the Wheatstone bridge, a second pair of leads precisely like the first is placed side by side with them throughout their entire length extending from the Wheatstone bridge to the coil in the capsule. By measuring at different temperatures both the resistance between the terminals of the coil and the resistance between the terminals of the dummy leads, the change in the re- sistance of the coil is obtained independent of any change in the resistance of the leads. In this experiment, the particular form of Wheatstone's bridge called the "slide wire" or "meter" bridge will be found most convenient. This apparatus is illustrated in Fig. 72. A comparison of Figs. 71 and 72 will make the construction clear. A long uniform wire is stretched over a divided scale. The ends of this wire are connected to a parallel copper rod in which are two gaps. In one of these gaps is inserted the resistance i?, to be measured while in the other gap is inserted a known re- sistance R^. One end of the bridge wire terminates in a sliding contact key K^ which can be moved back and forth along the slide wire. In the bridge wire circuit is a galvanometer for indi- cating the passage of current. W^hen a mirror galvanometer, telescope and scale are used, they must be placed in such relative positions that on looking through the telescope toward the galvanometer mirror, a re- flected image of the scale will be seen. In making this adjust- ment, place the telescope with the attached horizontal scale so as to face the galvonmeter at about one meter's distance. Now standing behind the scale and looking at the mirror attached to the gal- vanometer needle, move the eye about until a reflected image of the scale is seen in the mirror. If the telescope is now placed where the eye is situated, it is obvious that on looking through the telescope toward the mirror the same image will be seen, as soon as the telescope is properly focused. If the galvanometer mirror does not face the middle part of the scale, it should be adjusted with the control magnet attached to the galvanometer case. After focusing the telescope until the image of the scale is visible, slide the eyepiece back and forth until the cross hairs are not only dis- 152 Thermometry tinctly visible but also do not move with reference to the image of the scale when the eye is moved. After the telescope, mirror and scale are in adjustment make the electrical connections as shown in Fig. 72. With the re- sistance thermometer packed in a bath of melting ice and the terminals of the coil of platinum wire connected to the binding posts of the gap marked R^ in the diagram, balance the Wheat- stone bridge by moving the sliding key K^ back and forth until such a position is found that on first pressing the key K^ and im- mediately afterward pressing the key K^ no deflection is visible in the telescope. Note the resistance R^ and the lengths 4 and l^. Then from eq. (8), we have for the value of the resistance being measured R,=R,^. (9) In the same manner find the resistance of the dummy leads. If the temperature of the resistance thermometer when in the bath of melting ice is represented by t' , then the difference between the resistance just found is the value of R' in eq. (2). Proceeding in the same manner find the resistance of the platinum coil when immersed in a steam bath. If this tempera- ture be denoted by t" , the resistance will be the value of R" in eq. (3)- In making these temperature measurements one must press the keys for the shortest possible time else the resistance coil will be heated by the passage of the current. The sliding key K^ should never be moved while in contact with the wire, nor should it be pressed before the battery key K^. Since the bridge is most sensitive when the balance occurs with the slider in the middle of the wire, it follows that the known resistance R^ should never differ much from the unknown resistance R-^. By means of eqs. (2), (3) and (4) together with the values of R' , R" , t' and t" now obtained, the three constants a, b, and i?^ can be computed. On substituting the values of a, h and R^ in eq. (i), an equation is obtained which gives the relation between the temperature and the resistance of the coil for any temperature. Such an equation, containing experimentally determined con- Therm AL Expansion 153 stants whose value cannot be deduced from theory, is called an empirical formula. Substitute for t in this empirical formula tht values — 273", — 200°, — 100°, 0°, 100°, 200°, 300° and 400° and compute the corresponding values oi R ^. With these values plot a curve coordinating R and t. The accuracy of the preceding work should be tested by measuring the resistance of the thermometei coil at various known temperatures and comparing these ob- served values with the corresponding values given by the ctirve. Chapter XIII. EXPANSION OF SOLIDS AND LIQUIDS. ^^'hen the temperature of a body is changed and the pres- sure kept constant, in general, its linear dimensions will suffer a change which is a function of the change of temperature. If a body at 0° is heated through a small temperature range, the change in the length of any linear dimension is found to be di- rectl}' proportional to the change in temperature. Thus if a rod which has a length of l^ at 0° be heated successively to tem- •peratures t^° and t^" (where t^° and t^" are not far from 0°), then its length at t^° will be /, = /„(i+o^) (i) where a is a constant called the mean coefficient of linear expan- sion between 0° and t^" . Similarly the length of the rod at 1.^° will be l^ = l„(i + a.,t^) (2) where a^ is the coefficient of linear expansion between o°and t.° . Since ^1° is nearly equal to t^" , a = a^. Consequently, on being heated from T-c Volume r Thi!; RuvivRSnii,!; Engine; 173 has either entered or left the substance, the heat change is zero. Second. Place the cylinder in the tank of water of temper- ature T and allow the substance to expand. Being in the large tank of water the temperature of the working substance remains constant, but this fact requires heat to enter the substance. De- note this amount of heat by the symbol H^. During this process the work done by the working substance is represented by the area B C c b and the heat absorbed equals J/-,. Third. Transfer the cylinder to the insulating stand and allow the working substance to expand adiabatically until its tem- perature falls to(T — d T). During this process the work done by the working substance is represented by the area C D d c, and the heat change is zero. Fourth. Place the cylinder in the tank of water at the tem- perature {T — d T) and apply pressure to the piston until the sub- stance attains its original condition represented by the point A in the diagram. During this process, the work done on the sub- stance is represented by the area D A a d and an amount of heat that may be denoted by the symbol H^ has left the substance. Since the final condition of the working substance is the same as the original condition, the result of this cycle of four opera- tions is that an amount of work represented by the area ABCD has been produced at the expense of an amount of heat {H^ — H„) . Obviously the cycle could start at any point and proceed in either direction. If the direction is reversed, heat will be transferred from the cold to the hot body at the expense of mechanical work supplied from outside. This ideal engine is consequently per- fectly reversible. From the fact that the working substance must be colder than the source of heat while absorbing heat, and warmer than the condenser when giving heat to it, it follows that the work done will be actually less than that represented by the area ^4 BCD. In addition, in an actual engine there are always unavoidable thermal losses due to conduction and radiation which are not present in the ideal cylinder here considered. Consequently no actual engine is perfectly reversible and no actual engine can do the amount of work corresponding to a perfectly reversible cycle. 174 Gases and Soi^utions It will now be shown that a reversible engine working be- tween any two temperatures will transform a greater fraction of the heat absorbed into mechanical work than any other engine working between the same temperatures. That is, the eificiency of a reversible engine is the highest possible. Thus suppose a certain irreversible engine X be conceived to have a higher effi- Hi Hot H4 N W / ) \ Cold h; Hi Fig. 81. ciency than the reversible engine Y . Imagine the two' engines coupled together so that the irreversible engine X drives the re- versible engine Y in ,the reverse direction. B3' this process, at every stroke X is putting into Y a certain amount of mechanical work W, and by the expenditure of this work, Y will absorb from the cold body an amount of heat H^ and give to the hot body a greater amount H^. The engine X will absorb from the hot body an amount of heat H^, deliver a part of it H^ to the cold body and transform the remainder into an amount W of mechanical work. By the assumption that X is more efficient than Y it follows that the amount of heat H^ required to produce the me- chanical work W necessary to operate the engine Y is less than the amount of heat H^ delivered to the hot body. Consequently if the original assumption is correct, the combined self contained system enables heat to pass continuously from a cold to a hot body until the entire quantity of heat in the cold body is exhausted. Since this result is quite contrary to experience, it proves that the original assumption is false. Therefore no engine can be more efficient than a reversible engine. It follows that all reversible engines have the same efficiency. In the preceding paragraph it was shown that no irreversible engine can do the amount of work corresponding to a perfectly reversible cycle.Therefore a reversible engine is the most efficient possible ; that is, a reversible engine The Reversibi.]!; Engine 175 Fig. 82. working between any two temperatures will transform a greater portion of the heat absorbed into mechanical work than will any other engine. The value of this efficiency will now be obtained. Since the efficiency of all reversible engines is the same and is independent of the working substance, the efficiency can be determined from the consideration of a reversible engine using whatever working substance is most convenient. It will be simplest to consider the action of a perfect gas. Let a mass of perfect g^s go through the cycle of operations represented by A B C D, Fig. 82, between the temperatures T and (T — d T). Draw the lines B b and C c parallel to the pressure axis. The mechanical work done during the cycle is represented by the area {A B C D) =: area (B B C F) when the isothermals and adiabatics are drawn so close together that the figure A B C D may be considerd to be a par- allelogram. The work done W = (B B) (b c). (16) Let V denote the volume of gas when in the condition represented by the point B and let d v denote the small change of volume rep- resented by the line (be). Let p^^ be the mean pressure along the path B C and p.^ the mean pressure along the path D A. Then (BB)=p,-p, (17) represents the mean difference of pressure of the gas when at the temperature T and when at the temperature (T — d T). For the condition represented by the point B we have p, V = R T and for the condition represented by the point B, p^v = R (T ■— d T), since the volumes at B and B are equal. Dividing eq. (19) by eq. (18) p^ __ T—d T _ dT A" f -^ T dT dJT T (18) (19) --P-P. Pi —P2 = Pi (20) 176 Gases and Solutions Whence the work done in the cycle is W = (B B) (b c) = (/>! — p,) dv = p,iZdv (21) And since the heat, received from the source, expressed in dynami- cal units is ^1 d v, the efficiency of a reversible engine is W H,—H^ p,~dv dT (22) J^i J~li pi dv i' And since a perfectly reversible cycle is the most efficient method for the conversion of heat into work, it follows that the maximum- amount of work that can be produced by the passage of the quan- tity of heat J/i from the temperature T to (T — dT^ is W = IT, AJ^ (23) 1 • It should be noticed that this result applies to any reversible cycle employing any working substance. In deriving this result, the sole reason for considering Carnot's cycle is the great sim- plicity of this particular series of operations. XIvV. DETERMINATION OE THE COEFFICIENT OF EXPAN- SION OF A GAS B^ MEANS OF AN AIR THERMOMETER. Object and Theory of Experiment. — If a body is heated from 0° to 1°, its pressure remaining constant, then the ratio of the increase of volume to the initial volume is called the coefficient of expansion of the body. If a body is heated from 0° to i", its volume remaining constant, then the ratio of the increase of pressure to the initial pres- sure is called 'the coefficient of increase of pressure of the body. In the succeeding paragraph it will be shown that in the case of a gas these two coefficients are equal. And since it is easier to measure the pressure of a gas under constant volume than to measure the volume under constant pressure, in the present ex- periment the coefficient of expansion will be determined from Expansion oi? a Gas 177. observations of the change in pressure produced in a gas by changes in temperature, when tlie mass and volume remain con- stant. Let a given mass of gas be at an initial temperature of n° C. (To, absolute), pressure p^ and volume v^. When it is heated to t° C, [ (T^ + ^) absolute], let the pressure be represent- ed by p^ and volume by i\ From the fundamental law of gases eq. (8) p To 'I\ + t ' If the volume be kept constant eq. (i) becomes Po t Pt — Po or, if the pressure be kept constant, we have rp ^0 ^ 168, n = (1) (2) ■P" = , Po t (3) Fig. 83. from eq. (5) p. 154. Therefore the coefficient of expansion of a gas equals the "pressure coefficient." The apparatus best suited to the applica- tion of the above formula is some form of Jolly's Air Thermometer. This consists. Fig. 83, of a glass bulb B filled with air or other gas, connected to an open manometer tube M filled with mercury. Immediately below the bulb is a tube containing an index finger F made of colored enamel. The volume of the gas is made definite, by adjusting the plunger P until the mercury surface is brought to the point P. The pressure of the gas in the bulb is measured by the difference of level of the mercury at P and the mercury in the mano.- meter tube. The bulb is enclosed by a vessel in which can be placed water or ice. Z? is a drying tube used in filling the bulb. 12 178 Gase;s and Solutions On account of the temperature of the small amount of gas in the exposed part of the bulb being different from that in the bulb, and also on account of the change of volume of the bulb when its temperature is changed, eq. (3) cannot be used in its present simple form. The corresponding equation in which these facts are taken account of will now be derived. Let Pa = pressure in bulb when at 0° C. (T^ absolute) , p^ -z pressure in bulb when at t° C. (T absolute)^ ■Vg = volume of bulb at o" C. V ^ ^=. volume of bulb at t° C. M' = mass of gas in bulb when at 0° C. M" = mass of gas in bulb when at t° C. v' = volume of exposed part of bulb when bulb is at 0° C. v" =: volume of exposed part of bulb when bulb is at f C. Ill' := mass of gas in exposed part of bulb when bulb is at 0° C. m" = mass of gas in exposed part of bulb when bulb is at o" C. Without sensible error, the temperature of the exposed part of the bulb may be assumed to be constant and equal to that of the room. Let this temperature be denoted by t' C. {T' absolute). It follows that v' = v" approximately. Applying the fundamental law of gases we have p, Vo = -B M' r„ (4) p^v^ = RM"T (5) p„v' = R nf T' (6) p^ V' = R m" T' (7) Since the mass of gas in the apparatus remains constant M' +m' = M" + m" (8) By means of eqs. (4), (5), (6) and (7) eq. (8) becomes To T' T r ^ ' Representing by 7 the coefficient of cubical expansion of glass between the temperatures 0° and t° C, we have Vt=Vo{l+yt) (10) Expansion of Gases 179 Substituting this value in eq. (9) and remembering that T^T, + t. T'=T, + f and that T, = ^ (eq. 3) we obtain Dividing through b}- v„ and denoting the constant ratio v' -^- v„ by k we obtain ^^'hence solving for /3 and neglecting terms containing ^"- and y/8 we have for the value of the coefScient of expansion of a gas Po (« + «' +/c<)— j>o (r +/(;i) Manipulation and Computation. — If the value of the quotient {y' -^ z\^) is not known, it must be determined by finding the volume of the bulb and of the attached tube by weighing first when empty and then when filled with water up to the proper points. The bulb is now to be thoroughly dried by drawing through it air that has passed over fused calcium chloride. After the apparatus is assembled the bulb is filled with thoroughly dried gas. In this experiment air will be the gas used. To simplify this operation, a fine capillary tube C has been sealed into the bulb. This capillary is connected to a drying tube D and then the bulb partially evacuated by raising the plunger P. This operation draws dry air into the bulb. By alternately depressing and rais- ing the plunger a number of times, the air finally in the bulb will be tjuite dry. The end of the capillary C is now sealed off with a small blowpipe flame. Fill the vessel enclosing the bulb with snow or small pieces of ice and adjust the plunger until the mercury in the air ther- mometer tube reaches the index F. Observe the difference in height between the mercury in the manoineter tube M and the index F. This difference in height plus the height of the barometer, is the value of p,, in eq. (13). Substitute cold water for the ice and carefully heat it to about 30° C, by passing steam into it. This must not be done i8o Gases and Solutions too quickly else the glass may crack. Again adjust the plunger until the meniscus of mercury in the thermometer tube just touches the index finger, and again observe the difference of height between the mercury in the manometer tube and the index. This difference added to the barometric height is the value of Pj in eq. (13). The temperature of the warm bath is ^° C. The resulting value of /5 is the coefficient of expansion of air from 0° to 30°. Raise the temperat-ure of the bulb to about 50° C, and find the value of the coefficient of expansion for the range 0° to 50°. In the same manner find the value of the coefficient of expansion for temperature ranges o" to 70" C, and o" to 100° C. Test Questions and Problems. 1. Why is it usually unnecessary in. this experiment to re- duce the barometric height and the manometric height to 0° C? 2. Outline clearly the assumptions that have been- made and any considerations that have been neglected in the derivation of eq. (13) which limit the accuracy of the result. XLVI. DETERMINATION OF THE MAXIMUM VAPOR PRES- SURE OF A LIQUID AT TEMPERATURES BELOW 100" C, BY THE STATICAL METHOD. Object and Theory of Experiment. — When a liquid evap- orates in a closed space the vapor formed produces a pressure on the surface of the liquid and the enclosure which increases with the mass of vapor. The vapor pressure* reaches a maxi- mum value when the space is saturated. The maximum vapor pressure, or pressure of the saturated vapor, depends upon the temperature of the vapor. The object of this experiment is to determine the pressure of saturated aqueous vapor at tempera- tures from about 50" C, to 100° C. *The expression "vapor tension" is sometimes used instead of "vapor pressure" to denote the elastic stress exerted by a vapor. Careful writers, however, use the v/ord pressure to denote a push, and tension to denote a pu'.l. Since vapors and gases cannot exert a pull the term vapor tension is a mis- nomer. Vapor Pressurr i8i In the Statical Method to be used in this experiment, the vapor pressure is determined from an observation of the change in the height of a barometer column produced by the introduc- tion in the vacuous space above the mercury of a small quantity of the specimen. The apparatus, Fig. 84 consists of a baro- meter tube having its upper end enlarged into a narrow bulb, and its lower end joined to an open manome- ter tube M. Opening into the horizontal tube joining the barometer and manometer is an iron cylinder filled with mercury. The height of mercury in the two tubes can be varied by means of a plunger, P, in this cylin- der. A small finger, F, ma'de of colored enamel in the bulb of the barometer tube serves as a convenient fixed point from which heights can be measured. TlTe vapor being studied can be brought to the desired tem- perature by means of a glass water bath sur- rounding the bulb. If the barometer tube contains only mer- cury, water, mercury vapor and water vapor, then the sum of the maximum vapor pressure of mercury and of water, at the temperature of the experiment, equals the atmospheric pressure diminished by the difference of level of mercury in the two tubes. The difference between this pressure and the maximum vapor pressure of mer- cury at the particular temperature of the experiment gives the maximum vapor pressure of water at this temperature. But the vapor pressure of mercury is so minute at temperatures below 100'' C, that this correction is useless in any experiment made by this method by inexperienced students. Manipui,ation and Computation.- — For the purpose of in- troducing the specimen into the barometer tube, the bulb is terminated by a fine capillary tube. Screw the plunger down to near the bottom of its run and fill the apparatus up to the top of the bulb with mercury. The small amount of air clinging to i82 Gasss and Solutions the sides of the tube can be driven off by passing a Bunseu's burner flame up and down the length of the tube until the con- tained mercury is raised nearly to its boiling point. Allow the mercury to cool to the temperature of the room. Then slip over the end of the capillary a piece of rubber tubing about three inches long, depress the plunger until mercury is forced out of the end of the capillary. Fill the section of rubber tubing with boiled water. Now raise the plunger until about a cubic centi- meter of water is drawn into the bulb. Remove the piece of rub- ber tubing and seal off the end of the capillary with a blow-pipe flame. If this operation has been carefully performed, the ba- rometer tube will now contain only mercury and water. Observe the atmospheric pressure from the laboratory standard barometer. Fill the water jacket with water and pass steam into it until it reaches a temperature of 50" to 60" C. The current of steafn must not be directed either on the bulb or on the glass water jacket else the sudden expansion of the glass may crack it. By means of the plunger adjust the height of the mer- cury in the barometer tube until it is brought just into contact with the tip of the index finger. Stir the water in the water jacket, observe its temperature, readjust the plunger if necessary, and move the slider S until its cross wire is tangent to the men- iscus in the manometer tube. Read the difference of level between the cross wire of the slider 6' and the end of the index finger F. Add to this height the number of mm. of mercury equivalent to the layer of water on the surface of the mercury in the ba- rometer tube. The atmospheric pressure in mm. of mercury di- minished by this corrected difference of level is the maximum pressure of aqueous vapor for the particular temperature of the experiment. Take similar readings every 5° up to 100° C. Plot a curve with vapor pressures as ordinates and corresponding tempera- tures as abscissas. On the same coordinate axes plot another curve from the values given in Table 15. This method is liable to several errors. The surface tension of the dry mercury in the manometer tube is different than that of the wet mercury in the barometer tube. This will cause a rise of the column having the wet surface of o.i to 0.15 mm. Vapor Pressure 183 The fact that the lower part of the barometer tube is at a lower temperature than the upper causes the final result to be too low. This error will be of the order of 0.15 mm. If the position of the end of the index finger is read through the water jacket the refraction of the glass and water will introduce an un- certainty that can amount to 0.5 mm. This error is obviated by carefully measuring the distance from the end of the index to a fine scratch on the tube below the water jacket before the ap- paratus is assembled This scratch would then be used as the fiducial point from which heights are measured. The greatest limitation to the use of this method, however, is due to the large error introduced in the depression of the barometer column pro- duced by any impurity of the specimen. Test Questions and Problems. I. It is found that under the same atmospheric pressure, the boiling point of a solution is higher than that of the pure solvent. It follows from the definition of boiling point that the maximum vapor pressure of a solution is less than that of the pure solvent at the same temperature. The fraction obtained by dividing the difference between the vapor pressure of the pure solvent and that of the solution, by the vapor pressure of the pure solvent for that temperature is called the fractional lowering of the maximum vapor pressure of the solution. This fractional lowering of the vapor pressure is a constant quantity independent of the temperature, depending on the nature and concentration of the solution. Knowing the boiling point of the solution, this quantity can be obtained directly from the curve coordinating vapor pressures and temperatures for the pure solvent. Raoult has shown that the fractional lowering of the vapor pressure of a solution equals the ratio of the number of mole- cules of solute to the number of molecules of solvent in the solu- tion. It follows that if one knows the fractional lowering of the vapor pressure of a solution, together with the molecular weight of the solvent and the masses of solute and solvent in the solution, then the molecular weight of the solute can be deter- mined. By the method above indicated, show fully how the mole- cular weight of sugar could be obtained. i84 Gases and Soi^utions XLVII. DETERMINATION OF THE MAXIMUM vXPOR PRES- SURE OF A LIQUID AT VARIOUS TEMPER- ATURES, BY THE DYNAMICAL METHOD. Object and Theory oe Experiment.- — The object of this experiment is to determine the maximum vapor pressure of water at various temperatures from 50" C, to about 120° C. The dynamical method to be employed in this experiment is based upon the following two laws of vapors. First, a liquid boils when the pressure of its vapor equals the external pressure. Second, the temperature of the boiling point remains constant as long as there is any liquid to vaporize, provided the pressure re- mains constant. In Regnault's apparatus. Fig. 85, the specimen is enclosed in a boiler B, connected by means of an inclined tube with a large Fig. 85. Vapor Pressure 185 metal reservoir R, enclosed in a water bath kept at constant tem- perature. The reservoir is filled with air the pressure of which is varied by means of a pump connected to P. The pressure of the vapor is measured by means of the open manometer M. The temperature of the vapor in the boiler is obtained from thermo- meters T, placed in tubulures filled with mercury, projecting into the boiler. The inclined tube connecting the boiler and the res- ervoir is enclosed by a condenser C^ through which flows a stream of cold water. By means of this condenser, however vigorously or however long a time the liquid may have been boiling, the pressure of the vapor in the boiler is kept constant. Another object of the condenser is to prevent vapor from reach- ing the manometer. The air in the reservoir serves to equalize any sudden changes of pressure due to "bumping" or other ir- regularities in boiling. Manipulation and Computation. — Fill the boiler one- third full of water. To reduce "bumping" put a handful of clean pebbles or metal turnings into the boiler. Close the boiler and then connect a suction pump to P- Pump air out of the res- ervoir tmtil the pressure is reduced to about 10 cms. of mercury, i. e. until the difference of height between the mercury in the two arms of the manometer is about ten centimeters less than the barometric height. Close the stop cock in the tube connecting the pump and the reservoir. Start a stream of cold water flow- ing through the condenser. Place a Bunsen's burner under the boiler and after the thermometer indicates a constant temperature, observe the temperature and the pressure. Note also the temper- ature of the room and the barometric height. The barometric height diminished by the difference of level between the mano- meter columns equals the pressure of the vapor at the tempera- ture of the experiment. This pressure in centimeters of mercury is to be reduced to 0° C, assuming that the manometer scale is correct at 20° C. Air is now allowed to enter the reservoir until the pressure is about 10 cms. higher than before. This increase of pressure requires that a higher temperature be attained before the water will boil. When the temperature has reached the new boiling 1 86 Gases and Solutions point, a second series of observations is to be taken similar to the preceding. In the same manner, the boiling points corres- ponding to a number of different pressures are to be determined. For pressures less than one atmosphere, the boiling points should be observed at intervals of pressure of about lo cms. of mercury. For pressures above one atmosphere the intervals of pressure may be 15 to 20 cms. of mercury. Plot a curve with pressures as ordinates and temperatures as abscissas. This curve showing the variation of the pressure of saturated aqueous vapor with the temperature is called the steam line. XLVIII. DETERMINATION OF THE DENSITY OF AN UNSAT- URATED VAPOR BY VICTOR MEYER'S METHOD. Object and Theory oe Experiment. — Probably the most accurate method for determining the density of an unsaturated vapor is to allow a known mass of the liquid whose vapor density is to be determined, to vaporize in the Torricellian vacuum of a barometer, and then to observe the volume occupied by the vapor. The ratio of the mass of the liquid vaporized to the volume oc- cupied by the vapor is the density of the vapor at the temperature and pressure of the experiment. But in case the required ac- curacy does not exceed from three to five per cent, a method due to Victor Meyer will be found much more convenient. The apparatus used in this method is shown in Fig. 86. It comprises a gas measuring tube B, and a vapor chamber con- sisting of a long, glass tube terminating in a bulb B surrounded by a bath containing a liquid of higher boiling point than the substance under examination. The specimen is contained in a small bulb which can be supported in the upper cooler part of the vapor chamber by means of a rod R capable of a back and forth motion in a side tube. When the bath has attained a con- stant temperature, high enough to vaporize the specimen, the rod R is drawi-uback so as to allow the little bulb containing the Vapor Pressure 187 specimen to fall to the bottom of the chamber. Here it either breaks by concussion with the bottom, or bursts due to the ex- pansion of the contained liquid. When the contained liquid is vaporized, it pushes out of the vapor chamber an equal volume of air which is measured by means of the measuring tube B. In bubbling up through the water in the measuring tube the air expelled from the vapor chamber becomes cooled and so contracts according to Charles' law. It follows that the volume i88 Gases and Solutions of water displaced in the measuring tube equals the volume which would be occupied by the hot air displaced by the vapor, if the vapor were at the temperature of the air in the measuring tube and if it remained unsaturated at that temperature and pressure. Hence we conclude that the volume of water displaced in the measuring tube equals the volume the vapor would occupy at the temperature and pressure of the air in the measuring tube. Therefore the density of the vapor at the temperature and pres- sure of the air in the measuring tube equals the mass of sub- stance vaporized divided by the volume of water thereby dis- placed from the measuring tube. The temperature of the bath surrounding the vapor chamber must remain constant during the vaporization of the specimen, but its value need not be known. Since the densities of gases and vapors vary greatly with changes of pressure and temperature it is customary to reduce the values to what they would be at some standard pressure and' temperattu-e. The pressure usually adopted as standard is the pressure of 760 mm. of mercury, and the temperature adopted as standard is 0° C. Manipulation and Computation.^ — The substance to be selected for the bath will depend upon the temperature of vapor- ization of the specimen being examined. The following sub- stances will be found convenient to use : water, whose boiling point is 100°; analin, 182.5° C; bromonapthalin, 280° C. The specimen is enclosed in a thin glass bulb C which may be filled as "ilhistrated in figure. By placing a hot metal rod below C some of the contained air will be driven out and on cooling, the bulb will become partially filled with the specimen. By repeating this operation the bulb can be entirely filled. If the liquid is volatile the stem of the bulb must be sealed in a flame or plugged in some manner. The mass of the specimen is determined by weighing. The bulb is now supported in the cool part of the vapor chamber by the rod R. When the temperature of the bath becomes constant no more air will bubble up through the water in the trough V. When this state is attained, the measuring tube U, filled with water, is placed over the outlet of the discharge tube and the rod R is withdrawn, thus allowing the bulb to fall. The bulb will now be broken and its contents vaporized. The volume Vapor Density 189 of air entering the measuring tube is observed as well as its tem- perature. Note also the barometric height. The pressure of the moist air in the measuring cube is equal to the barometric pressure diminished by the sum of the pressure due to the column of water within B above the surface of V, and the pressure of aqueous vapor at the temperature of B. This latter ma}' be taken from tables. Let ;// be the mass of substance vaporized, and v and v^ the volume of the vapor when at the pressure, temperature and den- sity i*. i p and Pa, to- Po respectively. From the fundamental law of gases jpo ''o = Rm 2'o (1) pv = Rm{l\ + i)- (2) Dividing (i) by (2) Po ^'0 + t _ 76 m (273 - Vq 273 -r p Whence ^ m ^ 76 m (27 3 + t) .2^ The Relation between Density and Molecuear Weight. Charles and Gay Lussac have shown that the coefficient of thermal expansion of all perfect gases {i. e. those obeying B'oyle's Law) is the same. From this result in connection with the fact that the dimensions of the molecules composing a gas are minutely small compared with the distance separating them, Avogadro inferred that equal volumes of all perfect gases at the same tem- perature and pressure contain the same number of molecules. If the densities of two different gases under the same pres- sure and temperature be represented by P and p' respectively, and the masses of single molecules of the two substances be repre- sented by w and w' respectively, then from Avogadro's Law, the number of molecules contained in unit volume is N = ^=^. (4) w ir' It is impossible to measure the absolute mass of the mole- cule, but the above equation furnishes a method for determining 190 Gases and Solutions the relative masses of the molecules of two different gases. These relative masses are usually called Molecular Weights. The mole- cular weights of substances are computed in terms of hydrogen which h. arbitrarily assumed to have a molecular weight of two*. It is also customary to express densities of gases in terms of the density of hydrogen. Denoting the specific gravity of a gas referred to hydrogen hy p' a and putting w =^ 2 and P = i, eq. (4) becomes w' = 2 p'l (5) That is, the molecular weight of a gas equals numerically twice its specific gravity referred to hydrogen. This law furnishes an important method for determining the molecular weight of sub- stances. In Table 5, under the heading Molec. Wt. Calc. are given values of the molecular weight of certain gases deduced from their chemical behavior, and under the heading Molec. Wt. Obs. are given values deduced from their vapor densities. An inspection of these values shows that there are some substances which are abnormal in that they have different molecular weights at different temperatures. The mole- cular weights of the substances belonging to this class are less at high temperatures than at low temperatures. For instance, for temperatures up to about 450° C, iodine vapor gives a normal value for the molecular weight, while for higher temperatures it is much less. At about 1500° C, the molecular weight of iodine is about half of its normal value. This in- dicates that at these high temperatures the molecule dissociates *The reason for calling the molecular weight of hydrogen 2 instead of 1 is apparent from the following considerations. It is found by analysis that 1 part of hydrogen, by weight, combines with 35.4 parts of chlorine to form hydrochloric a'cid gas. Since it is found impossible to replace a fractional part of either this hydrogen or chlorine by another substance, it is assumed that the molecule of hydrochloric acid consists of a single atom of hydrogen combined with a single atom of chlorine. That is, the smallest possible mole- cular weight of hydrochloric acid is 1 -|- 35.4 =36.4. Since molecular weights are proportional to densities according fo Avogadro's law, we have density of H : density of HCl . : molecular weight of H : 36.4. But from experiment it is found that density of H : density of HCl : : 1 : 18.2. Therefore the molecular weight of hydrogen is taken as two. Rei,ati\i3 Humidity 191 — /. c. the molecule has changed from 1^ to 1 ~\- I. Similarly nitrog-en peroxide (N., O4) dissociates into N O2 + N Oj. On the other hand acetic acid has a normal molecular weight only at comparatively high temperatures. At low temperatures the molecule is associated — i. c. two molecules are combined into a single molecule. Chapter XV; HYGROMETRY. Hygrometry or Psychrometry is the art of measuring the amount of moisture in the atmosphere. The mass of water con- tained in unit volume of air is called the absolute humidity. The ratio of the mass of moisture contained in unit volume to the mass which would saturate the same space at the same temper- ature is called the hygrometric state or relative humidity of the atmosphere. Let p be the pressure of a mass iii of aqueous vapor at the temperature T contained in a given volume v of air. Let m' be the mass of vapor at the pressure p' necessary to saturate the same space at the same temperature. Then since for ordinary atmospheric temperatures aqueous vapor obeys approximately the fundamental law of gases up to the point of saturation, we have p V ^ R m T and p'v = R m' T. That is -^=p^ (approx) (1) Consequently relative humidity equals the ratio of the actual pres- sure of the aqueous vapor in the air to the maximum pressure at the same temperature. It thus appears that there are two general methods of de- termining the relative humidity of the atmosphere. The first re- quires the measurement of the actual mass of aqueous vapor contained in a given volume of air. This can be done by draw- ing a given volume of the air through a drying tube and weigh- 192 Hygrometry ing the drying tube. The mass of aqueous vapor required to sat- urate the same space at the same temperature can be obtained from tables. The more common method, however, is to determine the actual pressure of the vapor in the air, and then from tables find the pressure of saturated aqueous vapor at the same tem- perature. XLIX. DETERMINATION OF THE RELATIVE HUMIDITY OE THE AIR WITH DANIELL'S DEW-POINT HYGROMETER. Object and Theory oE Experiment. — The temperature to which the atmosphere must be cooled in order that the water vapor present may be saturated is called the dew-point. The object of this experiment is to determine the relative humidity of the atmosphere from an observation of the dew-point. Consider a mixture of air and aqueous vapor having the volume if' and temperature T. Let p" denote the pressure of the mixture and p the pressure of the water vapor contained in it. Down to the temperature of saturation, both aqueous vapor and air obey approximately the fundamental law of gases. Therefore V' p" — 7p- = const. — =- = const. So long as p" is constant -^- is constant, whence p is constant. Expressed in words this conclusion is, that down to the temper- ature of saturation, the actual pressure of the aqueous vapor con- tained in any portion of the atmosphere is a constant quantity. Consequently, the pressure of the aqueous vapor in any portion of air can be determined by cooling the air down to the dew point and looking up in tables the pressure of saturated aqueous vapor corresponding to this temperature. From the definition of rela- tive humidity and eq. (i) p. 191, it follows that for any portion of air at the temperature T, the relative humidity ^=-f (approx.) (1) Relative Humidity 193 where p and p' represent the pressures of saturated aqueous vapor at the dew-point and at the temperature T respectively. These quantities are given in tables. Manipulation and Computation. — Daniell's hygrometer consists of two glass bulbs connected by a bent tube as shown in Fig. 87. The lower bulb contains ether and a thermometer. The upper bulb is wrapped with a piece of muslin. In determining the dew point with this apparatus all of the contained ether is passed into the lower bulb and then the upper bulb is moistened with ether. The evapora- tion of the ether poured on the upper bulb causes the bulb to cool and a part of the vapor in the apparatus to condense. This in turn, induces evaporation at the surface of the ether in the lower bulb. By this means the temperature of the lower bulb is grad- ually lowered until dew is deposited on its surface. The temperature of the lower bulb is then read. The apparatus is ^ now allowed to remain until equilibrium ' '' is restored and the temperature begins to rise. The tem- perature at which the deposit of dew disappears is noted. The mean of the temperature of the naked bulb when the deposit ap- pears and when it disappears is taken as the dew-point. Note the temperature of the surrounding air by the thermometer at- tached to the wooden stand supporting the hygrometer. From Table 15 find the pressure of saturated aqueous vapor at the dew-point and at the temperature of the room. Make at least five determinations and take the average. Test Questions and Problems. I. State and discuss the errors to which this method is sub- ject. 13 194 Hygrome;try L. DETERMINATION OF THE RELATIVE HUMIDITY OF THE AIR WITH THE WET AND DRY BULB HYGROMETER. Object and Theory of Experiment.- — If two exactly sim- ilar thermometers, the bulb of one being naked and the bulb of the other being covered with a wet wick, are placed near one another in a current of air, the thermometer with the naked bulb will in- dicate the temperature of the air while the other will indicate a lower temperature. The difference between the indications of the two thermometers is due to evaporation at the surface of the wet bulb and depends upon the degree of saturation of the air. The relation between the relative humidity of the air and the indica- tion of the thermometers has never been obtained in an entirely satisfactory manner from purely theoretical considerations. But by comparing the indications of this hygrometer with the indica- tions of hygrometers of other types, tables have been constructed by means of which the relative humidity of the air can be readily determined from a single pair of simultaneous readings of the wet and dry bulb thermometers. The numbers in Table 17 were obtained from a comparison of simultaneous readings extending over several years' of the Daniell and the wet and dry bulb hygrometers. As an example, on placing these two instruments near one another the following simultaneous readings were obtained. Temperature of the air, 21° C, Temperature of the wet bulb, 19° C, Dew-point, 18° C. From the Daniell hygrometer readings, together with eq. (i) p. 191 and the values given in Table 15 we have the relative .hu- midity fl = L = i^ = 83%. p' 18.47 Relative Humidity 195 Whence for the wet and dry bulb hygrometer we have 0.83 = -^^ . 18.47 Consequently in the case of the wet and dry bulb hygrometer, corresponding to an atmospheric temperature of 21° C, and a wet bulb temperature 2° C, lower, the pressure at the dew point of the saturated aqueous vapor contained in the atmosphere equals 15.33 riini- o^ mercury. In Table 17, this number 15.33 i^ placed in the line numbered 21 " C, and the column numbered 2° C. Manipulation and Computation. — The wet and dry bulb hygrometer, sometimes called August's psychrometer, consists of two similar thermometers, one with a naked bulb and one with the bulb covered by an envelope of wet muslin. A current of air is caused to blow over the two bulbs with a fan or some other means. A convenientt arrangement for this purpose consists of a frame supporting the two thermometers. Fig. 88, capable of rotation by hand. Fig. 88. See that the muslin envelope about the bulb of the wet thermometer is kept thoroughly moist. Change rapidly the air in which the instrument is situated with a fan or the rotating de- vice shown in the figure. When the wet bulb has reached a stationary temperature read the two thermometers. From Ta- ble 15, find the pressure p' of saturated aqueous vapor for the temperature given by the dry bulb thermometer. From Table 17, find the pressure p of the aqueous vapor in the atmosphere at the temperature of the experiment corresponding to the difference be- tween the readings given by the wet and dry bulb thermometers. Then from eq. (i) p. 191, the relative humidity. //= A . P' Make not fewer than five determinations and take the aver- age. Before each determination be certain that the muslin en- velope is thoroughly moist, 196 Calorimetry Chapter XVI. CALORIMETRY. The art of measuring quantities of heat is called calorimetry. Unfortunately, there is no single quantity of heat that is univer- sally adopted as the unit. The quantity of heat adopted as the unit in scientific work is the amount of heat required to raise the temperature of one gram of water from 10° C. to 11" C. This unit is called the calorie or the gram-centigrade-degree thermal unit. When a larger unit is desirable, a unit one thousand times as great is taken. This is called the larger Calorie or the kilo- gram-centigrade-degree thermal unit. In the British system of units, the unit of heat adopted is the amount of heat required to raise the temperature of one pound of water from 50" F. to 51° P. This is called the British thermal unit or the pound- Pahrenheit- degree thermal unit. Throughout this book the calorie will be used exclusively. The number of thermal units required to raise the tempera- ture of unit mass of a substance from t° to {t -[- 1)° is called its specific heat at t°. The specific heat of bodies is slightly differ- ent at different temperatures, but the difference is so minute that it need not be considered except in the most refined measurements. With this understanding, it is customary to speak of the average specific heat of a body between t° and t^°, as the ratio of the num- ber of heat units required to raise any mass of.it from t° to t^°, to the amount of heat required to raise the temperature of an equal mass of water through the same temperature range. That is, the quantity of heat H required to raise the temperature of m grams of the substance of average specific heat c from t° to t^"' is H = mc {t^° — t°) (i) The quantity of water which requires the same amount of heat as a given body in order to change its temperature by one degree is called the water equivalent of the body. That is, if e Correction for Radiation 197 represents the water equivalent of a body, and ^^ the mean specific heat of water between t° and t^°, then the quantity of heat re- quired to raise the temperature of the body from t° to t^° is H=es{t^°^t°) (2) Equating eqs. (i) and (2) m c (3) Ordinarily, the specific heat of water may be taken as a constant equal to unity. In this case e = m c (4) that is, the water equivalent of a body equals the product of its mass and its specific heat. Although most simple in theory, calorimetric experiments require great care and many precautions for their successful oper- ation. One source of error whose elimination must be always provided for, except in the coarsest measurements is that due to radiation, i. e. absorption or loss of heat due to the neighboring bodies being at a different temperature than the body studied. The principal methods of eliminating this error are (a) to compute the amount of heat actually gained from, or lost to, the surroundings by the body during the experiment; (&)to determine the tempera- ture which the body would have attained if there had been no gain of heat from or loss of heat to the surrounding bodies ; (c) by the employment of an experimental method in which the temperature of the body is kept the same as that of its surroundings. The Correction for Radiation. I. Regnault's method is based on Newton's Law of Cooling. This law may be briefly stated as follows : The rate of cooling {i. e. the change in temperature per second) varies directly as the excess of the temperature of the body over that of its surround- ings.* The same law may be enunciated in another form which •a student familiar with analytic geometry wil perceive that this statement Implies that the relationship between the temperature and time of cooling may be represented by an equilateral hyperbola referred to its asymptotes. A-, rgS Calorimetry will be often found very convenient :■ — the amount of heat lost by a radiating body is proportional to the time it radiates, and to the mean difference in temperature, during this time, between the radiating body and the surroundings to which it radiates; also, the amount of heat lost in a given time for a given mean difference in temperature between a body and its surroundings, depends only on the nature and extent of the surface of the body. X" " "" 1 1 :::::;:iti=; ::: ::::: j^::::::: ± *- --_^,__H rp: = ==i-_ -. .. .^ in ±LJ Curve showing rate of change of temperature of heated body. Fig. 89. Imagine a body whose water equivalent is e' to have a definite quantity of heat given to it causing it to rise in temperature in a manner that can be represented by the curve ABC Fig. 89. The maximum temperature is reached when the body ceases to receive heat from the source faster than it radiates heat to the cooler sur- roundings. After this point is reached, the body falls in tempera- ture in a manner that can be represented by the line C D. During the time the body was below the temperature of the surroundings, it was absorbing heat from the surroundings; while during the time it was above the temperature of its surroundings it was losing heat to the surroundings. The Radiation Correction, now to be found, is the difference between the amount of heat lost by the body on account of radiation and the amount gained by absorp- tion while the body was rising from its original to its maximum temperature. Let H^ represent the amount of heat lost during the time T while cooling from the temperature t' to t", and let t^ represent CORRISCTION FOR RADIATION I99 the temperature o.f the surrouDclings. Then applying Newton's laws of cooling to the part C D oi the curve, we have Ih = rT[y2{t' + t")-t,] where r is a constant and t' and t" are taken so close together that the portion of the curve between tliem may be regarded as a straight line. Since H^ = e' {t'—t"), it follows that g' {f — t")- '^ ~ Tl%{t' +t")-t,-]- (5) In other words, the constant r is the amount of heat lost (or gained) per minute, per degree difference in temperature between the body and the surroundings. This is called the Radiation Con- stant of the body. We are now in a position to find the Radiation Correction for the interval represented by the curve ABC, that is, while -the body was rising to its maximum temperature. Let H' be the amount of heat lost by radiation during the interval T' while the body was above the temperature of the room, and H" the amount of heat received by absorption during the interval T" that the body was below the temperature of the surroundings. Let t^ be the average difference in temperature between the body and the room in the first case, while t^ represents the average difference in the second case. Then H' = rrt, and H" = r T" t,. Finally, the Radiation Correction is R^ H' — H" =r{T'ti — T" t^) e' {f—t"){T' t,— T" t,) When the curve B C \s not a straight line, the average differ- ence in temperature, t^, between the body and the room during the interval of time T' is found by dividing the area of the figure B C B hy the horizontal distance between the ordinates passing through the points B and C. This is the general method of find- ing the mean ordinate of a curved line. 2. Instead of finding the number of heat units lost by the body due to radiation while the temperature of the body is rising 200 Calorimetry to its maximum value, the effect of radiation can be accounted for if the temperature is determined to which the body would have attained if there had been no radiation. In the following modi- fication of a method due to Rowland this temperature can be ob- tained to a close approximation by a stmple graphical construc- tion. Imagine a body at a temperature below that of the room to be given a quantity of heat, H, such that its temperature will rise to a value above, that of the surroundings. During the time that the temperature of the body is lower than that of the sur- roundings, the body will absorb heat from the surroundings ; and during the time that the temperature of the body is above that of the surroundings, the body will lose heat to the surround- ings. Let the rise of temperature with respect to time before the ±::::::::::::::J::f:E±^::: ::::::::::::::::::::::::: J"-" ±: :::::: :"-[::■ V:- -j: -:-::-:::: ^ = :h-i m = ;:;::: E":: :: 40^ tt-j----^ x---i:-"-i, " T^t'-'- = = = 1 = = = = = """"" -^:::::^:^^ = ::::5.r:::::±::: :::::::::::::::::::: :::::::::::: :-T1 Tife-Pi \rFin?is 8 9 10 t1 Curve showing rate of change of temperature of the heated body. Fig. 90. quantity of heat, H, was added be represented by the line A B Fig. 90. The line B D represents the rate of change of tem- perature of the body while it is absorbing the heat H. During the time the temperature is rising from B to C the body is, in ad- dition, receiving heat from the surroundings, while during the time the temperature is rising from C to D heat is being lost to the surroundings. The line D B represents the rate of change of temperature of the body due to radiation alone. Correction for Radiation 201 Through the points B, C and D draw lines parallel to the temperature axis. Lay oS W Z oi a. length corresponding to any convenient interval of time. Draw W X parallel to the tem- perature axis. Through D draw a line parallel to X Z, and also a line parallel to the time axis. Produce A B until it intersects the vertical line drawn through C. From the diagram, the rate of fall of temperature due to radiation is (A' W)^{WZ) = Un e, therefore the fall of temperature due to radiation that has oc- curred during the interval of time T' is T ta.n Q = {b d). (7) Similarly from the absorption curve A B the rate of increase of temperature due to absorption is found to be {Bs)-^ {As) - tan$, therefore the increase in the temperature of the body due to ab- sorption of heat from the surroundings during the interval T" is T" tan # = (wz n). (8) Consequently, the temperature that the body would have at- tained if there had been neither loss of heat by radiation, nor gain of heat by absorption from the surroundings, would be repre- sented by the point D on the diagram, plus the number of degrees corresponding to the distance b d minus the number of degrees corresponding to tn n. That is, the total rise of temperature would be represented by the distance m d. The condition under which this method is most accurate is readily deduced from Newton's laws of cooling. Considering the part of the curve CD above the temperature of the sur- roundings, and letting t represent the rise in temperature above the maximum observed temperature, that would have occurred if there had been no loss by radiation, then the amount of heat lost by radiation is H' = et. (9) Again, from the laws of radiation, H'^rVt, (10) where r is the radiation constant and f, is the mean difference 202 CaIvORIMETRY in temperature between the body and the room during the in- terval of time T'. If the temperature corresponding to the point X on the dia- gram be denoted by t', and that corresponding to H^ be denoted by t", then as in eq. (5) p. 199, we have for the value of the radia- tion constant, _ e {f —t") . _ ^ tan e ^" r[y^ {f + 1") — /o] ~ Yz {f + t") — /„ (11) Eliminating H' from eqs. (9) and (10), and substituting for r its value from eq. (11) we obtain, T t-^ tan e Denoting by t-[ the temperature difference corresponding to the distance (in n) Fig. 90, we have from eq. (7), T' tan = t^. Therefore ^-V[ ^ (.+%_, J - (12) In the preliminary discussion of this method it was assumed that the cooling curve Z? 5 is a straight line and on this assump- tion it was shown that t = t^. Equation (12), shows that even when the temperature range of the experiment is too great to permit the assumption that the cooling obeys a straight line law, that t = t^ when ?g = J^ (^'+*") — ^o- I'^ other words, when the angle 6 is measured at a point on the coaling curve corresponding - to the mean difference in temperature during the interval T' be- tween the body and the room, then the number of degrees that must be added to the maximum temperature D in order to correct for radiation corresponds to the distance {m n) on the diagram. The preceding consideration indicates that this method fur- nishes an accurate and easily applied means of correcting for radiation whenever the difference between the maximum or mini- mum temperature of the body and the temperature of the room is not considerable. And even when this difference is large the method is applicable under conditions that can be realized in practice. Correction for Radiation 203 3. — Another method should be referred to, which though fallacious, is so plausible that the student must be warned against its use. In this method, first suggested by Rumford, the initial and final temperatures of the body are arranged so that the dif- ference between the room* temperature and the initial tem- perature of the body equals the difference between the room tem- perature and the final temperature of the body. The idea is that by this arrangement, the heat absorbed from the room while the body is colder than the surroundings equals the heat lost to the room while the temperature of the body is higher than that of the surroundings. That this idea is illusory is apparent from the following consideration. In a certain experiment performed in the Laboratory of Physics at Purdue University, a mass of lead was first heated to a high temperature and then immersed in a mass of water that was at a temperature lower than the room temperature. By previous trials the factors of the experiment were so arranged that the original temperature of the water was as much below the tem- perature of the room as the final temperature was above. Obser- vations of the temperature of the water taken every half min- ute after the introduction of the piece of lead are shown plotted in the curve H F A Fig. 91. Now since the rate at which heat is absorbed from the room by the water at any instant varies directly with the difference between the temperature of the water and of the room at that particular instant, it follows that the total heat absorbed by the water from the roof is proportional to the area E F H. Similarly the heat lost by radiation while the water was rising to its highest temperature is proportional to the area FAG. It follows that if the area E F H equals the area FAG then the heat gained by absorbtion from the room equals the heat lost to the room by radiation. This condition cannot be realized in an experiment of this sort because during the first part of the experiment when the hot body and the water are at a considerable difference of 0.5 1.0 Fig. 91. 2.0 204 CaiU+e)it, — t,) +R' (3) where e represents the water equivalent of the calorimeter and R' represents the radiation correction. From the discussion given on pp. 197-9, it is obvious that R' can be made very small by se- lecting proper values for f^ and t^. An inspection of eq. (2), shows that since c is very small compared with ;«!, even though R' is not entirely eliminated, this fact will cause an error in .j which may be entirely neglected. If t^ be about 10° below the temperature of the room and ^5 be about 20° above the temperature of the room, then R' will be so small that without sensible error, the value of the water equivalent of the calorimeter is given by the equation, ' = (/. - A) - "* ^'^ In this determination the greatest care must be taken in observing temperatures, — especially tg. Determine the mass m of the specimen whose specific heat is 2i6 Calorimetry' to be determined and place it in the heater until its temperature assumes a constant value t. While the specimen is heating, fill the calorimeter about two thirds full of water and determine its mass Wi. Assemble the parts of the calorimeter, placing one ther- mometer in the water contained in the inner vessel and another thermometer against the inner surface of the outer vessel. This second thermometer is to give the temperature of the surround- ings. The temperature of the water should now be observed reg- ularly at quarter minute intervals, and the temperature' of the surroundings should be observed every two minutes. The read- ings are taken at regular intervals but belong to two^ successive periods. first period. Observe the temperature t of the specimen in the heater and the temperature ^^ of the water in the calorimeter, then quickly transfer the specimen to the calorimeter. While stirring the water continuously, take temperature readings every quarter minute. While the heated specimen is giving up its heat, the water will rise rapidly to a maximum temperature ^2- This period may last for one or two minutes. This maximum tempera- ture is attained when the rate at which heat is radiated by the water to the air equals the fate at which the water receives heat from the specimen. The temperature may remain stationary at this value for an appreciable length of time. Thereafter the loss by radiation will exceed the gain of heat from the specimen. Second period. Continue to take readings of temperature and time for not less than five minutes during the cooling of the water in the calorimeter. During the entire time covered by these periods, the water must be stirred continuously and the readings of temperature and time must be taken without interruption. With these readings, plot two curves — one coordinating tem- perature and time fon the water in the calorimeter, and another coordinating temperature and time for the surroundings. A pair of such curves is shown in Fig. 89. From these curves and the data already obtained the value of the radiation correction R can be determined by Regnault's method, p. 197. The data are now at hand which when substituted in eq. (2) will give the value of the specific heat of the specimen. Spe;ci]?ic Heat 217 Test Questions and Problems. 1. Show how to determine the specific heat of ice by the method of mixtures. Take account of the heat absorbed during the experiment from tlie surroundings by the graphical method. 2. Assuming the specific lieat is known of a solid body which will produce no thermal action when immersed in a certain liquid, show how to determine by the method of mixtures the specific heat of the liquid. IvIV. deter:\iination of the specific heat of a solid by the method of stationary temperature. Object and Theory o^ Experiment. — The object of this experiment is to determine the specific heat of a solid by a modi- fied form of the Method of Mixtures in which the water equiva- lent of the calorimeter is avoided and the radiation correction is eliminated. This is accomplished by maintaining the temperature of the calorimeter throughout the experiment the same as that of the surroundings. Consider a body of mass vi, specific heat j and temperature f to be placed in a calorimeter containing w^ grams of water at the temperature of the surroundings t;^. Let now cold water be added to the calorimeter at a rate such that the temperature of the calorimeter remains constant. If the mass and temperature of this cold water be represented by m^ and t^ respectively, then the heat emitted by the specimen is ins (t — t^^) and the heat gained by the cold water added to the calorimeter equals in„(t^ — t^). Since the water originally in the calorimeter has not changed in tem- perature, the heat lost by the specimen must equal the heat gained by the cold water. That is in s (t ■ — #1) =^ m^ (^1 — t^) or ■* — «(/ — A) *■ •' Manipui,ation and Computation. — The apparatus used in this experiment includes a calorimeter of special design, Cj Fig. 2r8 Calorimetry about vertical axes. The water dropper consists of a reservoir, 99, and a heater H and a water dropper D capable of rotation. Fig. 99. R Fig. 100, having a valve V by means of which the flow of water through the orifice O can be regulated. Surrounding this reser- voir is an ice jacket /. By means of the thermometer T, the tem- perature of the water at the moment it issues from the water dropper can be observed. The calori- meter, Fig. ID I, is essentially the me- tallic bulb of an air thermometer into which projects a copper tube X for the reception of the water and specimen. Any change in the temperature of the calorimeter is indicated by an open man- ometer tube M. To prevent any effect due to changes in the temperature of the surrounding air, the calorimeter is placed in a water bath Y at the tempera- ture of the room. After the apparatus has been as- sembled ready for use, the specimen is weighed and placed in the heater: The Fig. 100. mixing tube of the calorimeter is un- screwed and weighed, first when empty, and then when I T 1 X n J 1 1^^ 1 f V Spe;cific Heat 219 filled with enough water at the temperature of the room to cover the specimen. The mixing tube is now replaced and the stop cock attached to the manometer is opened for an instant. By this means any difference of pressure between the inside of the air thermometer bulb and the outside air is equalized. \Mien the specimen in the heater has attained a satisfactory tem- perature, the water dropper is made ready for use by allowing cold water to escape until the thermometer in qE F M the escaping steam indicates a stationary tem- perature. Observe the temperature of the speci- men in the heater, the water in the mixing tube of the calorimeter, and the cold water in the water dopper. Now rotate the heater into po- sition over the calorimeter and quickly lower the specimen into the mixing tube. The heater Fig. ioi. is immediately rotated out of position and the water dropper rotated into place. By operating the valve V, cold -water is now allowed to fall into the mixing tube at such a rate that the index in the manometer tube of the air thermometer will remain stationary. The proper rate can be ascertained only by previous trials. It depends largely upon the conductiidtyof the specimen. The mixing tube with its contents is then weigKed. All of the data necessary for the computation of the specific heat of the specimen by means of eq. (i) are now at hand. IvV. DETERMINATION OF SPECIFIC HEAT WITH JOEY'S STEAM CALORIMETER. Object and Theory gi? Experiment. — A cold body placed in an atmosphere of steam will absorb heat until its temperature equals that of the steam. A certain amount of steam will be thereby condensed. If the steam is at the boiling point of water, the amount of heat lost equals the product of the mass condensed and the heat equivalent of condensation of steam. By "heat eauivalent of condensation" of steam is meant the number of heat 220 CaIvORIMETRY units given up by tlie condensation of unit mass of steam. This is numerically equal to the "heat equivalent of vaporization" of water, i. e. the number of heat units required to vapKjrize unit mass of water. The object of this experiment is to determine the specific heat of a solid from a measurement of the mass of steam condensed on the body as it rises in temperature to the boiling point of water. Joly's apparatus consists of a steam chamber, Fig. 102, enclosing a scale pan of a delicate balance. The scale pan is sus- pended from the balance beam by a thin wire passing through a small hole in the top of the steam chamber. Steam is first passed into the steam chamber and. the mass of steam which condenses on the scale pan is weighed. The apparatus is now allowed to Fig. 102. cool to the temperature of the. room. The scale pan is dried and upon it is placed the specimen whose specific heat is required. Steam is again passed into the steam chamber and the mass of steam which condenses on the specimen and on the scale pan is weighed. Let M and j be the mass and specific heat respectively of the specimen; e, the thermal water equivalent of the scefle pan and Specific Hicat 221 suspending wire ; ^i and t., the temperatures of the room and the steam respectively; ;;/' and ;/)", the masses of steam condensed by the specimen and b>- the balance pan respectively; h, the heat equivalent of condensation of steam. The amount of heat absorbed by the scale pan and suspending wire as they rise in temperature from f^ to t., is c{t.^ — ^j). This heat is supplied by the heat liberated through the condensation of the mass m" of steam. Therefore P^^- ^°3- c{t—Q^m"h. (I) Similarl)-, the amount of heat absorbed by the specimen together with the balance pan and suspending wire is Ms {t, — t^) + e {h — Q. This heat is due to the condensation of the mass of steam {in' -\- in"). Consequently M s (L — t^)+e (t^ — f J = (;«' + m") h (2) Subtracting (i) from (2), Ms (^2 — ^1) = m' h Whence W h Manipulation and Computation. — A common source of error in this method is an uncertainty in weighing produced by steam condensing on the suspending wire where it emerges from ■ the steam chamber. In the apparatus illustrated in the figure, this trouble is avoided by having the suspending wire pass through a small tube surrounded by a steam jacket, Fig. 103. By passing the steam through this jacket before it enters the steam chamber, the neighborhood of the aperture is sufficiently heated to prevent all condensation on the suspending wire outside of the chamber. In performing this experiment first weigh the specimen and then assemble the apparatus. Take care to have the suspending wire hang free. When water in a detached boiler is boiling vig- orously, note the temperature t^ of the inside of the steam cham- ber and then connect the boiler to the steam chamber with a good 222 CAI.ORIMETRY sized rubber tube. Steam will immediately be condensed on the object pan. After one or two minutes diminish the flow of steam to such an extent that the current will not disturb the object pan. Now bring the balance into equilibrium by counterbalancing the object pan. Without disturbing the counterbalancing mass on the mass pan of the balance, disconnect the boiler from the steam chamber and allow the chamber to cool to the temperature of the room, t,^. Dry the object pan and place upon it the specimen whose specific heat is required. Again connect the boiler to the steam cham- ber. After four or five minutes the specimen and the object pan will have acquired the temperature Jj of the steam. Diminish the flow of steam as before and add standard masses to the mass pan until the balance is again in equilibrium. The mass thus added equals the mass in' of steam condensed by the specimen in rising in temperature from t-^ to t^. Knowing h, the heat equivalent of vaporization of water, all of the data are now at hand for com- puting the specific heat of the specimen by means of eq. (3). Test Questions and Problems. I. When this method is employed in precise work a cor- rection must be applied to account for the difference in weight of the specimen when in air and when in steam. Assuming that under normal barometric pressure the specific gravity of air at 20° C. is 0.0012 and the specific gravity of steam at 100° C. is o . 0006, find the factor by which m' must be multiplied to give the actual mass of steam condensed by the specimen. Oil Testing 223 CHAin^ERXVir. CHANGE OF STATE. LVI. THE FLASH TEST, FIRE TEST AND COLD TEST OF AN OIL. Object and Theory of Experiment. — If an inflammable gas is mixed with air in proper proportion, the mixture will ex- plode on ignition. The air above a volatile oil is saturated with the oil vapor. If the temperature of the oil is slowly raised the proportion of oil vapor in the air will increase until, at a certain temperature, the saturated air will become an explosive mixture. This temperature is called the Flash Point of the oil. If the temperature of the oil be still farther increased, a point will be reached at which the oil will evolve vapor so rapidly that, when ignited, it will burn continuously. This is called the Fire Test of the oil. The Cold Test of an oil is the lowest temperature at which the oil will flow. The object of this experiment is to make a flash test, fire test and cold test of a sample of oil. The general method of determining the flash point is to grad- ually heat the specimen in a cup and at frequent intervals pass a small flame near the surface of the oil. In making a fire test, the specimen is heated in an op^^n cup and the temperature is noted at which the vapor will burn continuously when ignited. The flash point depends upon (a) the rate of heating, (b) the depth and diameter of the cup, (c) whether the cup is closed or open, (d) the quantity of oil used, (c) the size of the testing flame and its distance from the surface of the oil. Consequently the size and design of the testing apparatus and the method of carrying out a determination are explicitly described in the legislative en-, actments of the various states. 224 Change oi? State Manipulation and Computation. — The form of apparatus most commonly used in this country for the flash point is the "New York State Board of Health Tester." This consists, Fig. 104, of a seamless copper cup C covered by a glass plate per- forated with two holes, — one for the insertion of the thermometer and another for the testing flame. This cup is heated in either a water or air bath B by means of an alcohol lamp or small Bun- sen burner. Place the whole apparatus in a sheet iron pan filled with sand. In using this apparatus for the testing of illuminating oils, the New York State Board of Health publish* the following reg- ulations : "Remove the oil cup and fill the water-bath with cold water up to the mark on the inside. Replace the oil cup and pour in enough oil to fill it to within one-eighth of an inch of the flange joining the cup and the vapor-chamber above. Care must be taken that the oil does not flow over the flange. Remove all air bubbles with a piece of dry paper. Place the glass over on the oil cup, and so adjust the thermometer that its bulb shall be just cov- ered with oil. "If an alcohol lamp be employed for beating the water bath, the wick should be carefully trimmed and adjusted to a small flame. A small Bunsen burner may be used in place of the lamp. The rate of heating should be about two degrees per minute, and in no case exceed three degrees. "As a flash torch, a small gas jet one- quarter of an inch in length, should be em- ployed. When gas is not at hand employ a piece of waxed linen twine. The flame in this case, however, should be small. When the temperature of the oil has reached 85° F., the testing should Fig. 104. *Rep. of N. Y. State Board of Health, 1882. Oil Testing 225 commence. To this end insert the torch into the open- ing in the cover, passing it in at such an angle as to well clear the cover, and to a distance about half-way between the oil and the cover. The motion should be steady and uniform, rapid and without a pause. This should be repeated at every two de- grees' rise of the thermometer until the thermometer has reached 95", when the lamp should be removed and the testings should be made for each degree of temperature until 100° is reached. After this the lamp may be replaced if necessary and the testings con- tinued for each two degrees. "The appearance of a slight bluish flame shows that the flashing point has been reached. "In every case note the temperature of the oil before intro- ducing the torch. The flame of the torch must not come in con- tact with the oil. "The water-bath should be filled with cold water for each separate test, and the oil from a previous test carefully wiped from the oil cup.'' Make five determinations of the flash point and take the mean. , After each determination, remove the cover from the oil cup and blow the burnt gases out of the cup. After the flash point has been determined, remove the cover from the oil cup and continue to heat the oil at the rate of two degrees per minute. About every half minute test the oil with the small flame as above described. The lowest temperature at which the vapor of oil will burn continuously is the Fire Test. Remove the thermometer and smother the flame by placing on top of the oil cup a piece of asbestos board. Such a damper should always be at hand for emergencies. In the case of lubricating oils the method of finding the flash point and the fire test is exactly as above described except that the rate of heating should be about 15° P., per minute and the testing flame should be first applied when the oil is about 200° P. In making the cold test, a glass vial or boiling tube of about 100 c. c. capacity is one-fourth filled with the oil under investi- gation, and then placed in a freezing mixture of ice and salt. When all of the oil has congealed it is removed from the freezing 15 226' Changs of State mixture and thoroughly stirred with a thermometer until it is sufficiently softened to flow from one end of the tube to the other. The temperature at which this occurs is called the cold test of the oil. LVII. DETERMINATION OE THE BOILING POINT OF A SOLUTION. Object and Theory oe Experiment. — The object of this experiment is to measure the boiling point of a dilute solution of given concentration and also to measure the boiling point of a saturated solution of the same substance. The boiling point of a liquid is the temperature at which the liquid is in equilibrium with its vapor. The boiling point of a solution is higher than the boiling point of the pure solvent. If a current of steam be passed into an aqueous solution below its boiling point, steam will be condensed. in the solution until the heat thereby liberated -raises the temperature of the solution to its boiling point. Consequently the steam that passes through the solution will be at the boiling point of the solution and not that of pure water, Obviously if the steam is formed in the solution it- self instead of being conducted into the solution from outside, the same result will occur. In general, then, the steam from a boiling solution comes off at the boiling point of the solution and not that of the pure solvent. Just as soon, however, as the steam escapes into the space above the solution it becomes slightly cooled by expansion, by contact with the walls of the vessel, etc. This cooling will con- tinue until the steam becomes saturated, that is, until its tempera- ure becomes that of the boiling point of pure water. Consequently, in determining the boiling point of the pure solvent the thermometer is suspended in the space above the liquid, while in determining the boiling point of a solution the thermometer bulb must be immersed in the solution. BoiijNO Point 227 J. Y. Buchanan has recently utilized the principle described in the preceding paragraphs for finding the boiling point of a saturated solution. A quantity of the pure solute is placed in the bottom of a tall test tube containing a thermometer. A current of steam is sent through a glass tube extending to the bottom of the test tube until a saturated aqueous solution of the given so- lute is obtained. So long as any of the solute remains undis- solved and the current of steam is uninterrupted, the temperature of this saturated solution will be at its boiling point. Manipulation and Computation. — The apparatus used in determining the boiling point of a dilute solution consists of a small boiling flask provided with a cork fitted with a thermometer and condenser. Without the condenser the solution would grad- ually increase in concentration through the loss of steam. To pre- vent "bumping," a handful of clean dry pebbles or pieces of broken glass is placed in the boiling flask. Fill the flask about one-third full with a normal* solution of the assigned salt, im- merse the bulb of the thermometer in the solution, place the flask in a sand bath heated with a Bunsen burner and observe the maximum temperature attained. This is the boiling point of a normal solution of the given salt. In the same manner find the boiling point of a semi-normal and a deci-normal solution of the same salt. For finding the boiling point of a saturated solution, fasten a large boiling tube in a vertical position in a retort stand, and fill the tube to a depth of about one centinieter with crystals of the assigned salt. Suspend a thermometer in the tube so that the bulb just touches the layer of salt and then push half way through the layer of salt the end of a glass tube in which flows a current of steam. When the bulb of the thermometer is submerged in the solution formed by the salt and condensed steam, observe the temperature. This is the boiling point required. *For defination of a normal solution see p. 236. 228 Change of State I.VIII. DETERMINATION OF THE FREEZING POINT AND MELTING POINT OF A SUBSTANCE. Object and Theory of Experiment. — The object of this experiment is to determine the freezing point and the melting point of a given substance. Suppose that while a liquid is being gradually cooled to a temperature below its freezing point a series of simultaneous ob- servations is taken of the temperature and time. It will be no- ticed that the temperature of the cooling Hquid falls to a certain value and then rises to a temperature at w^hich the liquid begins to freeze. While the freezing process continues the temperature remains constant but when all of the liquid is frozen the temper- ature again falls. These facts are brought out clearly by a curve coordinating the observations of temperature and time. The stationary temperature corresponding to the horizontal portion of the curve is the freezing point of the liquid. With crystalline bodies, e. g. water, there is a well marked horizontal portion of the cooling curve which shows that these bodies have a definite freezing point. But with waxes and other amorphous bodies which on solidifying pass through an inter- mediate pasty condition, the portion of the curve in the neighbor- hood of the freezing point is somewhat inclined to the time axis. This indicates a somewhat indefinite freezing point. It is customary to call the mean ordinate of this line the freezing point. If instead of cooling a liquid until it freezes we heat a solid until it melts, a curve is obtained which is the obverse of the previously described cooling curve. For crystalline bodies the freezing point and melting point are equal. But for amorphous bodies which in the neighborhood of the freezing point pass through an intermediate pasty condition, the temperature at which fusion begins and the temperature at which solidification begins are not identical. Freezing Point 229 Manipulation and Computation. — The apparatus used in this experiment consists of a narrow test tube projecting nearly to the bottom of a ^\■ider tube. The substance being studied is placed in tlie space between the two tubes, and a thermometer togetlier with sufficient mercury to cover its bulb is placed in the inner tube. In determining the freezing point of a substance, this arrangement is placed in a jar containing a freezing mixture. In determining the melting point it is placed in a suitable hot bath. If the substance under investigation is liquid at ordinary room temperatures, place the apparatus containing the liquid in an appropriate freezing mixture. Note the temperature of the speci- men at 15 second intervals for a period extending from the time the apparatus was introduced into the freezing mixture to at least 5 minutes after the specimen solidifies. With time as abscissas and temperatures as ordinates plot the cooling curve of the substance. The mean ordinate of the nearly horizontal portion of this curve is taken as the freezing point of the substance. Then place the apparatus containing the solid substance in a bath of boiling water or other suitable liquid. Begin immediately to take observations of temperature at 15 second intervals and continue the observations for at least 5 minutes after complete liquefaction has occurred. From these observations plot a heat- ing curve coordinating temperatures and time. The mean ordinate of the nearly horizontal portion of this curve is taken as the melt- ing point of the substance. 230 Change of State LIX. DETERMINATION OF THE HEAT EQUIVALENT OE FUSION OF ICE. Object and Theory of Experiment. The Heat Equivalent of Fusion* of a substance is the number of heat units required to just melt unit mass of it. Imagine that on mixing m grams of ice at 0° C, with Wj grams of water at t-^" that the ice melts and the temperature of the mixture becomes t^° . During this operation, the ice has ab- sorbed the heat required to melt it and also to raise its tempera- ture after melting from o" to t.^° , while the calorimeter and its con- tents have lost heat. If during the experiment, there were no gain from nor loss of heat to the surroundings, then the heat gained by the ice would equal the heat lost by the calorimeter and contained water. That is, if we represent by e the water equivalent of the calorimeter, and by / the number of units of heat required to melt unit mass of ice, then in equating the heat gained by the ice to the heat lost by the water and the calorimeer, we obain mf + mt2= (m^ + e) {t^ — Q . That is, the heat of fusion would be ^^ K+^HA-0 _,^. (1) In most cases, however, the error due to radiation is too great to be neglected. This error may either be computed by Regnault's method or determined graphically by the modification of Rowland's method given on pp. 199-202. If the latter method *Prom the fact that the heat absorbed by a body during fusion does not change the temperature of the body, at the time when heat was considered to be a form of matter, it wg.s supposed that the heat absorbed during fusion exists in the melted body in a hidden or latent form. This heat ab- sorbed during fusion was then called the "latent heat" of fusion. But since it has been proved that heat is a form of energy, it follows that the heat ab- sorbed by the body during fusion does not exist in the melted body as heat but as mechanical energy. Consequently, the expression "latent heat" of fusion is now obsolete and has given place to the term "heat equivalent of fusion." Heat op Fusion 231 be selected, it is necessary to determine the temperature that the mixture would have attained if there had been no radiation or ab- sorption. Denoting this -corrected value by t^ we obtain the cor- rected equation J ^ {m, + e) {t, - /,') _ ^ , 1-2) Manipulation and Computation. Weigh the inner vessel of the calorimeter and the stirrer. The product of this mass and the specific heat of the material of which the vessel and stir- rer is composed gives the water equivalent e. Fill this vessel about half full of water at about 60° C, weigh, and then assemble the calorimeter. One observer will now note the temperatures indicated by the two thermometers of the calorimeter at half minute intervals. A second observer will carefully dry a piece of ice having a mass about one-fourth that of the water in the calorimeter, and at a given instant drop it into the warm water contained in the calorimeter. The first observer will continue to take readings of temperatures at half minute inter- vals for about five minutes after all of the ice has melted. During all of this time the water in the calorimeter must be stirred and the ice kept submerged. Under no circumstances must the temperature of the calorimeter fall so low that dew will be pre- cipitated on its surface. Now weigh the inner vessel of the calorimeter with its contents. The data for determining in and m^ are now at hand. The corrected temperature of the mixture can be determined graphically as follows. On a single pair of coordinate axes plot two curves — one coordinating temperature and time for the water in the calorimeter, and another for the air between the two ves- sels. This latter curve will usually be a straight line. Such a pair of curves is shown in Fig. 105. Through the point of inter- section of the two curves draw a line parallel to the temperature axis. Produce the cooling curve A B until it intersects the line F 5* at some point X. Through the lowest point D of the curve A B C D B draw a line parallel to the portion D B and produce it until it intersects the line P 5* at some point R. Then in the 232 Change of State manner given on pp. 199-202 it may be shown that t^ equals the temperature corresponding to the point R plus the number of de- grees corresponding to the distance (w x). That is, {t^ — ^2') is represented by the distance (x R). 45 40 35 30 25 20 15 10 :::::!:::!:=E:::-::::::5::|::: :::::::::::::::: ::::::::::::::::- ^=-3»,.JR^ 1 1 .. . .. ,.. :::::::::::::::::::::::::::;::: 15 ;:::-:::::::::::: ::;-.|:i = 2 4 6 S 10 12 Curve showing rate of change of temperature of calorimeter. Fig. 105. Test Questions and Problems. 1. Find the relation between the heat equivalent of fusion of a substance when the calorie is taken as the unit of heat, and when the British thermal unit is taken. 2. Deduce an expression for finding the heat equivalent of fusion of a body whose melting point is above the room tem- perature. Heat op Vaporization 233 Obviously the simple theory given in this experiment ap- plies only to a solid whose temperature is at its fusion point at the moment it is introduced into the calorimeter. In the general case not only will the temperature of the specimen be below its fusion point at the moment of its introduction to the hot water of the calorimeter, but in addition, its specific heat will be different in the solid and the liquid state. Even though neither of these spe- cific heats is known, by means of three experiments, similar to the above, in which the masses of the specimen and the water, as well as the original temperature of the water are different, the heat equivalent of fusion of a substance can be determined. We thus have three simultaneous equations containing but three un- known quantities — the required heat equivalent of fusion and the specific heats of the specimen in the solid and in the liquid states. By eliminating the specific heats, the heat equivalent of fusion can be determined. 3. For the same reason that heat is absorbed when a body is melted, heat is also required to dissolve a body in any solvent. Sometimes, however, this absorption of heat on solution is masked or even exceeded by an evolution of heat produced by a chemical combinatiori of the solute and solvent. Assuming that there is no chemical action between Potassium Nitrate and water, determine the heat equivalent of solution of this salt in water. LX. DETERMINATION OF THE HEAT EQUIVALENT OF VAPORIZATION OF WATER. Object and Theory of Experiment. If heat be applied to a liquid the liquid will rise in temperature until its maximum vapor pressure equals the external pressure on its surface; the liquid will then begin to boil, and farther addition of heat will not increase its temperature. The heat now absorbed by the body is doing internal work in converting the liquid into vapor. The 234 Change o]? State number of heat units required to vaporize unit mass of liquid is called the Heat Equivalent of Vaporization of the liquid. In order to condense a mass of vapor without changing its temperature, the same amount of heat must be abstracted that was absorbed in the vaporization. In other words, the heat of vaporization equals the heat of condensation. When the caloric hypothesis was in vogue, this quantity was called the "latent heat" of vaporization. The object of this experiment is to determine the heat equivalent of vaporization of water. Imagine that m grams of steam be condensed in Wj grams of water contained in a calorimeter of water equivalent e. L,et t de- note the temperature of the steam; t^, the temperature of the calorimeter and contents at the moment the steam began to enter ; t^, the temperature of the mixture; and v, the heat equivalent of vaporization of water. Then we will have the equation mv -\- m {t — ^2) = (wii 4" ^) (^2 — ^1) -\- R where R is the radiation correction. Whence, {m, + e){t^-tC)+-R (A—/,). Manipulation and Computation. The apparatus used in this experiment comprises a boiler in which the liquid is vaporized and a calorimeter contain- ing a copper worm in which the vapor is con- densed. The liquid in the boiler A, Fig. 106, is heated by means of an electric current pass- ing through a coil of platinum wire. The arm holding the boiler is attached to a vertical brass rod supported by the tubular column B. Be- low the clamp D there is a horizontal slit ex- tending through an arc of about 90° and from one end of this slit there is a vertical slit ex- tending about half way down the tubular col- umn. A pin in the vertical rod supporting the boiler extends through this slit. By means of this arrangement, the boiler can be quickly rotated into a definite plane and dropped in a vertical line so as to cause the outlet of the boiler to register with one end of the copper worm W contained in the calor- meter C. Fig. 106. Heat of Vaporization 235 Weigh the condensing worm, inner vessel of the calorimeter and stirrer and determine their total water equivalent, c, as in the previous experiments. Pour water into the inner vessel of the calorimeter until all of the convolutions of the condensing worm are covered. The temperature of this water should be below that of the room but must not be so low as to cause dew to be deposited on the calorimeter. Determine the mass m-^ of this water. Assemble the apparatus and adjust the position of the calorimeter until when the boiler is rotated into position and al- lowed to drop,, the outlet will accurately register with the opening W in the short section of rubber tubing on the end of the condensing worm. Raise the boiler, thus disconnecting it from the calorimeter, rotate it to one side and fill half full with water. Connect a storage battery to the terminals of the platinum spiral and regulate the current so that the water will boil freely but not violently. Now commence taking quarter minute observations of the temperature of the water in the calorimeter. About once per minute observe the temperature of the air betwen the two calori- meter vessels. At a given instant rotate the boiler into position, drop it into place and allow steam to flow into the condensing worm until the temperature of the water in the calorimeter rises to about 50° C. The water in the calorimeter must be stirred con- tinuously. Now disconnect the boiler from the calorimeter, ro- tate it to one side and continue taking temperature readings at one minute intervals for about ten minutes. Observe the temper- ature t of the steam in the boiler. Remove the condensing worm (from the calorimeter, carefully dry the outside and weigh. The difference between this mass and the mass of the worm, already determined, is the mass m of the condensed steam. Compute the value of the radiation correction, R, by Reg- nault's method in the manner given on p. 197. All of the data are now determined for substitution in the equation for the heat equivalent of vaporization. Test Questions and Problems. I. Show the relation between the value of the heat equivalent of vaporization when determined in calories and when determined in British thermal units. 236 ThURMOCHEMICAI, MBASUREM5NTS Chapter XVIII. THERMOCHEMICAL MEASUREMENTS. Ill physico-chemical operations where it is necessary to con- sider comparable quantities of different substances the unit of mass usually employed is the gram molecule. , The gram mole- cule is defined as the number of grams numerically equal to the molecular weight of the substance. For example a gram mole- cule of oxygen is 2 x 16 = 32 grams, and a gram molecule of CuSO^ is (63.3 + 32. -f 4 X 16) = 159.3 grams. A normal solution is a solution containing one gram mole- cule of substance dissolved in one liter of solvent. For example a deci-normal solution of sulphuric acid (written 0.1 n H2SO4) contains one-tenth of a gram molecule of sulphuric acid per liter, i. e. one liter of solution contains 9.8 grams of sulphuric acid. If Wj grams of a solute having the molecular weight w-^ is dissolved in m^ grams of a solvent having the molecular weight ■W2, the resulting solution will contain (Wj -f- w^) gram molecules of solute to (?;j2 -^ w^) gram molecules of solvent. Denoting by n the number of gram molecules of solvent in which is dissolved one gram molecule of the solute n = —2 — L (1) The masses of two elements which are capable of exactly replacing one another in chemical compounds are said to be chemically equivalent. Thus, since 31.7 gms. of copper will re- place I gm. of hydrogen from sulphuric acid to produce copper sulphate, 31.7 gms. of copper is chemically equivalent to i gm. of hydrogen. A gram equivalent of an element is the number of grams of an element chemically equivalent to one gram of hydrogen. It is numerically equal to the atomic weight of the element divided by the valence. A gram equivalent of a chemical compound is the TuERMOCITEMICAI, JVoTATlON 237 number of grams of the compound which contains a replaceable amount of an element equivalent to one gram of hydrogen. The value of the gram equivalent of a compound is numerically equal to the value of the gram molecule divided by the product of the number of atoms and the valence of the element in the molecule that is replaced during the reaction. Since some compounds may in different reactions have different elements replaced, it follows that the gram equivalent of a compound may have different values depending upon which element is replaced during the reaction. The value of the gram equivalent cannot be determined until the reaction in which the substance takes part is known. The notation ordinarily employed to express chemical reac- tions is incomplete in that it does not take account of the thermal changes occuring with the matter changes. For example, if carbon burns in a plentiful supply of oxygen, each atom of car- bon will unite with two atoms of oxygen to form one molecule of carbon dioxide. This fact is briefly expressed in the ordinary chemical formula C + O, = CO,. But on the union of two substances a part of the potential energy due to their separation is transformed into heat energy. The amount of the heat change depends upon the physical state of the substances entering into the reaction. Thus, if 12 gms. of car- bon in the crystalline form combine with 32 gms. of oxygen to form one gram molecule of carbon dioxide, there will be an evolu- tion of 94300 calories of heat. If, however, the carbon is in the amorphous form the heat of reaction is 96400 calories. According to the very convenient system of notation devised by Julius Thom- sen, these reactions are stated as follows : (C.,,.,. O,) = CO, -f 94300, (Ca,nom' O,) = CO, + 96400, If, however, the carbon dioxide is formed by the oxidation of carbon monoxide, we will have (CO, O) = CO, + 67960. Similarly, the fact that the solution of one gram molecule of carbon dioxide in a large amount of water developes 5880 calories of heat is expressed by the formula 238 Thermochemical Measurements (CO,, Aq.) = CO, Aq. + 5880. Again, the heat set free in the neutralization of one gram mo'ecule of carbon dioxide gas with the equivalent amount of so- dium hydroxide in dilute aqueous solution is (CO,, 2 NaOH Aq.) = Na, CO3 Aq. + H,0 + 26060 while the heat of neutralization of one gram molecule of carbon dioxide in dilute aqueous solution, with the equivalent amount of sodium hydroxide also in dilute equeous solution is (CO, Aq., 2 Na OH Aq.) = Na, CO3 Aq. + H,0 + 20180 LXI. DETERMINATION OE THE MOLECULAR HEA.T OE SOLUTION OE A SALT. Object and Theory op Experiment. — The heat of solution of a substance is the number of heat units developed, (+ or — ), by the solution of unit mass of it in a given mass of solvent. The molecular heat of solution is the number of calories of heat de- veloped by the solution of one gram molecule of the substance in a given mass of solvent. The molecular heat of solution varies with both the temperature of the components and with the con- centration of the resulting solution. Consequently both must be specified. The object of this experiment is to determine the mole- cular heat of solution of an assigned salt for various concentra- tions of solution. Let m^ grams of the given substance whose molecular weight is w^ be dissolved in m^ grams of a solvent having the molecular weight tw,. Let the temperature of both solute and solvent before mixing be ^,° C, and let the temperature of the re- sulting solution be t^° C. Then if the specific heat of the solution is s, the quantity of heat developed by the solution of the solute is [(wi + OT,) s -\- e] (t^ — ts) -\- R calories where e is the thermal water equivalent of the calorimeter, thermo- meter and stirrer and R is the radiation correction. Consequently the molecular heat of solution of the given substance for the par- Heat of Solution 239 ticular temperature and concentration employed in the experiment, expressed in calories, is ^=^\ \_{'>i,+m,)s + e]{i,- (,) + /? I (1) Manipulation and Computation. — The apparatus used in this experiment consists of a large battery jar. Fig. 107, pro- vided with a cover through which project two thin walled test tubes and a larger thin walled copper tube closed at its lower end. A thin platinum tube provided with a thermometer is placed in the copper air jacket. This constitutes a calorimeter for mixing the two substances contained in the two test tubes. The jar is filled wih water which can be agitated by means of a stirrer S. Fill the jar with water at the temperature of the room. Weigh out into the two test tubes such a mass lUj^ of the salt under inves- tigation and such a mass Wj of water that there are exactly 50 gram molecules of solvent to one of salt. [See eq. (i) p. 236] Place the two test tubes in the w.ater bath and when their temper- atures becomes that of the bath, note the tem.- perature and empty into the platinum tube. While continuously stirring the solution in the platinum tube, observe its temperature at 15 second intervals for at least 5 minutes. If the maximum temperature change of the solution occurs within 30 seconds after the two sub- stances are mixed it will be unnecessary to de- termine the radiation correction R; otherwise compute K by Regnault's method. Take ■p jQy the value of the specific heat ■ of the so- lution from Table 24. If it is not pos- sible to compute the value of e determine it experimentally by the method of mixtures. All of the data are now at hand for computing the molecular heat of solution of the salt at the given temperature for the given concentration. In the same manner determine the molecular heat of solution of the assigned salt for n = 100 and n = 200. If the salt is suf- ficiently soluble in water determine its molecular heat of solution for « = ID and n = 20. 240 Thermochemical Measurements LXII. DETERMINATION OF THE HEAT OF DILUTION OF A SOLUTION. Object and Theory of Experiment. — The heat of dilution of a solution is the amount of heat developed. (+ or — ) when a definite mass of solution of given concentration is farther diluted by the addition of a definite mass of the solvent. Usually the heat of solution is expressed in calories of heat developed when a so- lution containing one gram molecule of solute to n gram molecules of solvent is farther diluted by the addition of n' gram molecules of the solvent. Let Wj and ■W2 be the molecular weights of the solute and sol- vent respectively. Then a solution containing one gram mole- cule of solute and n gram molecules of solvent will have a mass of (Wi + n w^) grams. Denote this mass by M'. Let this solu- tion be farther diluted by the addition of n' gram molecules of the solvent. This added mass equals n' uf grams. Denote this mass by M". Suppose that the temperature of both the solution and the added solvent before they were mixed was t^, and that after they were mixed the temperature of the mixture was t.^. Then if the specific heat of the final mixture is j and the thermal water equiva- lent of the calorimeter, thermometer and stirrer is e, the heat of dilution of the solution expressed in calories is V = [{W + M") s + e] {t-t,) (I) Since the change of temperature is generally very small and oc- curs very quickly the correction for radiation is usually negligible. . It is unnecessary to employ such large quantities of material as above considered. Suppose one ^-th of each of the above quantities is taken. Denote {M' -^ x) by m' and (M" -^ x) by m". Then the value of the heat of dilution of the solution ex- pressed in calories is V = x[{m' + m") s + e] {t^ — t,). (2) Hp;at 01? Neutralization 241 Manipulation and Computation. — The apparatus and ex- perimental method employed in this determination are the same as described in the experiment on the determination of tlie mole- cular heat of solution of a salt. Test Questions and Problems. I. The necessity of knowing the specific heat of the final mixture can be obviated by arranging the temperature of the original solution and of the pure solvent so that the temperature of the final mixture is the same as the temperature of the original solution. Deduce the formula that would be employed if the de- termination were made in this manner. LXIII. deter:\iination of the heat of neutraliza- tion OF AN ACID AND A BASE. Object and Theory of Experiment. — The heat of neutral- ization of a given acid and base is the amount of heat evolved in the formation of one gram molecule of a normal salt. A nor- mal salt is one containing no replaceable hydrogen. Let ?:i'i, zv.^ and w^ be the molecular weight of the given acid, the given base, and of water respectively. Suppose that in order to produce one gram molecule of normal salt, the reaction requires «i molecules of the acid to u., molecules of the base. Let each gram molecule of the acid and of the base be dissolved in n^ and »„ gram molecules of water respectively. Then the solutions .con- taining equivalent amounts of the acid and base will have the masses {u^ w^ -\- n^ Wg) grams and (Mj ^2 + '"^2 ^3) grams re- spectively. Denote these two masses by the symbols M.^ and M^ respectively. Let the temperature of both solutions before mix- ing be ^2 °^- a.nd the temperature of the system after mixing be ^1 °C. Then if the specific heat of the final mixture is Sj the ther- mal water equivalent of the calorimeter and accessories is e and the heat lost by the apparatus by radiation is R, then the heat of neutralization of the given acid and base, expressed in calories, is N=[(M, + M,)s + e]it,-t,)+R. (i) 16 242 Thermochemical Measurements It is unnecessary to employ such large quantities of material as above considered. Suppose one x- th of each of the above quantities be taken. Denote (M^ -^ ;tr) by Wi and {M^ -^ x) by m.. Then the value of the heat of neutralization of the given acid' and base, expressed in calories, is N = x[(m^ + m,) s + e] (t, — t,) + R. (2) Manipulation and Computation. — The apparatus and ex- perimental method employed in this experiment are the same as described in the experiment on the determination of the molecular heat of solution of a salt. In order to avoid precipitation, the solutions of the acid and base are made very dilute. Make the solutions of a concentration of one gram equivalent of the acid and base to 200 gram mole- cules of water. The rise of temperature on mixing such dilute solutions as these will be so small that the radiation correction can be neglected. The specific heat of such dilute solutions may be considered to be unity. In case a precipitate is formed in the mixture, its heat of precipitation (numerically equal to the heat of solution of the precipitated salt) must be subjected from the value of N given by eq. (2). I.XIV. DETERMINATION OE THE HEAT VAEUE OE A SOLID WITH THE COMBUSTION BOMB CALORIMETER. Object and Theory oe Experiment. — The object of this experiment is to determine the amount of heat developed by the complete combustion of a unit mass of coal. The heat value of a solid or liquid is expressed either in B. T. U.s per pound or in calories per gram. The method to be employed in this experiment is to burn a known mass of the given substance in a strong steel bomb filled with oxygen under high pressure. During the combustion the Ix)mb remains immersed in a water calorimeter and tlie heat de- veloped is obtained by the ordinary method of mixtures. Thus suppose that by the combustion of in grams of substance, the Heat Vai,ue of Fuels 243 bomb together with the calorimeter its accessories and the con- tained water rise in temperature from t./ to t^° C. Let the mass oi water in the calorimeter be in,^ and the total thermal water equivalent of the apparatus including the calorimeter, bomb, ther- mometer and stirrer be denoted by c. Then if the radiation cor- rection is R, the heat developed by the combustion of one gram of the substance is ff = (,», + e) {t, - t,) +R ^^j^^jg^ pg^ gj.^^^_ The superiority of this method is that since in it complete combustion is attained and all the products of combustion remain in the apparatus, the quantity of heat developed is readily com- puted. Manipui,ation and Computation. — The apparatus used in this apparatus comprises a water calorimeter, a combustion bomb, a press for molding the specimen into a small coherent pellet, a Fig. 108. 244 Thbrmochismicai, Measurements retort for generating oxygen and an electric battery for igniting the specimen. Hempel's combustion bomb consists of a soft steel or cast iron capsule D, Fig. io8, closed by a massive plug C. The inside surface of the bomb is coated with enamel. The plug is pierced by two passages^ — one, JH for filling the bomb with oxygen and another for the introduction of an insulated conductor KF. The gas passage is controlled by the compression valve A. The rod KP- is insulated from the metal plug by the rubber packing M and the asbestos packing N. G is a metal rod screwed into the plug. A little basket B made of incombustible material is sus- pended by means of heavy platinum wires from the ends of the rods G and F. The ends of the rods G and F are connected by a thin platinum wire. In preparing a specimen of coal for a determination, the coal is first pulverized and then molded into a compact coherent pellet by means of a screw press, Fig. 109. The mold of the press con- sists of a block of steel, X Fig. no, bored out to the required t V f : X M X size, with 109. The upper portion of this hole is cylindrical and is fitted a cylindrical plug A. The lower portion of the hole is reamed out to a conical form and is fitted with a conical plug B. On the conical surface of the plug are two narrow channels which extend from one face to the other. A loop of cotton or linen thread, FSB, is made over the coni- cal plug and its ends laid in these slots. About i .25 gms. of pul- verized coal is packed about this loop of thread, the cylinder A is put in place, the plunger P placed on top, and the screw turned down until the specimen has been compressed into a compact Heat Value oif Fuels 245 pellet. The screw is now raised, the mold slipped into the upper horizontal guides and the screw again depressed until the pellet is forced out through the bottom of the mold. With a sharp knife pare down the pellet until it weighs about one gram. Cut off one end of the thread close to the specimen. Remove any loose par- ticles of coal by means of a small brush, place the pellet on a watch glass and weigh. Do not touch the pellet with the fingers but handle by means of the thread. Unscrew the head C, Fig. 108 of the combustion bomb, mount it in a retort stand, connect the terminals of the battery to the binding posts K and L, and regulate the resistance in circuit so that the current will bring the platinum wire connecting G and F to a red glow. Without disturbing the resistance in circuit open the switch and disconnect the battery terminals from the binding posts K and L. Place the specimen of coal S in the basket B and tie the free end of the thread to the thin wire connecting G and JP. Now, without disturbing the specimen, remove the head of the combustion bomb from its support and screw it tightly into the bomb. The combustion bomb is now ready to be filled with oxygen. Fig. III. Fill the gas generating retort, R Fig. iii, two-thirds full with a mixture consisting of equal parts of potassium chlorate and manganese dioxide. Put a tightly wound toll of copper or brass wire gauze into the tube leading from the retort, and con- nect the retort and a pressure gauge G to the combustion bomb in the manner shown in the figure. The pressure gauge and com- 246 Thejrmochemicai. Measurements. bustion bomb are immersed in a vessel of water for the purpose of detecting any leak in the bomb and also for the purpose of cooling the oxygen coming from the hot retort. Open the gas valve in the combustion bomb and apply a Bunsen flame to the upper part of the gas generating retort until the gauge indicates a pressure of about i kg. per sq. cm. (14 lbs. per sq. in.). If the flame be now removed, the heat already given to the retort will generate enough oxygen to raise the pressure to about 5 kg. per sq. cm. (70. 5 lbs. per sq. in.). Now loosen the flange coupling P so as to allow the mixture of oxygen and air contained in the apparatus to escape. By tightening the coupling P and repeating this operation the entire apparatus can be freed of air. Now tighten the couplings and slowly heat the retort un- til the gas pressure rises to about 12 kg. per sq. cm. (170 lbs. per sq. in.). Close the gas valve on the combustion bomb and im- mediately afterwards disconnect the bomb at the coupling H from the remainder of the apparatus. Cool the bomb to about the tem- perature of the room and carefully dry it with a towel. Place the bomb in a water calorimeter containing m^ gms. of water at about the room temperature. Connect the terminals of the previously arranged electric circuit to the binding posts K and L, Fig. 108. Before closing the switch in. the battery circuit, take temperature readings of the continuously stirred water at quarter minute intervals for at least five minutes. At a given in- stant of time close the battery switch so that the electric current will ignite 'the specimen. The switch should be closed for but a moment else the heating effect of- the current will need to be taken into account. While continuously stirring the water, continue to take quarter minute temperature readings for at least ten minutes after ignition. The temperatures should be read to hundredths of a degree. Take the bomb out of the water, open the valve, unscrew the head, wash out the inside and oil the screw threads. From a curve coordinating temperature and time find by the grapical method described on p. 199 the highest temperature that would have been attained by the calorimeter if there had been no loss of heat by radiation. Let ^1' represent this corrected temperature. Then instead of eq. ( i ) we can write Hkat Vai^ui; 01? Fuels 247 // ^ {mi+e)(/\ —t., ) calories per gram. (2) tn In this equation the thermal water equivalent of the appara- tus 1' is still unknown. This constant can be determined in any one of three different ways, (a) By taking the sum of the prod- ucts of the masses and the assumed specific heats of the various parts of the apparatus. In an apparatus like this composed of so many different materials of uncertain composition, this method is unreliable. (&) Experimentally, by the method of mixtures. The large amount of water required in this experiment and the difficulty of obtaining temperatures accurately make this method unsuitable for inexperienced observers. (c) By means of a supplementary experiment in which a definite amount of heat is developed in the apparatus by the combustion of a known mass of substance having a known heat value. There are a number of substances whose heat values are accurately known which can easily be obtained pure. This last method is the one that will be employed in this experiment. Suppose that when using the same apparatus as before, the burning of m' gms. of a substance of heat value H' raises the temperature of the apparatus and of Wj gms. of water from ti° to ^3° C. Let t^ be the temperature that the calorimeter would have attained if there had been no loss of heat by radiation. Then we will have jf, ^ {fn^+e){t'^ — ti) calories per gram. (3) m' Eliminating e between eqs. (2) and (3) we have H= (JH^ + m,- ^}) (LlZzh.) (4) Napthalin in a suitable substance to use in this supplementary experiment. Make a pellet of somewhat smaller mass than the mass of coal already used and proceed exactly as in the experi- ment with coal. All of the data for determining H by means of eq. (4) are now at hand. Before putting away the apparatus, dig the remaining solid substance out of the gas retort, rinse out the combustion bomb with water and carefully oil the threads of the bomb and all parts 248 Thermochbjmicai, Measurements of the press. Be certain that no water or oil is left inside of either the retort or combustion bomb. If oil or any other organic sub- stance is heated in the retort with the oxygen producing mixture an explosion is liable to occur. LXV. DETERMINATION OF THE HEATING VALUE OF A GAS WITH JUNKER'S CALORIMETER. Object and Theory of Experiment. — The object of this experiment is to determine the number of heat units developed by the combustion of unit volume of a given sample of gas. In Junker's method the heat developed by a constantly burning flame is determined by measuring the heat absorbed by a steady stream of water enclosing the flame. Fig. 112. Heat Value ot? Fuels 9. 249 Fic. 113. The apparatus consists of a gas pressure regulator, R Fig. 112, an accurate gasmeter M and a calorimeter C of special design. The calorimeter consists of a combustion chamber, A, Fig. 113, enclosed b)' a water jacket B traversed by a large 250 Thermochemical Measurements number of tubes for the passage of the products of combustion. The water jacket is surrounded by a closed space L filled with air. After traversing the pressure regulator and meter the gas is burned in the burner Q. The products of combustion after passing through the tubes traversing the water jacket escape through the vent Y. The temperature of the gas as it enters the burner and the temperature of the products of combustion as they leave the calorimeter are given by the thermometers T" and T'". A stream of water flows from the supply pipe D into a small res- ervoir kept at constant level by means of the overflow pipe 0. From this regulator the water passes down the tube B through the control valve V, thence through the water jacket B^ thence through G and the discharge nozzle H into the measuring vessel U. The temperature of the water as it enters and as it lea,ves the calorimeter is given by the thermometers T' and T. Water vapor formed by the combustion of the gas condenses on the inside of the combustion chamber and escapes through the outlet / into the measuring vessel W. The flow of water and of gas is so adjusted that the temper- ature of the products of combustion escaping at Y is approximate- ly the same as the temperature of the gas entering the burner at P. Let V represent the volume (reduced to standard conditions) of the gas burned during a certain time. Let the mass of water which passes through the calorimeter during this time be denoted by m^, and let its temperature on entering and on leaving be rep- resented by f and t respectively. Let the mass of steam condensed during the combustion be represented by m^, and let the temper- ature at which it condenses and the temperature of the condensed steam as it leaves the calorimeter be denoted by t^ and t^ respec- tively. Then the heat value of the gas, H, is given by the equation ^= ""' ^^^^'^ — \m,h + m^ {t, — t^)\ (1) where h is the heat equivalent of vaporization of water. If m^ and 1112 are measured in grams, v in liters and temperatures in °C., then H is given in gram calories per liter or kilogram calories per cubic meter. Heat Value of FueI/S 251 Manipulation and Computation. — After assembling the apparatus connect D to the water supply so that any leak in the calorimeter will make itself evident. The flow of water into the apparatus must always be sufficiently great to overflow through the pipe 0. With the gas valve at the burner P closed, connect the gas regulator to the gas supply and observe if the index of the meter moves. If it does, seek out the leak and remedy it. With the water still flowing through the apparatus, take the burner out of the calorimeter, light the gas and replace the burner. If the gas is lighted while the burner is inside the combustion chamber there is danger of an explosion. Have the top of the burner from 12 to 15 cms. above the lower opening to the combustion chamber. The clamper Z should be from one-half to completely open de- pending upon the draught required for the flame; Arrange the flow of water by means of the valve V , and the flow of gas by means of the valve P so that the thermometers T" and T'" indicate practically the same temperature. For ordinary illuminating gas the proper rate of flow of water is from i . o to 1 .5 liters per minute. After all of the thermometers indicate nearly stationary tem- peratures, note simultaneously the gas meter reading and the temperatures indicated by the thermometers T and V . Then im- mediately place suitable vessels U and W so as to catch the warmed water escaping from H and the condensed steam es- caping from /. Note the temperatures of the ingoing and the outgoing water every 15 seconds until two or more liters of water have flowed into the vessel U. Then remove the vessels U and W and at the same time take the gas meter reading. Note the tem- perature ^2 of the condensed steam in W. Determine m^ and m^ by weighing. From the difference between the two gas meter readings to- gether with the temperature and pressure of the gas passing to the burner, the value of v is found by means of the fundamental law of gases. The temperature is given by the thermometer T". The pressure is the sum of the barometric reading and the height of mercury corresponding to the difference in the level qf water in the manometer V. 252 Thbrmochemicai, Measurements All of the data are now at hand for substitution in eq. (i). By substituting a properly designed lamp for the gas burner, Junker's calorimeter can be used for finding the heating value of a liquid. ' Test Questions and Problems. I. Show that the multiplying factor for transforming kilo- gram calories per cubic meter to British thermal units per cubic foot is 0.1 1235. LXVI. DETERMINATION OF THE MOLECULAR WEIGHT OF A SUBSTANCE BY THE BOILING POINT METHOD. Object and Theory oE Experiment. — A solution of a non- volatile substance has a higher boiling point than the pure solvent. It has been found that the elevation of the boiling point of a solution is proportional to the ratio of the number of molecules of solute to the number of molecules of solvent in the solution. Thus if the boiling point of a pure solvent be measured, and also the boiling point of a solution of definite concentration, it is evidently possible to determine the molecular weight of the solute. The object of this experiment is to determine the mole- cular weight of a substance from a measurement of the difference between the boiling point of a pure solvent and of a solution of known concentration of the substance. The relation between the molecular weight of a soluble non- volatile substance and the elevation produced in the boiling point of a pure solvent by the addition of the given substance can be obtained from the consideration of a perfectly reversible engine in which the solution is used as the working substance. Imagine an engine consisting of a cylinder fitted with a piston made of a material which is permeable to a certain pure solvent but is impervious to a given solute. Suppose that the space below the piston is filled with a very dilute solution of the substance, and the space above the piston contains some of the pure solvent. When the pressure on the piston equals the os- MoivECULAR Weight 253 motic pressure of the solute, there is equilibrium between the sol- \ent above the piston and the solution below. But if the pressure on the piston is greater than the osmotic pressure, solvent will bt forced out of the solution until the two pressures are equal. If, however, the pressure on the piston is less than the osmotic pressure, solvent will enter the solution making it more dilute. Thus the volume occupied by the solute depends upon the pres- sure to ^\■hich the piston is subjected in a manner analogous to that in which the volume occupied by a gas depends upon the pressure to which it is subjected. ' Let the dilution and volume of the dilute solution in the cylinder be so gerat that on farther dilution the thermal change of the solution and the volume change of the solute are negligible. Let w(i, zv and 7^2 l^e the mass of solute, molecular weight of solute and mass of solvent, respectively, in the solution. Let the number of gram molecules of solvent in which is dissolved one gram molecule of solute, be denoted by M. Let h be the heat equivalent of vaporization of one gram of the solvent. Let T be the boiling point of the pure solvent and (T -[- d T) the boil- ing point of the solution, at the pressure p. Starting with the solution in the cylinder at the pressure p and absolute temperature T, let the engine go through a complete reversible cycle consisting of the following five operations. First. Depress the piston until there osmoses through its semipermeable head that amount of solvent in which is dissolved one gram molecule of solute. The work required to be done on the engine to effect this separation of (w m^ -> «ii) grams of pure solvent from the solution is, from eq. (15) p. 171, equal to R T. There is no thermal change during this operation. Second. Allow the M gram molecules of solvent separated from the solution during the first operation to evaporate at the temperature T and pressure p. This requires the absorption at the temperature T of (tc 111,^ h -f- in,) calories of heat. The work done by the engine during this operation is, from eq. (14) p. 171, equal to MRT. - Third. Keeping the pressure constant, heat separately the solution and vapor to the temperature (T -{- d T). From eq. (13), 254 Th]5rmochemical Measurements p. 171, the work done by the engine during this operation equah MR (dT). The heat absorbed equals e{dT) calories, where e is the thermal water equivalent of the solution and vapor. Fourth. Bring the vapor into contact with the solution and permit it to condense without change of temperature. During this operation (zv m^ h -f- m^) calories of heat will be liberated at the temperature (T -|- d T), and an amount of work will be done on the engine equal io M R (T-\- d T). Fifth. Cool the solution to the temperature T. The heat liberated equals e{dT) approximately. Since there is but an infinitesimal change of volume, the work change can be neglected. The cycle is now complete and the working substance is in its original condition. The net result of the cycle is that (w m^ h -=- ?»i) calories of heat have been carried from a body of temperature T to a body of temperature {T -\- d T) by an ex- penditure of work equal to RT. Since the cycle is reversible, the arnount of work required to transfer {wm^h^^ m^) calories from T to (T ~{- d T) is, from eq. 23, 176, equal to w Mz h d T ~i,r, T' Whence, R T — '^""2 ^ d T ...^ ~ m, ~W ^ ' Since for one gram molecule of substance R ^ i . 98 calories, we have for the moleciilar weight of the substance \ h ) {m^d T) ?«2 {cLT) ^ ' where m^ and m^ are the masses of the solute and solvent, respec- tively, composing the solution, {d T) is the elevation of the boil- ing point of the solvent produced by the addition of the solute, and k is a constant depending upon the heat equivalent of vapor- ization and upon the absolute temperature of the boiling point of the pure solvent at the pressure p. Computed values of k are given in the following table for some of the commonly used solvents when under ordinary atmos- pheric pressure: IMoivECULAR Weight 255 SOL\'ENT. Y^o h k Acetone 329.6 125-3 1717 Benzol 352.6 93-5 2633 Carbon bisulphide 319- 83.8 2404 Chloroform 334- 58-5 3776 Ethyl Alcohol 351-3 214-3 1 140 Ethyl ether 308. 90-5 207s Water 373 -o 536- 514 Manipulation and Computation. — Beckmann's apparatus, to be used in this experiment, consists of a boiling tube A contain- ing the solution together with a Beckmann differential thermo- meter B. The boiling tube is surrounded by a jacketing arrange- ment containing a liquid of a slightly higher boiling point' than the solution being examined. To prevent the evaporation of the solvent in the boiling tube and the liquid in the jacket, both the 114. boihng tube and the jacket are provided vfith condensers D. The entire apparatus is supported on a sand bath or wire gauze. Heat is supplied by one or two small Bunscn burners. In order to prevent the form of explosive boiling called "bumping," a thick platinum wire is sealed into the lower end of the boiling 255 Thermochemical Measurements tube and in addition, the tube is filled to a depth of about two centimeters with clean glass beads. The quantity of mercury in the bulb of the differential ther- mometer must be adjusted so that when the bulb is immersed in the boiling pure solvent, the mercury thread will come to rest near the bottom of the scale. Heat the bulb 4 or 5° above the boiling point of the pure solvent and then give the tube of the thermometer a sudden slight jar. The mercury in the upper part of the reservoir will separate from the mercury thread thus leaving the proper amount of mercury in the bulb. The proper temperature can be easily obtained by immersing the bulb in a boiling solution of the solvent to be vised in the experiment. After the boiling tube is dried and the dried beads replaced it is weighed. Sufficient solvent is now introduced to cover the bulb of the thermometer and the boiling tube is again weighed. The entire apparatus is now assembled, the solvent heated to boiling and the thermometric reading noted. The boiling should be as brisk as is consistent with quiet uniform ebullition. Before taking a reading, the tube should be given a few light sharp taps so as to avoid the adhesion of the mercury to the side of the cap- illary tube. Repeat the determination of the boiling point of the solvent until a series of consecutive consistent results is obtained. After the boiling point of the solvent is determined, a weighed amount of the substance whose molecular weight is re- quired is introduced into, the boiling tube and the boiling point of the resulting solution is determined. After this determination has been verified, additional weighed amounts of substance are in- troduced and the boiling point of the solution determined for each concentration. Each determination should be repeated before the succeeding concentration is taken up. It should be noticed that although the vapor of the solvent comes off at the boiling point of the solution, it immediately cools to its saturation temperature. The vapor then begins to condense and continues to condense until the boiling point of the pure sol- vent is attained. Consequently the thermometer bulb must be immersed in the solution and not in the vapor coming off from the solution. Molecular Weight 257 Since there is a part of the pure solvent in the form of vapor and an additional part condensed on the walls of the apparatus, the amount of solvent actually present in the solution is some- what less than that actually put into the boiling tube. The solution is therefore more concentrated than the weighing would indicate. When water is the solvent employed, and the apparatus is of the form and dimensions used by Beckmann, this correction amounts to about 0.5 gm. The specimens of substance should be in the form of little pellets in order that they may be introduced easily into the boil- ing tube. If a volatile solvent is employed, the specimens should be introduced through the condenser without opening the boiling tube to the air. The greatest practical difficulty in this experiment coftsists in securing the proper rate of boiling. If the boiling is too slow the temperature of equilibrium will not be attained. On the other hand, if the boiling is so vigorous as to cause the vapor to form in large bubbles which explode violently on reaching the surface of the liquid, the thermometer will rise above the true point while the bubble is forming and suddenly fall below it after the bub- ble explodes. When condensed vapor drops into the solution, the thermometer suddenly falls. Test Questions and Problems. 1. Discuss the errors to which the boiling point method is subject. 2. From an inspection of eq. (2), show what properties should be possessed by a solvent in order that the greatest ac- curacy may be obtained. 3. If a given substance were soluble to a nearly equal de- gree in 'water and in ether, show what considerations would af- fect the choice between these two solvents. 17 258 Thurmochumicai, Measurements LXVII. DETERMINATION OE THE MOLECULAR WEIGHT OF A SUBSTANCE BY THE FREEZING POINT METHOD. Object and Theory op Experiment. — Usually the freezing point of a solution is different than that of the pure solvent. When a dilute solution partially congeals, if the solid separating out is the ice of the pure solvent the freezing point of the solu- tion is lower than that of the pure solvent. This is the case to be considered in this discussion. It has been shown by Coppet and Raoult that if n molecules of any substance exist dissolved in m molecules of a solvent, the lowering of the freezing point of the solution is a constant quantity. By using quantities of substances proportional to their molecular weights, if there is no association or dissociation, solutions are produced in which the ratio of the number of molecules of solute I0 number of molecules of sol- vent is known. This suggests the possibility of determining the molecular weight of a soluble substance from the lowering pro- duced in the freezing point of a given solvent by the addition of the soluble substance. Or, if the molecular weight is already known, this principle furnishes a means of determin- ing the amount of association or dissociation of a substance in solution. The object of this experiment is to determine the mole- cular weight of a substance from the lowering which it produces in the freezing point of a given solvent. In the present experi- ment water will be used as the solvent, and the molecular weight of sugar and sodium chloride will be determined. The relation between the molecular weight of a soluble sub- stance and the lowering of the freezing point of a pure solvent produced by the addition of the given substance can be obtained from purely thermodynamical considerations. As in the previous experiment, consider an engine consisting of a cylinder fitted with a semipermeable piston. Let the working substance be a very di- MoivECUI.AR WUIGIIT 259 lute solution of the substance whose molecular weight is required. Let ^Hi and 1112 be the masses of solute and solvent, respec- tively, composing the solution. Let w be the molecular weight of the solute and let f be the heat equivalent of fusion of one gram of the solvent. Also let T and ( T — d T) be the freezing points of the solvent and solution respectively. Beginning with the solution in the cylinder at the absolute temperature ( T — d T) with an indefinitely small quantity of ice, let the engine go through a complete reversible cycle consisting of the following five operations. First. Allow that amount of pure solvent in which is dis- solved one gram molecule of solute to freeze at the temperature (T — d T). Since in this operation (tt/Wj ~^ '''^i). grams of solvent are frozen, the amount of heat liberated at the temperature (T — d T) equals (if m^ f -^ w^) calories. The work done by the working substances equals p(dv), where p is the atmospheric pressure, and dv is the difference between the volume of the sep- arated solvent when in the solid and when in the liquid condition. Second. Heat the solid and solution to T°. The heat ab- sorbed equals e (d T), when e is the thermal water equivalent of the solid and solution. The work done by the working substance is a positive quantity of negligible magnitude. Third. Permit the solid to melt at the temperature T. In this process (w m,_ f -^- Wi) calories of heat are absorbed at the temperature T. The work done on the working substance is p(dv). Fourth. ■ By means of the semipermeable piston, return the pure solvent to the solution at the temperature T. Since the dilution and volume of the solution are so great the change in the concentration produced in this operation can be neglected. Con- sequently there is no thermal change, and the work done by the working substance during the osmosis of the solvent back into the solution is, from eq. (15) p. 171, equal to R T. Fifth. Cool the solution to ( T — d T). The heat liberated equals e(dT), approximately. The work done by the working substance is a negative quantity of negligible, magnitude. The working substance has gone through a complete rever- 26o Thermochemicai. Measurements sible cycle and is now in its initial condition. The net result of the cycle is that by the transfer of (w mj / -f- w^) calories of heat from a body of temperature T to a body of temperature ( T — d T) the working substance has performed an amount of work equal to R T calories. From eq. (23) p. 176, the amount of work de- veloped by a reversible engine by the passage of the quantity of heat {w'm^f-^m-i) from the temperature T to the temperature (T — dT) is w w-if d T nil T Whence RT = ^-± (1) m^ T Since for one gram molecule of vapor R ^ i . 98 calories, we have for the molecular weight of the dissolved substance ^_/L98_7-x^^^_ ^^j«^ (2) V / )m,{dT) " m^{dT) ^ ' where m.^ and Mj are the masses of the solute and solvent, respec- tively, composing the solution, {dT) is the depression of the freezing point of the solvent produced by the addition of the so- lute, and k' is a constant depending upon the heat equivalent of fusion and upon the absolute temperature of the freezing point of the pure solvent. The values of k' computed for some of the commonly used solvents are given in the following table. SOLVENT. 7^0 h k' Acetic acid 290.0 43-2 3855 Benzol 277.9 29.1 5255 Formic acid 281. 5 55-6 2822 Nitrobenzene 278.3 22.3 6877 Water 273.0 79-4 1858 Manipulation and Computation. The apparatus used in this experiment consists of two concentric test tubes, A and B, Fig. 115, immersed in a freezing mixture contained in a jar C. Through a cork in the inner test tube passes a Beckmann differen- tial thermometer T and a stirrer 5". For convenience of intro- ducing the specimen, the inner test tube is usually provided with a side tube. Fill the glass jar C with a freezing mixture composed of MoLECui^AR Weight 261 Df one part of coarse salt to about five parts of snow or chopped ice. Following the method given on p. 140 the quantity of mercury in the bulb of the thermometer must be adjusted so that when the bulb is immersed in freezing water, the top of the thread will come to rest near the upper end of the scale. Invert the thermometer, and by means of a sudden slight jar cause the mercury in the upper reservoir to fall to the end of the reser- voir connected to the capillary. Heat the bulb until the thread in the capillary joins the mer- cury in the reservoir. Immerse the bulb in pure water contained in the freezing tube A. Place the freezing tube in the freezing mixture, and as soon as ice particles begin to form remove the thermometer from the ice cold water. Let it warm slightly in the air and then separate the mercury in the upper reservoir from the thread in the capillary by means of a sudden sHght jar. On again immersing the bulb in the ice cold water, if the top of the thread comes to rest near the upper end of the scale the thermometer is in proper adjustment. If a crystal of the liquid is not present, the liquid must usu- ally be considerably undercooled before crystallization will occur. For the purpose of inducing crystallization, an infinitesimally small crystal of the liquid can be introduced by the following method due to Beckmann. Select a glass tube about 15 cms. long having a bore of about 3 mm. diameter. On one end fix a short section of rubber tubing fitted with a stopper of glass rod. Fill the tube with the solvent being used (water in this experiment), plug the upper end with the glass rod and immerse the entire tube in the freezing mixture. See V, Fig. 115. When the contents of the tube have become frozen, warm the outside of the tube with the hand until the rod of ice can be pushed beyond the end sufficiently to protrude about two millimeters. If the tube be now replaced in the freezing bath, the rod of ice will be frozen fast to the con- taining tube and the "vaccination point" will be ready for use. In using this device, when the temperature of the liquid in the freez- FiG. 115. 262 Thermochemicai, Measurements ing tube A is slightly below its freezing point raise the stirrer 5* out of the undercooled liquid, open the side tube and touch the stirrer with the "vaccination point." By this means crystallization is induced on the wet stirrer which is communicated to the re- mainder C/f the liquid in the freezing tube so soon as the stirrer is set into motion. Now that the thermometer is in adjustment and the "vaccin- ation point" is prepared the freezing tube A is weighed, first when empty and then when containing enough water to cover the bulb of the thermometer. This gives m^. Assemble the appara- tus and allow the water to cool until the mercury In the thermom- eter stands about one degree below the position attained when previously placed in ice water. Now "vaccinate" the contents of the freezing tube A in the manner described in the preceding paragraph. When freezing begins, the temperature will rise un- til the freezing point is reached. During this rise, tap the ther- mometer frequently with a lead pencil so as to prevent the sticking of the mercury in the fine bore. Note the thermometer reading to o.ooi°. Remove the freezing tube with its accessories from the apparatus, warm it with the hand until all of the ice in it is melted, replace tube in apparatus and take another reading of the freezing point. If these readings differ by more than 0.003° repeat until a series of consecutive concordant readings is ob- tained. Drop into the freezing tube a weighed amount of the_ substance whose molecular weight is desired, stir until dissolved and note the freezing point of the solution. Warm the solution and again observe the freezing point. Drop an additional weighed amount of the substance into the solution, note the freezing point; warm the solution and again observe the freezing point. In this manner obtain the freezing point at five different concentrations and compute the molecular weight of the substance at each con- centration. For salt (Na CI., Mol. Wt. 58.5) begin with a cencentration of about 0.1 gms. salt to 10 gms. water, and inciease the concen- tration to 0.2 : 10, 0.3 : 10, 0.4 : 10 and 0.5 : 10. For sugar Dissociation 263 (C12H22O11, Mol. Wt. 342) use concentrations about three times those used for salt. Discuss the results of the experiment and compare them with the values given by theory. Explain any departure from theory that is too great to be due to experimental error. IvXVIII. DETERMINA1I0N OF THE DEGREE OE DISSOCIA- TION OF A SUBSTANCE IN SOLUTION BY THE FREEZING POINT METHOD. Object and Theory op Experiment. — There is a large class of bodies which when dissolved have their molecules split up into smaller particles called ions. Ions must not be confused with atoms. An ion is always electrically charged and may con- sist of one atom or a group of atoms. For example KCl splits up into one K ion and one CI ion, the K ion being positively charged while the CI ion is negatively changed : CuSO^ splits up into one Cu ion (-(- charged) and one ion whose composition is SO4 ( — charged) : K4Fe(CN)„ splits up into four K ions (+ charged) and one ion whose composition is Fe(CN)e ( — charged). Molecules and ions produce the same effect in lowering the freezing point and in raising the boiling point of a solution. Consequently if the molecular weight of the undis- sociated substance is known, it is possible to determine the frac- tion of the molecules which are dissociated in a solution of given concentration. The object of this experiment is to determine, by the freezing point method, the fraction of the whole number of molecules of a dissolved substance which is dissociated in aqueous solution. In eq. I p. 260 it is shown that if in„ grams of pure solvent whose freezing point is T° (absolute) and whose heat equivalent of fusion per gram is / has dissolved in it lU-^ grams of solute of 264 Thjjrmochemicai,' Measurements molecular weight iv, then if there is no dissociation, the freezing point will be lowered by the amount (rfr)=^i_('i:^ill)=^L^.l ;" ' (1) m2 w \ f J OT2 w U ' ■ But if some of the solute is dissociated, the depression of the freezing point will have some greater value (dV). Let the number of ions into which a molecule of the ,giv'm solute divides be denoted by n, and let a represent the fraction of the whole number of molecules of solute which are dissociated. Then for every gram molecule of solute, there are (i ■ — a) undissociated molecules and n a ions. Since an ion produces the same effect in lowering the freezing point of a solution as does an undissociated molecule, each gram molecule of the solute may be considered to consist of (i — a -\- na) effective molecules. Consequently (dT) : (dV) : : 1 : (i—a + na). Whence ,.M .{a.T) {n — l) ^ ' where the value of {d T) is given in eq. (i) and {d-T') is ob- tained by experiment. Manipulation and Computation. — Use the same ap- paratus described in the preceding experiment on the determina- tion of the molecular weight of a substance by the freezing point method. Make up a series of solutions of known concentration of the substance under investigation. For example dissolve 4 gms. of NaCl in 200 gms. of water. This gives 200 c.c. of 2% solution. By means of a measuring cylinder, take 100 c.c. of the 2% solution and add to it 100 c.c. of water, thus giving 200 c.c. of 1% solution. In this manner make up solutions of percentage strength 2.0, i.o, 0.5, 0.25, o.i, etc. Proceeding in precisely the same manner as described in the preceding experiment, find the freezing point of the pure solvent and of each of the solutions. Then by means of eq. (2) com- pute the fraction of the whole number of molecules of Na CI that are dissociated in solutions of the given concentrations. With these results plot a curve coordinating dissociation and concentra- tion for NaCl. Heat Conduction 265 CHAPTER XIX. HEAT CONDUCTION. A surface connecting points which at a given instant have the same temperature is called an isothermal surface. In heat con- duction it is assumed that the direction of heat floiv is always nor- mal to the isothermal surface at that point. Imagine a long homogeneous rod of uniform cross section and with square ends to have one end maintained uniformly throughout its area at the constant temperature t^. Imagine far- ther that there is no loss of heat from the sides of the rod. From this assumption it follows that all of the isothermal surfaces are equal in area to the cross section of the rod and that the rate of heat flow across each section is the same. After the thermal con- dition of the rod has attained equilibrium, let t^ denote the temper- ature of the cold end, / the length of the rod and A its cross sec- tion. Under the given conditions, it has been found by experi- ment that the quantity of heat H crossing each section of a rod of given material varies (i) directly with the difference between the temperatures of the two ends, {t^ — fj) ! (2) directly with the area of cross section, A ; (3) directly with the time, T ; (4) inversely with the length, /. Expressing these experimental results in analytical form, we have the equation, e Obviously the constant of proportionality k equals the amount of heat which will flow in unit time across a plate of given material having unit thickness and unit length when there is no lateral loss of heat and the difference between the temperatures of the 266 Heat Conduction opposite faces is one degree. This constant k is called the abso- lute thermal conductivity of the given material. If, however, there is loss of heat from the sides of the rod, the isothermal surfaces will not be planes normal to the length of the rod nor will they be equal in area to the cross section of the lod. In addition, the amount of heat passing any given section will diminish with the increase of the distance of the section from the hot end of the rod. It follows that eq. (i) is inapplicable when there is loss of heat from the sides of the rod. But by con- sidering an element of the rod so small that the isothermal sur- faces bounding its ends are parallel planes normal to the length of the element, an equation applicable to this element can be derived which is of the same form as eq. (i). Thus, consider an indef- initely small element of the rod having a length dl and area dA. If the opposite faces of this elemeiit have a very small tempera- ture difference dt, then during the indefinitely brief time dT the element will transmit the very small amount of heat ^M- kdtdAdT , dl ^ ' where k is the absolute thermal conductivity of the material com- posing the rod. Relative Conductivity 267 LXIX. DETERMINATION OF THE RELATIVE THERMAL CONDUCTIVITY OF TWO SUBSTANCES. Object and Theory of Experiment. — The determination of the absolute thermal conductivity of a substance is a task of con- siderable difficulty. Usually, however, it is sufficient to know the relation between the absolute thermal conductivity of one sub- stance with reference to another taken as standard. This ratio is called the relative conductivity of the first substance with ref- erence to the second. The object of this experiment is to de- termine by Voigt's method the conductivity of iron relative to copper. Let a rectangular plate formed by brazing together two triangular plates of the same thickness, one of copper and one of iron, be arranged in a horizontal plane as in Fig. 116. Let the / u \ / n V/irt ^-fiST N 1 w E^rfEE:' ^ s_ Fig. 116. Fig. 117. edge R S he uniformly heated to a constant temperature by means of a metal tube carrying a current of steam. If the upper surface of the composite plate is coated with wax which melts at a definite temperature, then when the plate is in thermal equilibrium there will be a sharp line UVW separating the melted from the solid film of wax. This locates the isothermal line of the plate correspond- ing to the melting point of the wax. If the plate were homogeneous and if there were no lateral loss of heat, this isothermal line would be parallel to the heated edge R S. But with a composite plate as here considered, there 268 Heat Conduction will be a point of discontinuity in the isothermal line at the inter- section V of the two substances. At this point the two parts into which the isothermal line is divided will be inclined to one another at an angle depending upon the conductivities of the two sub- stances composing the plate. At the edges of the plate the isother- mal line will be curved backward due to lateral loss of heat, but toward the center both portions of the isothermal line will be straight. Imagine three isothennal lines similar to U V W and let the distance between the first and second, and between the second and third, correspond to the very small temperature difference d t. Consider the heat flow through two indefinitely small elements, one on either side of the joint separating the two substances com- posing the plate. In Fig. 117, let A V and V B he the small ■elements, each normal to the isothermal lines bounding its ends. Denote the lengths of the elements by d /i and d l^. Let the depth of each element be d x, and let their widths parallel to the isother- mal lines bounding their ends be denoted by d w^ and d w^. Imagine that the areas of cross section of the two elements are such that for a constant temperature difference dt between their ends, there will be the same amount of heat flow through the two elements during the time dT. From eq. (i) p. 266 we can write , „ k, dl ■ dx ■ dw, ■ d T , d Hy= —^ i and dly k^ dx ■ dt ■ dw^ ■ d T dU dH,=^ Since d H-^ = d H^ k^ ^J^^k^ ^"^g . (1) d l^ d l^ If the element shown in the position ^ F, Fig. 1 1 7, be caused to assume a series of positions extending from B to A, its length will be different in each position. If the temperature difference between its ends is kept constant, then in order that the same quantity of heat d H^ will flow through it in unit time, the area of cross section of the element must increase in the same propor- tion as does its length. And since the length of the element varies dwi A E dl, -- A V dw^_ = cot. Relative Conductivity 269 directly with the distance A V , and A V varies directly with A B, we will have = cot. < di^ A V Similarly Substituting these values in eq. (i) we obtain ^1 cot. i^i. = ^2 cot. ^2 (2) Manipulation and Computation. — After the apparatus is assembled, pass a Bunsen flame back and forth underneath the plate until the entire plate is sufficiently warm to melt paraffin. Then rub the entire upper surface with a lump of paraffin contain- ing 10% of pure turpentine. When the upper surface of the plate is covered with an even coating of this indicator, allow the plate to cool to the temperature of the room. Pass steam through the heating tube and be careful that the plate receives no heat from the exhaust or from any other out- side source. Air currents must be carefully guarded against. When the line of demarcation between the melted and the solid wax assumes a stationary position, trace through the wax with a small sharpened wooden stick the straight portions of this line. Disconnect the steam pipe from the boiler and when all of the v/ax is solid draw through the wax two lines, M N and P Q., Fig. 116, normal to the Hnes previously traced. Then from the figure, . — — = cot. A, and — ^ = cot, (4,. MN ' P Q ^' Again heat the plate with a Bunsen flame, renew the coating of wax and repeat the experiment. In this manner obtain five values of the relative Ihermal conductivity of the two substances composing the plate. 270 Thermodynamics Chapter XX. THERMODYNAMICS. LXX. DETERMINATION OF THE MECHANICAL EQUIVA- LENT OF HEAT BY PULUJ'S METHOD. Object and Theory of Experiment. — The great general- ization called the First Law df Thermodynamics may be enun- ciated in the following form: whenever mechanical energy is converted into heat or heat is converted into mechanical energy, there is a constant relation between the work done and the heat produced or lost. This constant ratio is called the Mechanical Equivalent of Heat. It is defined as the number of units of energy which are equivalent tO' one unit of heat : e. g. the number or ergs equivalent to one calorie, or the number of foot pounds equivalent to one British thermal unit. The object of this exper- iment is to determine the Mechanical Equivalent of Heat by means of an experiment in which a measurable amount of mechanical energy is directly transformed into a measurable amount of heat. In the form of Puluj's apparatus used in this experiment, Fig. 118, a vertical spindle capable of rotation about its axis is surmounted by a vulcanic clamp. A, holding a thin steel cup of the shape of a truncated cone. Accurately fitting the inside of this conical cup is a similar hollow truncated cone attached to the center of a large wooden disc. A light cord supporting a mass M passes over the pulley P and has its other end fixed to a peg in the edge of the wooden disc. By means of a hand wheel, not shown in the figure, the outer cone can be rotated. By means of a watch and the counter C, the number of revolutions made in a given time is readily obtained. The mass M is so adjusted that the wooden disc with its attached steel cone will remain station- ary when the outer cone is rotated. At the rubbing surface be- Joule's Equivalent 271 tween the two cones mechanical energy is converted into heat. The work absorbed at the rubbing surface is readily obtained. Naturall}- the work absorbed at the rubbing surface between the two cones while the outer cone makes n turns and the inner cone remains stationary, is the same as would have been spent if the Fig. 118. outer cone had remained fixed and the inner cone had been made to revolve n times by the descent of the mass M. L,et the radius of the disc be r cms. ; then, if the cord supporting a mass of M grams leaves the disc tangentially, the mass would have fallen 27r I' n cms. Therefore the total work absorbed at the rubbing- surface is W ^ 2Tr rn(Mg -{- x) ergs, (i) where Mg is the weight of the mass M, and x is the friction of the pulley p. The heat produced by the energy absorbed in the rubbing cones is H = e (t^ — ti) -{- R calories (2) where e represents the water ec^uivalent of the cones and con- tents, ti and /j the original and final temperature of the cones and contents, and R is the radiation correction. 272 Thermodynamics If the mechanical equivalent of heat be represented by J, then the number of absolute units of work which must be spent to produce one heat unit is '- H- e{t^ — t,)^R ^^> Manipulation and Computation. — By means of a large caliper and scale, measure the diameter of the grooved wooden disc. Remove the cones and stirrer from the supporting frame, carefully clean and weigh them' separately. Fill the inner cone to within about i cm. of the rim with mercury at a temperature of 2 to 5° C, below the temperature of the room and weigh again. Replace the outer cone in its frame and so carefully center it that it will rotate about its own axis. Assemble the remainder of the apparatus being careful that the bulb of the thermometer is sub- merged in the mercury and does not touch any part of the ap- paratus. One observer will jplace himself, watch in hand, at the hand wheel, while a second will be ready to observe the thermometer and the revolution counter. The second observer will record the observations of temperature, time and number of revolutions. The hand wheel is now steadily rotated at such a rate that the string supporting the mass M is wrapped sufficiently about the grooved wooden disc to insure the string being tangential to the edge of the disc. So soon as it is certain that everything is working smoothly, simultaneous readings of the watch, thermometer and counting wheel are recorded. Simultaneous readings of temper- ature and time are now taken every half minute until the cones have risen in temperature about 5° C, above that of the room. Then stop the rotating, read the index of the counter and continue the half minute temperature readings until the cones have cooled to within two degrees of the temperature of the room. During all of the time temperature readings are in progress, the mercury in the inner cone must be constantly stirred. The water equivalent of tlie thermometer is obtained experi- mentally by the method of mixtures. The water equivalent of the cones, stirrer and mercury is computed by taking the sum of the products of the masses of each part and the corresponding . Joule's EquivaivEnt 273 specific heat obtained from tables. The sum of these water equiva- lents is the value of c given in eq. (2). A curve is plotted in the usual manner coordinating temperature and time and the radiation correction R is determined by Regnault's method. All of the data are now at hand for determining the amount of heat developed in the rubbing cones. Before the total work absorbed at the rubbing surface of the cones can be obtained, the value of the force of friction of the pulley supporting the mass M must be determined. The force of friction developed in the pulley is proportional to the resultant pressure on its bearings produced by the horizontal force P, Fig. 119, and the downward force Mg. In the apparatus used in this K.^ 2+x Fig. 119. I' Fig. 120. experiment the force of friction of the pulley P is so small that without sensible error it may be assumed that F is so nearly equal to Mg that the resultant p is inclined to both F and Mg at an angle approximately equal to 45°. Whence, since the components of P and Mg in the direction of the resultant are P cos 45° and Mg cos 45" respectively, it follows that p ^ 2 Mg cos 45° dynes = Mgy 2 = 1. 414 M grams weight. (4) Now hang the cord over the pully as in Fig. 120, and attaching to each end a weight equal to yi p {i. e. a mass of 0.707 M grams), find what additional weight x is required to produce uniform motion. This weight x expressed in dynes, is the force of friction of the pulley P for the particular load Mg used in the experiment. All of the data are now at hand for substitution in eq. (3)- 274 Tables TABLE 1. — Conyersion Factors. Length. 1 centimeter = 0.39371 inch 1 inch = 2.53995 cms 1 meter = 3.28089 feet 1 foot = 0.30479 meter 1. kilometer = 0.62138 mile 1 mile := 1.60931 km. 1 micron = 0.001 mm. 1 mil = 0.001 inch = 0.000039 in. = 0.00254 cm. Area. 1 sq. cm. = 0.15501 sq. in. 1 sq. m. = 10.76430 sq. ft. 1 sq. in. =6.45137 sq. cms. 1 sq. ft. = 0.09290 sq. meter Volume. 1 cu. cm. = 0.06103 cu. in. 1 cu. m. = 35.31658 cu. ft. inter = 1.76077 pints 1 cu. in. = 16.38618 cu. cm. 1 cu. ft. = 0.02832 cu. m. 1 quart = 1.13586 liters Mass. 1 gram = 15.43235 grains Ikff. \ = 2.20462 lbs. 1 grain = 0.06480 gram lib. (7000 grs.) = 0.45359kg. Angles. 1 radian = 57.2956 degrees , ) 1 degree = 0.01745 radian Density. 1 gm. per c. c. = 62.428 lbs. per cu. ft. = 0.03613 lbs. per cu. in. lib. per cu. ft. = 0.01602 gm. per c. c. = 5.787X10-' lbs. per cu. in. Force. 1 dyne = 0.0000723 poundal 1 gm. weight = 0.0022 lb. wt. 1 poundal = 13826 dynes 1 lb. wt. = 453.59 gms. wt. 1 cm. gm. unit = 2.373X10-' ft. lb. units Moment oe Inertia. 1 ft. lb. unit = 421402 cm. gm. units Tables 275 Stress 1 dyne per sq. cm. = 0.067197 poundal per sq. ft. 1 gm. wt. per sq. cm. =2.0482 lbs. wt. per sq. ft. 1 in. of mercury at 0°C. = 34.5328 grams per sq. cm. = 0.49117 lbs. per sq. in. 1 poundal per sq. ft. =14.8816 dynes per sq. cm. i lb. wt. per sq. ft. =0.48824 gms. wt. per sq. cm. 1 cm. of mercury at 0°C. = 13.5956 grams per sq. cm. = 0.19338 lbs. per sq. in. Work or Energy. 1 erg =2.373X 10"" ft.poundals 1 joule = 10'' ergs = 23.7302 ft. poundals 1 gm. cm. = 7 . 233X lO-° ft. lbs. 1 ft. poundal = 421403 ergs. 1 ft. lb. = 13825.5 gm. cms. = 1.35485 joules 1 H. P. hour = 2685600 joules Power. 1 watt = 10' ergs, per sec. =23.73 ft. poundals per sec. = 44,23 ft. pounds per min. 1 force de cheval=75kg. m. per sec. = 0.986 horse power 1 ft. poundal per sec. = 421403 ergs, per sec. 1 ft. lb. per min. = 0.13825 kg. m. per min, 1 horse power = 745.96 watts =1.01387 force de cheval. °C =i (°F—32). Thermometric Scales. j OF ^^ °C + 32. Unit Quantity of Heat. 1 gm. calorie = 0.0)3963 B. T. U. | 1 B. T. U. = 253 gm. calories Mechanical Equivalent op Heat.* 1 gm. calorie = 4.19 joules = 426.9 kg. m. = 1400.6 ft. lbs. * Computed with the value of g at Greenwich 1 B. T. U. = 1055 joules. = 778.1 ft. lbs. Logarithms. log,„ N =: 0.43429 loge N. 1 loge N = 2.3026 logi„ N. 276 Tabi^es TABLE 2.— Mensuration. A - area of curoe : S = surface of body : V - volume of body. ParaI/LEI/OGRAM, sides a and 6; heiglit, h; angle between sides 0°. A = bh = ah sin 0. Triangle, sides a, b and c; lieig-ht, h. Denote, X (« + &+ c) by s. A = ^ bh ^ }4 be sin e -.[s (s—a) (s—b) {s—c)']y^. CiRci^E, radius r. A = ir r'- Circumference = 2 tt n Area of segment of circle „ r' sin 180 2 Paraboi2 5 10 20 50 100 200 6379 9418 13108 15400 16320 16690 16860 17065 1 5 10 20 40 80 100 160 3285 6665 7318 7458 7436 7421 7439 7450 1 2 4 8 20 50 100 200 -152 -156 —111 2 173 278 335 375 TABLE 29.— Heat of Dilution of Various Salts. From Julius Thomson, Thermochemische Untersuchungen. The following table gives the heat in gram calories developed when a solution of one gram molecule of an anhydrous salt in n gram molecules of water is farther diluted by the addition of n' gram molecules of water. w + n' 20 50 100 200 CaCU 10 10 10 60 10 10 20 20 20 20 5 10 20 1639 1606 940 2307 412 3152 1148 2225 3308 904 3242 404 279 1068 664 5317 1203 318 2355 4052 776 41 3526 364 324 1380 —1056 832 6809 1111 377 2515 Cu CI2 4510 Cu(N63)2 Cu SO 2 729 116 Mg-Cl2 3761 Mg-(N03)2 Mg-SOi 370 393 Ui CI2 1584 2('NaCl) —1310 2(NaC2H3 02).. 2nCl2 936 7632 Zn(N03)2 ZnSOi 1071 390 292 Tabi,e^ TABLE 30.— Heat of Solution of Chemical Subbtances. From Julius Thomsen, — Thermocliemisclie Untersucliungen. n indicates the number of gram molecules of water in whicli is dissolved one gram- molecule of the suhstance. The solutions are made at ordinary room temperatures' The heats of solution are expressed in gram calories. Suhstance Formula H at of Soln. Ammonia gas Carbon dioxide gas. . . Hydrochloric acid gas . Sulphuric acid Nitric acid Calcium chloride Cupric sulphate. Ferrous chloride. " sulphate Mercuric chloride Nickel sulphate " nitrate Potassium chloride " chlorate. ' ' dichromate . . . . hydrate " nitrate. " sulphate " permanganate. Silver nitrate Sodium acetate " chloride " hydrate " nitrate " sulphate Zinc chloride . Zinc sulphate. 6H2O H^O 5 H2 O.... NH3. . CO2.... HCl... H2 SO4 HNO3. CaCla. CaCla Cu SO 4 CUSO4 Fe CI2 FeCls. 4H2 O.... Fe SO4.7 H2O.... HgCl2 NiS04. 7H2O.... Ni(N03)2 .6H2O.. K CI KCIO3 KaCraOj KOH KNO3 K2SO4 K2 Mn2 Ob 2 (AgN03) NaC2H302 . 3H2O Na CI NaOH Na NO3 Na2S04 .H2O Na2 SO4 • IOH2O.. Zn CI2 Zn SOi- HaO...... Zn S04- 7H2 O.... 200 ISOO 300 1600 300 300 400 400 400 350 400 400 300 800 400 200 400 400 250 200 400 1000 400 400 100 200 200 400 400 300 400 400 — 8430 — 5880 —17315 17850 7480 17410 — 4340 9320 — 2750 17900 2750 — 4510 — 3300 — 4250 — 7470 — 4440 —10040 —16700 13290 — 8520 — 6380 —20780 —10880 — 4810 — 1180 9940 — 5030 — 1900 -18760 15630 9950 — 4260 Tabi,es 293 TABLE Sl.^Heat of Neutralization of dilute Acids and Bases. From Julius Tliomsen-Tliermooliemisclie Untersuchungen. The dilution. employed is one gram equivalent ot acid or Ijase in 200 gram mole- cules of water ; e.g. (NaOH + 200HjOl, (Hj SO4 + 400 H,0). The mixtures are made at ordinary room temperatures. The heats of neutralization are expressed in gram calories. (a.). Monobasic acids. Substance Q (NaOH Ag , Q Ag) HBrOa HCIO3 HBr HCl HFl HCIO HPH2O2 HPO3 HNO3 HCIO 13780 Chloric acid 13760 Hydrobroiuic acid 13750 Hydrochloric acid , 13740 Hydrofluoric acid 16270 Hypochlorous acid . . . 9980 15160 Metaphosphoric acid 14380 13680 14080 (b) Dibasic acids. Substance (3 Na OH Ag , Q Ag) Hydrofluosilicic acid Phosphorous acid. . . . Selenic acid Selenous acid Silicic acid Stannic acid Sulphuric acid Sulphurous acid Tetraboric acid H2 SiFlo H2 PHO3 H2 SeO, H2 SeOs H2 Si03 H2 SnOs H2 SO4 H2 SO3 H2 B2O4 26620 28450 30390 27020 5230 9570 31380 28970 20010 (c) Bases. Q (Q. H2SO4 Ag) (Q. 3 HCl Ag) (Q. 2 HNO3 Ag) 2 KOH Aq 31290 27500 27540 2 NaOH Aq 31380 27490 27360 2 LiOH4<2 31290 27700 2 TIOH Aq 31130 27520 27380 Sr(OH)2Ag 30710 27639 Ca(OH)2 Aq 31140 27900 294 Tabi^es TABLE 32. — The Atomic Weight and Talencies of the principal Elements, According to- the latest determinations. H = 1. Name Aluminium . Antimony. . . Arsenic Barium Beryllium. . . Bismuth Boron Bromine .... Cadmium . . . Caesium .... Calcium Carbon Cerium Chlorine .... Chromium . . Cobalt Copper Didymium . . Erbium Flviorine .... Gallium Gold Hydrogen . . . Indium Iodine Iridium Iron Lanthanum. Lead Lithium Magnesium . Manganese . Mercury .... Molybdenum Sym- Ai; Sb. As. Ba. Be. Bi. Bo. Br. Cd. Cs. Ca. C. Ce. Cl. Cr. Co. Cu. D. E. F. G. Au. H. In. I. Ir. Fe. La. Pb. Li. Mg. Mn. Hg. Mo. Valer.ce 4 ■ 3,5 3,5 2 2 3,5 3 1,5 2 1 2 4 3,4 1,5 4,6 2,4 2 3 3 1 4 1,3 1 4 1,5 2,4,6 2,4,6 3 2,4 1 2,4,6 2 2 2,4,6 At. Wt. 26.99 120.29 74.92 136.76 9.03 207.64 10.94 79.77 111.95 132.58 39.99 11.98 139.9 35.37 52.09 58.74 63.30 142.32 165.89 18.98 68.85 196.8 1.00 113.4 126.56 192.9 56.0 138.6 206.47 7.01 24.31 54.93 199.71 96.18 Name Nickel Niobium . . . Nitrogen. . . Osmium.. . . Oxygen .... Palladium . Phosphorus Platinum . . Potassium . Rhodium. . . Rubidium . . Ruthenivim. Samarium . Scandium. . Selenium . . Silicon Silver Sodium .... Strontium. . Sulphur .... Tantalum .. Tellurium .. Terbium . . . Thallium. . . Thorium . . . Tin Titanium... Tungsten . . Uranium < Vanadium. . Ytterbium . Yttrium. . . . Zinc Zirconium . Sym- bol Ni. Nb.- N. Os. O. Pd. P. Pt. K. Ro. Rb. Ru. Sm. Sc. Se. Si. Ag. Na. Sr. S. Ta. Te. Tb. Tl. Th. Sn. Ti. W. U. V. Yb. Y. Zn. Zr. Valence 2,4 5 3,5 2,4,6 2 2,4 1,3,5 2,4 1 2,4,6 1 2, 4, 6 3 3 2,4,6 4 1 1 2 2, 4, 6 5 2,4,6 3 1,3 4 2,4 2,4 4,6 4,6 3,5 3 2 3 4 At. Wt. 58.56 93.8 14-2 191.0 15.96 105.74 30.96 194.85 39.03 103.24 85.25 101.3 150.02 43.17 78.80 28.33 107.67 23.00 87.37 31.98 182.14 126.7 160. 2v:i3.71 231.09 117.7 47.85 184.04 239.0 51.26 172.73 89.02 65.11 90.40 Tabi,es 295- TABLE 33. — Degree of Dissociation of Varlons Substances in 1% Aqueous Solution. From Arrhenlus' Electr'ocliemistry. The results obtained from the electrical conductivity method are given under the eading a' and the values from the freezing point method are given under the heading ' '. The solution consists of 1 gm, of substance to 100 gms. of water.- Non-Electroly tes Methyl alcohol Ethyl alcohol.. Glycerol Mannitol ...... Cane sug^ar . . . . Phenol Acetone Ethyl ether Ethyl acetate . . Acetamide a' a" 0.00 0.06 0.00 0.06 0.00 0.08 0.00 0.03 0.00 0.00 0.00 0.16 0.00 0.08 0.00 0.10 0.00 0.04 0.00 0.04 Bases Barium hydroxide . . . Calcium hydroxide. . . Lithitim hydroxide . . Sodiujn hydroxide . . . Potassium hydroxide Ammonia Methylamine Ethylamine Propylamine Analin 0.84 0.80 0.83 0.88 0.93 0.01 0.03 0.04 0.04 0.00 0.85 0.80 1.02 0.96 0.91 0.03 0.00 0.00 0.00 0.17 Acids Hydrochloric acid. . Nitric acid Chloric acid Sulphuric acid Hydrogen Sulphide Boric acid Formic acid Butyric acid Oxalic acid Malic acid a' a" 0.90 0.98 0.92 0.94 0.91 0.97 0.60 0.53 0.00 0.04 0.00 0.11 0.03 0.04 0.01 0.01 0.25 0.13 0.07 0.08 Salts Potassium chloride . . Potassium nitrate... Potassium acetate . . . Potassium carbonate Potassium sulphate . . Sodium nitrate Barium chloride Eead nitrate Copper sulphate Mercuric chloride. . . . a' 0.86 0.81 0.83 0.69 0.67 0.82 0.77 0.54 0.35 0.05 0.82 0.67 0.86 0.63 0.56 0.82 0.81 0.51 -0.03 0.11 39.6 Index INDEX Acceleration, Instantaneous.... 50 Alcoholimeter 74 Area by planimeter 56 Areometer 74 Air Thermometer 170 Association 191 August's Psychrometer 194 Backlash of screw 15 Balance, The 33 Ballistic pendulum 50 Barometer scale. Verification of 16 Belt, Friction of 86 Beckmann's apparatus for mo- lecular weight determina- tions 255-258 Beckmann's Thermometer. .. .140 Barometric Headings, Reduc- tion of 155 Boiling Point of a, solution. .226 Bomb, Hempel's combustion . . . 244 Brittleness 100 Bunsen's effusiometer ........ 81 Burette^ Calibration of 63 Calorimetry 196 Calorimeter, The water 213 The steam 219 The combustion bomb 244 Junker's 248 Calibration of » thermometer 141, 148 Calibration of a set of stand- ard masses 40 Carnot's cycle 172 Cathetometer, The, described. 13 Adjustment of a 17 Chronograph, The 43 Coefficient of linear expansion . 157 Coincidences, Method of 45-55 Computation, Rules for 4 Coulomb's Method for deter- mining viscosity 134 Conductivity, Relative thermal. 167 Cold test of an oil 220 Curvature of spirit level vial. . 19 Curves, Rules for plotting 6 Curvature of spherical sur- faces 26 Damped vibrations 121 Daniell's dew point hygrometer 192 Datum circle of planimeter.... 62 Density 70 Densimeter 75 Density of unsaturated vapors. 186 Density and molecular weight. Relation between 189 Depressed zero, of thermome- ters 137, 147 Deviation measure 3 Dissociation 190, 263 Dividing Engine, The 9 Double weighing 38 Eccentricity of a 'divided circle 29 Efficiency of a reversible engine .171 Effusiometer, Bunsen's 81 Elasticity 99 Elastic limit 100 Emissivity, Thermal 204 Empirical equations 47-104 Errors, Discussion of 2 Errors, of thermometers 137 Eyepiece Micrometer, The 11 Expansion of solids and liquids 153, 157, 1G2, 164 Expansion of gases 176 I'^xpansion of gases. Work done during ; 170 Index 297 Flash test, fire test and cold test of an oil 223 Flexure, Study of 104 Flexure, Young's Modulus by.. 108 Filar Micrometer Microscope. . 10 Fieezing points 228 Friction, CoeflScient of 84 Friction of u belt 80 Friction of a journal 88 Fundamental law of gases.... 107 Fusion, Heat equivalent of . . . .230 Gauss' method of weighing 38 Gases and solutions, Properties of ...: 107 Graduation of a scale 15 Gram molecule 168, 230 Gram equivalent 230 Guillaume's hypsometer 145 Heat equivalent of fusion 230 of condensation. .. .219 of vaporization . 220, 233 Heat of dilution 240 solution 238 neutralization 241 Heat value of a solid 242 gas or liquid . . . 248 Hydrometer 74 Hygrometry 191 Hypsometer 145 Humidity 192 Jolly's air thermometer 17T Joly's steam calorimeter 219 Junkers gas calorimeter 248 Joule's equivalent 270 "Latent heat" 230, 233 Level testing 19 logarithmic decrement 120 Jlic'.ometer screw gauge, The. 9 Mechanical equivalent of heat, 270 Melting points 228 Mohr-Westphal balance, The . . 72 Moment of Inertia 91 Molecular weight and density. Relation between 189 Mueller's optical lever 158 Molecular weight by boiling point method. .2.52 by freezing point method 2.')S New York State board of health oil tester 224 Normal salt 241 Optical Lever 23-102 Oil tehter 224 Optical lever 23, 102, 158 Osmotic pressure KM Parallax 17 Passages, Method of 44 Pendulum, Ballistic 50 Pendulum, Borda's 54 Percentage deviation 3 Planimeter, Theory of 56 Plotting of curves 6 Platinum resistance thermome- ter 148 Psychrometry 191 Precision of observations .... 3 Poiscuille's method for deter- mining viscosity 129 Pyknometer 60-71 Radian 117 Radiation correction 197, 199 constant 199 Regnault's hypsometer 145 method of deter- minating vapor pressure 184 method of deter- mining expansion of liquids 162 Resilience 119 Restitution, Coefficient of 52 Resistance thermometer 148 Reversible engine. Efficiency of the 171 Rider for balance 33 Salinimeter 75 Scale for plotting. Best 7 Sensitiveness of a spirit level. . 19 298 Index Sensitiveness of a ba/lance 37 Significant figures 4 Simple Rigidity by vibration method 112 Simple Rigidity by statical method 116 Siemens' pyrometer 148 Stress 100 Solution, Normal 236 Boiling point of 227 Solvent, solute, solution de- fi.ned 169 Specific Gravity 70 Speed, Instantaneous 49-51 Spherometer J The -. 9 Specific heat. 196, 208, 212, 217, 219 Steana calorimeter 219 Stereometer, The "67 Strain 100 Telescope- and scale, Adjust- ments of 25 Tenacity 100 Thermometer, The air 176 The mercury in glass 14) The platinuin re- sistance 148 The Beckmann. .140 The weight 160 Thermometers, Errors of 137 Calibration of 141, 148 Thermodynamics, Laws of.... 171 Thermal units 196 water equivalent. . . 196 . . «missivity 204 Thermochemical measurements.236 Vapor pressure 180, 184 density 186 Van';-, Hofl'slaw 169 Vaporization, Heat equivalent of 233 Vacuum, Reduction of weight to 37 Velocity, Instantaneous 49-50 Vernier caliper. The 13 Viscosity by Poiseuille's method 129 Viscosity b y Coulomb's method 134 Volume, Determination of... 63-64 Volumenometer 67 Weighing 34 Wheatstone's bridge .149, 151 Weight thermometer 166 Weight dilatometer 164 Wet and dry bulb hygrometer.194 Water eqviivalent, Thermal 196, 215, 247 Young's Modulus by stretch- ing 101 Young's Modulus by bending.. 108 Zero Circle of planimeter .... 62 Illustrations 299 Ili^ustrations 301 303 aS3.l 0769 Il,L,USTRATIONS 305 igia 20 lUvUSTRATIONS 307 A B t" ill ~^"''""- 1 ' ^' ^''' - liJhM iiiinEiijmfjhiiJi. Illustrations 309 Battery Battery Illustrations 311 IlIvUSTRATIONS 313 Ili^ustrations 315 IlvtUSTRATlONS 317 Illustrations 319 .y / ni^S^ w ^fefesrzi^S*** s ILLUSTRATIONS 321 2( IlvLUSTRATIONS 323 IlvLUSTRATIONS 325 Illustrations 327 Illustrations 329 IlvLUSTRATIONS 33^ v©-^' '•m-