CORNELL UNIVERSITY LIBRARY CHEMISnrRYJ-lBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1891 BY HENllY WILLIAMS SAGE Cornell University Library 3 1924 031 189 883 olin.anx Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 1 89883 THE ELEMENTS OF PHYSICAL CHEMISTRY. BY , r J- LIVINGSTON R. MORGAN, Ph.D., Of the Department of Physical Chemistry, Columbia University. FIRST EDITION. FIRST THOUSAND. NEW YORK : JOHN WILEY & SONS. London : CHAPMAN & HALL, Limited. 1899. 31 loLf lopyright, 1899, \ Copyright, BY J. L. R. MORGAN. ROBERT DRUHMOKD, PRrNTER, NEW YORK, PREFACE. The object of this book is to present the elements of the entire subject of Physical Chemistry in one volume, together with the important and but little known applications of it to the other branches of chemistry. Many persons have found it difificult to t)btain a comprehensive outline of the subject, owing to the length of time which it has been necessary to spend upon the separate volumes devoted to special portions of it. To all such this volume may be of value. It is especially intended as a text-book for either class-work or self-instruction, and although the cal- culus is used in the derivation of some of the laws, still much can be done without any training in the higher mathematics. In general, references are given, so that any one wishing to make an extended study of any special portion may do so with little difficulty. The amount of the subject included, however, em- braces that which is likely to be useful to all chem- IV PREFACE. ists, and is that which is being given to all chemical students at Columbia University. No claims of originality are made for the major por- tion of the text, for Ostwald's Lehrbuch der all- gemeinen Chemie and Analytical Chemistry, as well as Le Blanc's Electrochemistry and Nernst's Theo- retical Chemistry, have been very freely used. Some things, however, will be found in a new form, which it is hoped is simpler and clearer than the usual one. The arrangement as well as the scope of the volume is new, which I think makes the subject more logical and clear. J. L. R. M. Department of Physical Chemistry, Havemeyer Laboratory, Columbia University, December, 1898. CONTENTS. CHAPTER I. PAGE Introductory Remarks i I. Physical chemistry. 2. Energy. 3. The factors of energy. 4. Methods for the determination of the atomic weight. CHAPTER n. The Gaseous State 10 5. Definition of a gas. 6. The gas laws. 7. The specific gravity of gases. 8. Methods of determining the specific gravity, 9. Abnormal vapor-densities. Dis- sociation. 10. The kinetic theory of gases. 11. Varia- tion from the gas laws. The theory of Van der Waals. 12. Specific heat. The first principle of thermodynamics. 13. Determination of the specific heat of gases. 14. The second principle of thermodynamics. 15. The cycle. Entropy. 16. The factors of heat energy. CHAPTER III. The Liquid State 55 17. Distinction between liquids and gases. 18. The specific gravity and its determination. 19. Connection between the gaseous and liquid states. 20. Vapor-pres- sure and boiling-point. 21. The heat of evaporation. 22. Volume relations. 23. Refraction of light. 24. Sur- face-tension. V VI CONTENTS. CHAPTER IV. PAGE The Solid State 69 25. Remarks. 26. Atomic heat. Law of Dulong and Petit. 27. Changes in the state of aggregation. CHAPTER V. Solution 74 28. Definition of a solution. 29. Gases in liquids. 30. Liquids in liquids. 31. Solids in liquids. 32. Osmotic pressure. 33. Electrolytic dissociation or Ionization. 34. Solution-pressure. 35. Vapor-pressures of solutions. 36. The relation between osmotic pressure and the de- pression of the vapor-pressure. 37. Increase of the boiling-point. 38. Depression of the freezing-point. 39. Division of -a. substance between two non-miscible solvents. Depressed solubility. CHAPTER VI. The Role of the Ions in Analytical Chemistry . . .122 40. Ions. 41. The color of solutions. 42. The action of indicators. 43. The solubility product. 44. General analytical reactions. 145 CHAPTER VII. Thermochemistry 45. Definition. 46. Application of the principle of the conservation of energy. 47. The heat of formation. 48. Chemical changes at a constant volume. 49. Chemical changes at a constant pressure. 50. Relation between results for constant volume and constant pressure. 51. The effect of temperature. 52. The ions in thermal reactions. CHAPTER VIII. Chemical Change. A. Equilibrium . „ 150 53. Reversible reactions. 54. The law of mass action. 55. Equilibrium between gases. 56. Dissociation of gases. 57. Equilibrium in liquid systems. 58. Dissocia- CONTENTS. VII PAGE tion of double molecules. 59. Solid solutions. 60. Ions and tlie law of mass action. 6i. Hydrolytic dissociation. 62. Equilibrium in non-homogeneous systems. 63. Phys- ical equilibrium. 64. Gases and solids. 65. Dissocia- tion of a solid into more than one gas. 66. Non-electro- lytic dissociation in solution. 67. Electrolytic disso- ciation or ionization and solubility. 68. The effect of temperature upon incomplete equilibrium. B. Chemical Kinetics ig4 69. Application of the law of mass action. 70. Reac- tions of the first order. 71. Catalytic action of hydrogen ions. 72. Reactions of the second order. 73. Reactions of the third order. 74. Incomplete reactions. 75. Reac- tions between solids and liquids. 76. Speed of reaction and temperature, CHAPTER IX. Phases 209 77. The phase rule. 78. The equilibrium of water in its phases. CHAPTER X. Electrochemistry. A. The Migration of the Ions 215 79. Electrical units. 80. Faraday's law. 81. The migration of the ions. 82. Determination of the relative velocity of migration. B. The Conductivity of Electrolytes 224 83. The specific conductivity. 84. The molecular con- ductivity. 85. Determination of electrical conductivity. 86. General rules. 87. The conductivity of organic acids. 88. The relative velocity of migration from the conduc- tivity. 8g, The absolute velocity of migration of the ions. go. The basicity of an acid. 91. The conductivity pf neutral salts. 92. The dissociation of water. 93. The temperature coefficient. 94. Conductivity of difficultly soluble salts. 95. Other solvents than water. The di- electric constant and dissociating power. VIU CONTENTS. PAGE C. Electromotive Force 245 96. Determination of electromotive force. 97. Types of cells. 98. Chemical theory of the cell. 99. Electro- lytic solution-pressure. 100. Theoretical formula for the difference of potential between a metal and a solution of one of its salts. lOi. The Lippmann electrometer. 102. Concentration-cells. 103. Dissociation by aid of the electromotive force. 104. The electromotive force be- tween liquids. 105. Solution-pressures of the metals. 106. Cells with inert electrodes. 107. Processes taking place in the cells in common use. D. Electrolysis and 'Polarization 283 108. Decomposition values. 109. Theory of polariza- tion, no. Primary decomposition of water, in. Elec- trolytic separation of metals by graded electromotive forces. Index 293 ELEMENTS OF PHYSICAL CHEMISTRY. CHAPTER I. INTRODUCTORY REMARKS. I. Physical Chemistry is that branch of the science of chemistry which has for its object the study of the laws and theories governing chemical phenomena. Other titles for this subject are also in use (General Chemistry, Theoretical Chemistry), but, since the sub- jects treated lie in that border-land between Physics and Chemistry, and since many purely physical methods are used, the term Physical Chemistry is to be preferred, and consequently is in general use. The subject of Physical Chemistry is usually divided into two part : Stoichiometry and Chemical Energy, the latter term including the laws of afifinity and all other allied subjects. As it is difficult, however, to make a sharp distinction, we shall consider both together, dividing the subject only into minor portions in the form of chapters. 2 ELEMENTS OF PHYSICAL CHEMISTRY. 2. Energy. — Since much of our work has to do with the different forms of energy, it will be well first to recall some points which later we shall use con- stantly. Force is whatever changes, or tends to change, the motion of a body by altering its direction or its mag- nitude. A force acting upon a body is measured by the momentum it produces in its own direction in the unit of time. The unit of force is the dyne. A dyne acting upon a gram for one second would give it the velocity of one centimeter per second. Since a weight is used in this definition, only such forces are comparable which are measured at the same place. To avoid this difificulty it is simply necessary to mul- tiply tlje results by g, the acceleration due to gravita- tion at that place. At Paris a body falling freely for one second acquires the velocity of 981 centimeters. In other words, the weight of one gram at Paris is equal to 981 dynes, or i gram-centimeter is equal to 98 1 dynes. Work. — The unit of work is that work which is done when unit force is overcome through unit dis- tance. This is called in the absolute system the erg. We have then dynes X cm. = ergs. The maximum work is that amount which can be produced under ideal conditions. Energy is the power of doing work which a body INTRODUCTORY REMARKS. 3 possesses by reason of its state. Mechanical energy is of two kinds, kinetic and potential — the former being due to motion, the latter to position. As energy is measured by its power to do work, its unit is also the erg. Since it is impossible to remove all the energy from a body, we have no way of determin- ing how much is contained in it. The excess of energy which a body contains in a given state over what it contains in a certain standard state is called the intrinsic energy. It is determined by allowing a body to go from the given state in such a way that the difference in energy all appears in one form, for example, as heat. If we measure this heat we have the intrinsic energy of the body in terms of heat energy. 3. The factors of energy. — It has been found that the different forms of energy can each be written in the form of the product of two factors. The one factor determines the amount of energy which can and must be absorbed by a body in going from one state into another. This is called the capacity factor of the energy. Upon the other factor depends the possi- bility of the transfer of the energy from one body to another in such a way that if the value of this factor for both the bodies is the same no transfer takes place. This is called the intensity factor of the energy. Bodies with the same value for the intensity factor are in equilibrium, i.e., no lasting transfer of energy takes place between them. If the intensity 4 ELEMENTS OF PHYSICAL CHEMISTRY. factor has a different value in two states of a sub- stance, an exchange of energy will take place between them until the intensity factor has become of the same value for both and equilibrium is established. Examples. — For heat energy we have, under certain conditions, for the capacity factor the capacity of the body for heat, while the temperature is the intensity factor. If two bodies at the same temperature are brought in contact, there is no change in them. Should however the temperature be different, there is an exchange of heat energy of such a nature that the tersperatures become the same. Bodies contain- ing different amounts of heat, when brought together, remain unchanged provided their temperatures are the same. Volume energy is equal to the work required to cause a decrease of volume, or that which is done by an increase of volume. The intensity factor is the pressure, while the volume is the capacity factor. Imagine a gas enclosed in a cylinder which is provided with a movable partition. If on each side of this par- tition we have the gas, under different pressures, then the partition will move to the side with the smaller pressure, until the two become equalized. As long as the pressure is the same the volumes may have any relation without causing the partition to move. For kinetic energy, equal to \/2Mv\ the capac- ity factor is the mass, and the intensity factor is the velocity. INTRODUCTORY REMARKS. 5 For electrical energy the amount of electricity is the capacity factor, the electromotive force being the intensity factor. In the same way all other energies may be divided into factors, and it has been found that the study of the different energies is much simplified by the process. In general, then, for all energies we have (1) E = ci, where c is the capacity factor and i the intensity factor, E being the energy itself. If the energy is increased by an infinitesimal amount, then dE = d{ci) = cdi -\- idc ; and from this we may find the equations for c, i, dc, and di. They are : dE (2) '^ ~'~d' ^'^'^ = o, c = const.) ; , V ,. dE , , (3) di = — {dc = o, t = const.) ; dE (4) i z= --J- [di =0, «' = const.) ; (5) dc = -^ {dt = o, z = const.). If in a system there are two kinds of energy which are so related that a change in the one causes a corresponding change in the other, i.e., one is a 6 ELEMENTS OF PHYSICAL CHEMISTRY. function of the other, then if the two kinds of energy are £, and E^ dE, = dE, or d{c,i,) = d{c,i^; i.e., if c = const. (6) c^di^ = c^di, ; This method of deriving an equation giving the relation between two kinds of energy in one system is very useful, and we shall have occasion to use it later. 4. Methods for the determination of the atomic weight. — Since we shall consider the molecular weights of substances rather deeply, and as these are found from the atomic weights, it will be well first to understand the general methods of the determination of the latter. First method. — We find how much of an element or compound of known atomic weight will unite with a certain amount of the element whose atomic weight is to be determined. If by experiment it is found that u grams of U (the unknown) combine with k parts of K (the known), then U: K::u: k, or k INTJiODUCTORY REMARKS. 7 Examples (Berzelius). — 25 grams of Pb when oxi- dized with HNO, give 26.925 grams of PbO. 25 grams of Pb combine, then, with 1.925 grams of O, and since the atomic weight of O is 16, that of Pb is ^^16X^5^ 3_ 1.925 On the other hand 21.9425 grams of PbO, when reduced by hydrogen, give 20.3695 grams of Pb. Here 20.3695 grams of Pb are combined with 1.5730 grams of O; ht;nce 16 X 20.3695 U = ^-^^ = 207.2. I-5730 Second method. — We find how much of an element or compound, of known atomic weight, will react with a compound of elements, the atomic weight of one being known. Let k^ grams of K^ correspond to ^, grams of the compound U -\- K^. If K^ and K., are the known atomic weights, then U, the unknown one, is found from the proportion (t7+/ro -.K^-.-.K-.K; i.e., Examples. — 5.414 grams of NaCl are necessary to completely precipitate 10 grams of Ag (Pelouze). 8 ELEMENTS OF PHYSICAL CHEMISTRY. The atomic weight of Na, since K^ = 107.94 and K= 35-45. is 107.94 X 5.414 ^= JO - 35-45 = 22.99. From 13.6031 grams of CaCO,, by ignition, Erd- mann and Marchand found 7.6175 grams of CaO. This amount had been united with 5.9856 grams of CO,. K=CO,^ 44, K,= 0= 16, k, = 5.9856, and k, = 7.6175 ; hence „ 44 X 7-6175 £ ^"■>= 5.9856 -^6 = 40.01. Third method. — A known amount of a compound, containing the element whose atomic weight is to be determined, is transformed into another compound with elements whose atomic weights are known. If K^ and K^ are the atomic weights of the elements which are combined with the unknown one U, and k^ parts of the one have given k^ parts of the other, then (U^K:):{U-\.K^::k,:k, or KK,-KK, klK,-K^ ""- K-K ~ K-K ~ '• Example (Struve). — From 100 parts of BaCl, 112.0938 parts of BaSO. can be obtained. Here K, = CI, = 70.908, K^ = SO. = 96.061 ; hence 1 12.0938(70.908 — 96. 061) 100 — 1 12.0938 ^^^ - I'ool ri2.0Q^8 - 96.061 = 137.07. mTRODUfTORY REMARKS. 9 The possible error in these three methods depends principally upon the atomic weights which we substi- tute in the equations. Since in the first method there is but one known atomic weight, that method is the most accurate, the second being next and the third last.* * For any further information as to these methods and the determination of the probable error see Ostwald, Lehrbuch d. allg. Chem., I , pp. 15-25. CHAPTER II. THE GASEOUS STATE. 5. Definition of a gas. — A gas is distinguished by its property of indefinite expansion. In other words, a gas is limited in volume only by the walls of the vessel which contains it. 6. The gas laws. — These are the result of experi- ence as to the behavior of gases in general under varying conditions. Boyle's (Mariotte's) law: At constant temperature the volume of any gas is inversely proportional to the pressure under which it exists. If V represents the volume at the pressure p, and v^ any other volume of the same gas at the pressure/,, then (7) pv = p^v^ — constant for constant temperature. If the temperature is different for the two cases the product /z' is no longer equal to the product /,z',. The effect of a change in temperature upon the volume of a gas is given by the law of Charles {Dalton, Gay-Lussac) : The volume of any gas 'increases by the 1/27J part of its volume at 0° C. for every in- THE GASEOUS STATE. II crease of its temperature equal to i" C. If % is the volume of the gas at o° C, the volume at any tem- perature is (8) Vt = vli + at), where « is the coefficient of expansion of all gases, i.e., 1/273. If the temperature were decreased continuously from 0° we would finally reach a point at which the volume is equal to zero. Provided that the law of Charles holds for such a low temperature, this point will be 273 centigrade degrees below 0° C. This point is called the absolute zero, and the temperature measured from it in centigrade degrees is the absolute temperature. This is designated by the letter T, the temperature according to the centigrade scale being distinguished by the letter t. Between these two terms we have, then, the relation r=^ + 273 or t= T- 273. By the aid of equation (8) it is now possible for us to find the value of the product of volume and pres- sure at any temperature if we know it for 0° C. As- sume in the equation p^v^ = p„v^ = constant the constant temperature to be 0° C, the pressure/, to be 760 (mms. of Hg), and the volume % to be that of a certain weight of the gas measured under 12 ELEMENTS OE PHYSICAL CffEMISTEY. these conditions. /, is then any other pressure and w, the corresponding volume for this same tempera- ture of o° C. If the product pv is constant for any one tempera- ture an increase of the latter will cause one in the former, for both pressure and volume increase with the temperature. If one of the factors remains con- stant, the entire change will take place in the other, and the value of the product fv for this temperature can be found. This value of pv, however, will be constant for that temperature (Boyle's law), so that it is indifferent whether the temperature changes only one factor or both. Equation (8) gives us the change in volume when the pressure remains constant, so that by combining it with the expression for Boyle's law (7) we obtain an equation giving the value of pv for any temperature. This is pv = p^vli + at), which holds for constant volume, for constant pres- sure, or for a variation in both pressure and volume. Since « = 7T7 and t z= T— 273, pv = ^T. 273 p V This term -^ is however a constant, for/, = 760 273 (mms. of Hg.), and v^ is the volume occupied by a certain definite amount of gas under this pressure at the temperature of 0° C. THE GASEOUS STATE. 1 3 If we assume v^ to be the volume of i gram of gas we have, pv = R'T (i.e., ^-^ for i gr. = R'\, \ 273 / where v is the volume of i gram of gas at the tem- perature T and the pressure / and R' is the specific gas constant. R' is a constant for any one gas and is inversely proportioned to its specific gravity. Jf V and v^ refer to the mol * (9) pv = RT (^ for I mol = R\, where R is the molecular gas constant, which is the same for all gases. R' multiplied by the molecular weight of the gas gives R. Equation (9) is the equation of state for gases. In it /, the pressure, is to be expressed as a weight in grams upon the square centimeter, v^ the volume in cubic centimeters, and T the absolute temperature in centigrade degrees, i.e., T ^ i -\- 273°. The molecular gas constant R may be calculated as follows : R = ^^-^ ; V. = 22400 cc. ; 273 P, = 7^ X 13-59 = 1033 grams per sq. cm. ; hence „ 22400 X 1033 - o o R = — 2 1^ ^ = 84800 gr.-cms. * From here on we shall call the molecular weight in grams the mol, as has been proposed by Prof. Ostwald. 14 ELEMENTS OF PHYSICAL CHEMISTRY. Daltons law refers to the pressure exerted by the single gases in a mixture of gases. The pressure exerted upon the watts of a vessel, con- taining a mixture of gases, is equal to the sum of the pressures which the single gases would exert were they alone in the vessel. The exact meaning of this will be seen from the following example: When into a closed exhausted vessel we introduce one gram of gas it will exert a certain pressure upon each square centi- meter of the surface. If now we introduce a second gram of the same or a different gas, this second gram will exert exactly the same pressure upon the vessel that it would have exerted had the first gram not been there. Upon the walls, however, the pressure is now doubled. One law still remains to be considered which has had and still has great value in chemistry; it is Avogadro' s law or hypothesis : All gases under the same conditions of pressure and temperature contain in unit volume the same number of molecules. This law is of particular value for the reason that by its aid we are able to determine the molecular weight of gases, or of substances in gaseous form, from their densities. Since equal volumes contain the same number of molecules, the weights of these volumes must be re- lated in the same way as are the weights of the mole- cules themselves. The density of a gas is the weight of the unit of volume. If the molecular weights of two gases are THE GASEOUS STATE. Ij tn^ and ;«, and their densities are d^ and «?■„ then we have the relation m^ : in, : : d^ : d, or m, m, m„ — _ — constant. «. «a dn Since the atomic weight of hydrogen is assumed to be I (molecular weight = 2), its density is also consid- ered as unity, and we have for any gas m 2 'd^l or m = 2d; i e., the molecular weight of any substance in the gaseous state is equal to twice its vapor density referred to hydrogen as unity. 7. The specific gravity of gases. — The specific gravity of a gas is the ratio of its weight to the weight of an equal volume of the standard gas, both being weighed under the same conditions of tempera- ture and pressure. In physics air is generally employed as the standard, but its composition varies so that oxygen has been chosen for the chemical standard. Since this is very easily obtained in the pure state and its atomic weight has been accurately determined, its advantages as a standard are apparent. One liter of O weighs 1.430 11 grams at 0° and 76 l6 ELEMENTS OF PHYSICAL CHEMISTRY. centimeters pressure. Its specific volume, i.e., the volume of i gram at any temperature and pressure, is ^^A^^Xl+^cc. P or I 76(1 + at) , 76(1 +«/) V = -^ — 6gQ.2S<— i — ! ' cc. 0.00143011 P P Its density, i.e., the weight of i cc. at any tempera- ture and pressure, is the reciprocal of the specific volume, i.e., d= —= 0.0014301 1 —rr^-, r grams. V ^^ 76(1 + at) ^ The molecular volume, since O = 32, is F= 32 X 699.25^^^1+-^ grams. P If w is the weight of a gas, and g is the weight of an equal volume, under like conditions, of the stand- ard gas, i.e., O, the specific gravity of the gas is w s = — . In order to preserve the relation between the mo- lecular weight and the density it is customary to cal- culate the latter as based upon an ideal gas, the weight of which is 1/32 that of O. In this way we avoid actually using hydrogen as a standard in the experiment, and our results give directly the molecular THE GASEOUS STATE. Xf weight. The weight, then, of any number of cc. of this ideal gas for any pressure and temperature is 0.0014^011 pv S = -^ ^- , r grams. 32 76(1 + a() ^ s is then found from the equation w76(i 4- 0.00367^) (i + 0.00367^) s = —^—^ — ■ -^ ' ' = 1701000^^^—1 ^—^\ 0.00004469 />v This equation may be used in any of the regular methods of determination by substituting the values of t, p, and V. t is given in centigrade degrees, / in centimeters of Hg, and v in cc. If great accuracy is desired, it is necessary to read the barometer always at the same temperature; other- wise the column and the scale will change with the temperature and the reading be incorrect. For the sake of uniformity the reading is always reduced to what it would be at 0°. This is done by aid of a formula, so that the actual reading may take place at any temperature and then be corrected. The coefiS- cient of expansion (i.e., the increase in volume at 0° C. for an increase of temperature of i" C.) for Hg is 0.0001813. If yS is the coefificient of expansion of the scale (which is correct at 0° C), then the height of the barometer, at any temperature, reduced to 0° is B, = Bl\ - (0.00018 1 3 - ysyj, /? for glass is 0.000009-0.000010, for brass = 0.00002, and for steel = 0.000012. 1 8 ELEMENTS OF PHYSICAL CHEMISTRY. 8. Methods of determining the specific gravity.— Here we shall consider the subject both for gases and for the vapors generated from substances by heat, since the determination for the latter is of importance for the ascertaining of the molecular weight. For convenience we shall divide the subject into three groups of methods. A. The weight of a certain volume of the gas. — This may be used both for gases and for the vapors given off by substances under high temperatures. a. For gases. — The weight of a certain volume is found by weighing first the apparatus with the gas, and then exhausting it and weighing it alone. For this purpose a glass balloon {v = 1/2 liter) is used, into which two tubes, provided with stop-cocks, are fused. The gas is allowed to pass into one tube and out of the other, at a certain pressure and tempera- ture, until all air is removed and the vessel filled with gas, when the cocks are closed and the balloon weighed. Let this weight be P^. The vessel is next exhausted and weighed. If it weighs P^ grams, then that volume of gas weighs P^ — P., =1 P grams. The volume of the vessel is not known accurately, so that instead of using the general formula it is simpler to carry out the same experiments for O under the same conditions. If this volume of O weighs P' THE GASEOUS STATE. . I9 grams, then the density based upon O/32 is equal to P The molecular weight of the gas is, then, equal to this density. /?. For gases generated by heat from substances (Dumas). — Here the process is slightly different and our general formula is to be used. A flask (& = 1/8 liter) which has very thin walls is used. The substance, in solid or liquid form, is placed in this and the whole surrounded by a temperature-bath. The substance goes into the gaseous state and drives out all the air. As soon as the last traces of substance have disappeared the neck of the flask is closed by heating, the gas being at the temperature t and the pressure /. After the bulb has cooled it is weighed. Let this weight be P grams. The temperature and pressure of the balance t' and p' are also observed. Finally, the neck of the flask is broken off under water and the water allowed to fill the flask. The weight of this, i.e., flask, water, and neck, we will assume to be P" . If the weight of the empty flask before the experiment is P', then, since the walls are very thin and the internal and ex- ternal volumes are the same, we need not correct for the displaced air. The volume of the flask is then V=(P" - P')cc. The weight P is equal to that of the flask plus that cf the enclosed gas minus that of the volume of air 20 ELEMENTS OF PHYSICAL CBEMISTRY. displaced, P,, since that would buoy it up and decrease the weight. The weight of the gas is then found from p = P' J^ D - P,; i.e., Z> = P- P' + P,. I cc. of air at o° and 760 mm. of Hg weighs 0.00129 gram; at t'° and p'° mm., as in the weighing, \l weighs „, 0.00129 / P' = =^-— -^ grams. I + 0.00366?' 760 ^ The weight of v cc, i.e., the air displaced by the balloon, is then P,'F = P,. The rimount of gas at the temperature of closing / and the pressure / fills the same volume as it does at 0° and 760 mms. Hg; its weight is D' = D— — (i + 0.00366?) grams. P This when divided by v is the term w in the gen- eral equation, which can be solved for s after substi- tuting the temperature t and the pressure p. For very exact results there are certain other cor- rections to be made, but for them the student is referred to some practical laboratory manual, as this is hardly the place for anything more than the principle of the method. This refers as well to all the other methods mentioned in the book. THE GASEOUS STATE. 21 This method of Dumas has been further modified, so that it is possible to work under diminished pressure and consequently at lower temperatures, which is of great importance in case the substance decomposes at a higher temperature. The flask for this purpose is the same as that used before except that the neck is connected with an air-pump and manometer. When the pressure has reached the desired point the tem- perature is increased until the substance goes into the gaseous form, when, as before, the neck is sealed. The calculation is the same as before except that the pres- sure under which the gas exists is equal to that of the barometer minus that of the manometer, B. The volume occupied by a certain weight. — The method of Gay-Lussac, as improved by Hofmann, is very simple, and is intended for the determination of the specific gravity of the gases from solid or liquid substances. A weighed amount of substance is brought into the Torricelli vacuum of a barometer- tube, which is surrounded by a temperature-bath. The substance is weighed in a very small bottle {v = 1/2 cc.) which is fitted with a glass stopper. This bottle is introduced into the mercury in the trough and allowed to rise in the tube. When the temperature has risen sufificiently the stopper is blown out and the substance assumes the gaseous form. The volume of this gas is measured, the tube being graduated, and the temperature and pressure noted. The gas is under a pressure equal to that of the atmos- 22 ELEMENTS OF PHYSICAL CHEMISTRY. phere minus that of the column of mercury, which, when reduced to o" C, we will assume to be h mms. of Hg. If this volume at t° and p — h mms. is V, then that at o° and 760 will be 760 vi _|- If the substance weighs £■ grams, then i cc. of the vapor at 0° and 760 mm. will weigh g/y 160,0' grams. This will give the specific gravity by comparing it with the weight of i cc. O/32, or the term Fmay be used and ^/F substituted for w in the general formula. The method of Victor Meyer differs from that of Hofmann in that the gas as it is formed in a flask dis- places the air, and the volume of this at a lower tem- perature is measured. This volume is of course the same that the substance itself would assume at that, temperature, provided that no condensation takes place, for the coefficient of expansion of all gases is the same. The apparatus. Fig. i, consists of a wide tube with a bulb at one end, which is fastened into a temperature-bath. The top of the tube is connected with a burette filled with water and so fixed that the gas may replace the water. There is also an arrange- ment t by which the weighed substance, in a small flask s as before, can be dropped in the tube. When the temperature has reached a certain point the sub- stance is inserted and goes into gaseous form. By THE GASEOUS STATE. 23 this the air in the upper part of the tube is driven into the burette and the amount of it may be read. This amount of course is the volume occupied by the weighed amount of substance in gaseous form at the temperature of the water. ^^^^^ Fig. I. If the weight of the substance is g and the measured volume at temperature f is V, then the specific gravity s may be found by substituting — for w in the general formula. In the experiment the pressure is made equal to that of the atmosphere by moving the bulb n until the level on the two sides is the same. 24 ELEMENTS OF PHYSICAL CHEMISTRY. C. The value of this method (Bunsen's), which is adapted only to gases, is that but very small amounts of substance are needed for the experiment. It depends upon the relation of the velocity of outflow of a gas through a small aperture to the specific gravity of the gas. The formula is derived as follows: Let / be the difference between the pressure under which a gas exists and that of the atmosphere, and v be the volume of the gas which flows out in the unit of time. The total work done by the change in volume is then pv. Since this work is used to force the gas out, it is equal to the increase of kinetic energy of the gas, i.e., pv = i/imc', where c is the increase of velocity in the direction of the flow, and m is the mass of the gas. If equal volumes of different gases are considered, flowing through the same-sized openings under the same pres- sure, then pv — \/2m^c^, pv — \/2m^c^, or or, since the masses of equal volumes are proportioned to their densities t, : f, : : Vd^ : Vd^. The apparatus used by Bunsen is very simple. A glass tube having a very fine opening at one end, below which is a stop-cock, is clamped into a vessel of mercury. In this tube is a piece of glass rod to act THE GASEOUS STATE. 2$ as a float, while on the outside of it are two marks close together. The gas is now passed into the tube at a certain pressure, and the tube lowered into the mercury until the float is just at the lower mark. The cock is then opened and the time necessary for the passage of the float from the lower to the upper mark observed. Since the time is inversely proportioned to the velocity, we have By carrying out the experiment for one gas and then for hydrogen we can find the density of the gas in terms of hydrogen. The marks are placed quite close together and the fine opening is made by a needle- point, so that the pressure may be considered as con- stant throughout the experiment and the time of outflow is large enough to prevent experimental error. 9. Abnormal vapor-densities. Dissociation. — The specific gravity or density of the gases generated by heat from substances has been found in many cases to be too small, i.e., the molecular weights thus deter- mined are smaller than the theoretical values. There are but two possible explanations for this: either Avo- gadro's law does not hold, and the correct molecular weight cannot be determined from the density, or the substance decomposes in such a way that there are more molecules present than there should be. Since Avogadro's law has been found to hold in all other cases, the latter explanation is the accepted one. If 26 ELEMENTS OF PHYSICAL CHEMISTRY. a substance decomposes in such a way that from each molecule of the gas there are two others formed, then the vapor density must be one half what it should be. For each volume of the undecomposed gas we shall have two volumes of the gases formed from it, at the same temperature, since by Avogadro's law we have the same number of molecules in equal volumes. These two volumes, however, will weigh the same as the original one volume, so that the weights of one volume of the mixed gases will be one half that of one volume of the undecomposed gas. This decomposi- tion is called dissociation. Thus NH,C1 dissociates according to the scheme NH.Cl^ NH,+ HCl, where the sign ^ means that the reaction may go in either direction, according to the conditions. That these two gases, NH3 and HCl, are actually present in the vapor of NH,C1 can be shown in the following way (Pebel and Than): A lump of solid NH,C1 is placed in a tube upon an asbestos plug and the temperature of the tube increased until the NH,C1 volatilizes and dissociates in NH, and HCl. Since NH3 is the lighter gas, it diffuses more rapidly through the asbestos plug than the HCl; consequently on one side of the plug we shall have an excess of HCl and on the other an excess of NH,. The presence of these two products may be shown by passing a cur- rent of dry air through the parts of the tube on each THE GASEOUS STATE. 27 side of the plug and then over moistened Htmus paper, which will show the nature of the gases present. There are certain conditions under which alone those three products may exist together, which we shall study to a greater extent later. These condi- tions are summed up in the equation (10) Kp=pj,, where / is the partial pressure of the undissociated gas, while/, and/, are the same for the products of the dissociation. K is the dissociation constant of the gas, which depends only upon its nature and tem- perature. Although this equation was deduced theo- retically, for the present we may consider it as an experimental fact. For the partial pressure we may substitute the proportional term concentration, i.e., the number of molecules divided by the total volume. We have then (i i) Kc = c^c,. If NH.Cl is present in a closed vessel provided with a stop-cock and manometer, and the vessel is heated, NH^Cl gas is given off, which dissociates into NH, and HCl until equation (10) is fulfilled. If we observe the temperature and pressure, and allow some of the gas to escape, both factors decrease. As soon as the cock is closed again, however, the temperature as well as the pressure rises, until at the temperature before observed we have exactly the same pressure. 28 ELEMENTS OF PHYSICAL CHEMISTKY. For every temperature we find, then, a certain corre- spending pressure and vice versa. Further, if we introduce one of the products of dis- sociation the dissociated portions unite to form the undissociated portion. This is evident from equation (lo). If A is increased, then the product/,/, is larger than before and the equation does not hold. The only way possible for it to hold is for the pres- sure of the dissociated portions to decrease, i.e., for them to unite to form undissociated portion, thus increasing the left side and decreasing the right. Since iTis a constant, the increase on the left side must take place in the undissociated portion, i.e., if one of the products of dissociation is added to a system the un- dissociated portion increases. A solid substance going into gaseous form has a certain pressure (vapor-pressure) at which it sends its molecules into the vessel. This pressure depends upon the temperature. The volatilization continues until the counter-pressure of the molecules in the gas formed just equals the vapor-pressure of the substance. If there are already molecules of the substance in the gaseous space, then the amount volatilized will be smaller for any one temperature than it would other- wise be, but will continue until the same pressure is reached. If the substance going into gaseous form dissociates, then the products behave just as the molecules do in the previous case. If, then, NH.CI is volatilized into THE GASEOUS STATE. 29 a space containing NH,; the product pv at any one temperature, will be reached more rapidly than is the case in the absence of NH, gas, and less NH,C1 will volatilize. Since gases dissociate, it becomes important to determine to what extent the dissociation takes place. This it is possible to find from the relation of the vapor density to the dissociation, i.e., the dependence of both upon the number of molecules. If, for ex- ample, a is the percentage of the gas dissociated, i.e., the degree of dissociation, and we start with one mole- cule of the gas, then i — a is the undissociated por- tion. If there are n molecules of the products formed from one molecule of the gas the total number of molecules present at any time is «a + (i — «) = [i + (« — 0«]- The ratio, then, of i to i + ('^ — O"^ ^^^^ t)^ ^^ same as that of the vapor density as it should be to the vapor density as it is, i.e., I A I -f (k — l)a 9 ■ Where 9 is the vapor density as it should be (i.e., as it is without dissociation), and A is what it actually is, the degree of dissociation is then ■d-A a = {n — i)A' 30 ELEMENTS OF PHYSICAL CHEMISTRY. If a substance dissociates completely into two products its vapor density is l/2 what it should be: if into three, 1/3. etc. Examples. — NH,C1 v.d=^ nearly 1/2 what it should be; hence NH.Cl = NH, + HCU CO(NH,)ONH, z/.^= nearly 1/3: hence CO(NH,)ONH. = CO + 2NH,. 10. The kinetic theory of gases. — This theory was developed by Kronig (1856) and Clausius (1857). The fundamental idea of it is that the pressure of a gas is due to the motion of its molecules, i.e., to the kinetic energy which they possess. The molecules are assumed to be in rapid motion in straight lines, and cause the pressure by their collisions with the walls of the vessel. By the aid of this assumption it is possible to derive the gas laws and many other relations. We shall simply derive two gas laws, so as to become familiar with the theory. If the mass of a gas-particle is m and its velocity is c, then each collision with the parallel walls will exert a force equal to 2mc. If the molecule is perfectly elastic it receives back again the same velocity in the opposite direction. Assume a hollow cube full of such particles, and let the side of the cube be / and the number of mole- cules be 11. The motion takes place in all directions, so that we THE GASEOUS STATE. 3 1 shall take advantage of that law of mechanics by which any velocity may be divided into three com- ponents, parallel to the edges of the cube, the diagonal of which is the velocity. If c is the velocity and u, V, and w are the components, then where u, v, and w may assume any value from o to c, according to the direction. If we consider any mole- cule which moves in any direction the velocity com- ponent u will cause the pressure u 2mu—., u where y is equal to the number of wall collisions in the unit of time, for this number is proportioned di- rectly to the velocity and inversely to the distance, which is / the edge of the cube. Since the action upon two parallel cube faces is the total action upon all cube faces for any one mole- cule is tn , . . „ ., mc' 2-(«» + Z-" X W) = 2-j There are, however, n molecules; hence the total action of all upon the whole cube is 2 mnc' 32 ELEMENTS OF PHYSICAL CHEMISTRY. For the pressure, however, we wish that upon the unit of surface of one face, so we must divide this ex- pression by the surface of the cube, 6/'; we find then ^2mnc' mnc' I', however, is the volume of the cube, i.e., the volume which is occupied by the gas; therefore zmnc" _ mnc^ ^ 6v ~ 3v or J>v = i/^mnc*. m, n, and c are, however, all constants for constant temperature, so that we find pv = const., which is Boyle's law. If we consider two gases under the same pressure and at the same temperature the kinetic energy of the single molecules must be the same, i.e., m , in' ,. — .c = — .c . 2 2 If, further, we assume the volumes to be the same, then pv = pv' ; and since pv = i/'^mnc', , m , m' 2/l—nc' = 2/i—n'c'\ ^ 2 TME GASEOUS STATE. 33 which when combined with the previous one gives i.e., Avogadro's law. These equations which we have derived hold for vessels of any shape, for all vessels can be considered as being made up of a number of cubes. It is neces- sary that the molecules be perfectly elastic; this, how- ever, must be so, for it is impossible that they can be anything else. An incompletely elastic body loses energy by a collision and this energy appears as heat. Heat, however, in a body is caused by the motion of the molecules. The kinetic energy of one molecule cannot be transformed into anything else, for Clausius has proven that the atoms have no motion which is not proportional to that of the molecule. Conse- quently, since it is impossible for energy to be trans- formed in the case of a single molecule, that molecule must be perfectly elastic. II. Variation from the gas laws. The theory of VanderWaals. — Boyle's law does not hold accurately for any gases at high pressures. The error for pres- sures as high as two atmospheres is still so small that it may be neglected, and consequently equation (9) holds. At higher pressures, however, the error is considerable. In the case of hydrogen the product pv is always higher than it should be. This was ascribed by Natterer to the fact that the molecules themselves occupy some of the volume. If this cor- 34 ELEMENTS OF PHYSICAL CHEMISTRY. rection for the volume is b, then, instead of (9), we have {pv —U) = RT, where ^ is a constant for each gas. From the formula Natterer calculated the value of b and found it to be equal to 0.00082, at which point it remained constant, for pressures varying from looo to 2800 meters of mercury. At high pressure other gases behave just as hy- drogen; at other pressures, however, the variation is different. As the pressure decreases they vary less and less, until finally a minimum of variation is reached, when Boyle's lav/ holds. If the pressure is still decreased, the variation takes place in the opposite direction from what it does at high pressure, i.e., the product pv becomes too small. This shows that there is some factor in the behavior of these gases which is absent in the case of hydrogen. Van der Waals ascribed this to the mutual attraction of the molecules, which acts in the same direction as the pressure, mak- ing that larger than it seems to be. Imagine a thin layer of a gas. In this a molecule will be attracted by the others with a force proportional directly to their number, and inversely to the square of the dis- tance between them and the single molecule. The THE GASEOUS STATE. 35 square of the distance between them, however, is directly proportionate to the square of the density or inversely to the square of the volume. If / is the pressure and a the specific attraction, then (12) [pJ^^-)(^-b)^RT, which is the equation of Van der Waals. R is here equal to «(/„ + "w(^o ~ '^)' instead of a/'„z'„, as in (9). By this law we can understand just how gases vary from Boyle's law. For small pressure and large a volume the term — and b disappear in contrast to/ and V, and the gas follows the simple law. With in- creasing pressure, it a is large the volume is smaller than it should be, for the pressure is larger by the term -5. As this pressure increases and a becomes insigificant in value the term b, the portion of the volume occupied by the molecules, must be taken into account. All this is shown by (12) in the form a ab pv= RT \-~ +bp V V or, for constant temperature, a ^ ab , , ab When V is large and p is small -^ and bp disappear as 2,6 ELEMENTS OF PHYSICAL CHEMISTRY. compared with - ; when - = -^ -\- bp we have the minimum of the variation of /z/ and Boyle's law holds. These constants have been determined for the dif- ferent gases, and a few of the values found are given below: a CO, = 0.0115 SO, =0.039 Air = 0.037 H = 0.0000 And for b{= , X molecular volume) : Air = 0.0026 CO, — 0.003 H = 0.0067 12. Specific heat. The first principle of thermo- dynamics. — When heat energy is applied to a body the temperature rises. The ratio of the amount of heat supplied to the consequent rise in temperature is called the capacity of the body for heat, i.e., -j: = k. This term, k, depends naturally upon the original temperature of the body, its pressure, etc. The specific heat of any substance is the capacity for heat of the unit of mass, i.e., — — — 2- ^ m m dt ' It has been observed that the specific heat of a gas THE GASEOUS STATE. 37 depends upon the conditions under which it is deter- mined. If the gas is allowed to expand under its previous pressure the specific heat, Cp, is different from that obtained when the pressure varies and the volume remains constant, c^. Before considering the reasons for this, and finding the relation between the two values, it will be necessary for us to inquire into the nature of heat energy and its possible transforma- tions. ^ Mayer in 1841 was the first to develop the subject of the theory of heat and its transformations (or Thermodynamics) as we know it to-day. Before that date heat was considered as an actual substance, which could be made to enter or leave a body. Mayer first recognized it as an energy, which could be obtained from any other energy or turned into the latter; his principal work was to determine the equiva- lent of heat energy, i.e., a factor by which heat energy can be given in units of mechanical energy. He was led to this conclusion by the point already mentioned, that the specific heat of a gas for constant pressure is always larger than the specific heat of the same for constant volume. His reasoning was as follows: Since the specific heat at constant pressure is always greater than that for constant volume, there must be some condition in the former state which absorbs this extra heat. In the case of constant pressure the volume increases, and work must be done to overcome the atmospheric pressure, which would keep the volume 38 ELEMENTS OF PHYSICAL CHEMISTRY. constant. It is a logical consequence, then, to con- sider that this extra heat is simply the amount neces- sary to do the mechanical work of expansion. In other words, a certain number of calories are found to be equal to a definite number of mechanical units; this value for one calorie is the mechanical equivalent of heat. This term may be calculated as follows: The difference between the heats necessary to raise the temperature of i gram of air i° C. under the two conditions is Cp — f » = 0.0692 cal. This 0.0692 cal. is the heat which is equivalent to the work necessary to expand i gram of air 1/273 of its volume at 0°. Imagine i gram of air at 0° enclosed in a tube with a cross-section of i square centimeter. It will occupy the space of 773.3 cms. if the pressure is 76 cms. of Hg. ; for i gram of air occupies under these conditions 773-3 cc, which is the specific volume of air. An increase of temperature of 1° C. will expand this volume 1/273, ^•^•t 2-83 cms. The weight of the atmosphere, 1033 grams, will then be raised through this distance. The work necessary to do this is 1033 X 2.83 = 2923.4 gr.-cms., which is equivalent to 0.0692 calorie. For one calorie we have 2027 5:^6^2 = 42245 gr.-cms., which is the mechanical equivalent of heat. THE GASEOUS STATE. 39 That this reasoning is correct is evident, but as a further proof of it we have Joule's later determina- tion of the same factor in an altogether different manner. He transformed a known amount of me- chanical work into heat by friction, and found from the heat developed that I cal. = 42355 gr.-cms. From Mayer's work the great consequence is what is known as the first principle of thermodynamics. The energy of the world is constant, i.e., when energy seems to disappear it is simply transformed into another form. If, for example, a gaseous body is heated its internal energy is increased ; if the volume increases, however, a certain amount of this heat is used to overcome the atmospheric pressure, and the increase of the internal energy is less than would otherwise be the case. If U is the internal energy and the amount of heat dQ is supplied to the body, then dQ = dU-{-dW, where dW is the amount of work done by the body by virtue of the heat absorbed, and dU is the corre- sponding increase in the internal energy. If we con- sider the work as overcoming a resistance it is more readily handled. Suppose the gas to increase its volume, the pressure remaining constant ; then dJV = J>dv, where / is the intensity factor and v the capacity 40 ELEMENTS OP PHYSICAL CHEMISTRY. factor of volume energy. Substituting this value of dWm. the former equation, we have (13) dQ^dU-\-pdv, where dU and pdv are expressed in heat units. If these two are given in mechanical units, then (13) becomes / \ jn dU-\-pdv (13a) dQ = Xi— . where A is the mechanical equivalent of heat, i.e., the value of I calorie. We shall use the form (13), how- ever, always remembering that dU and pdv are ex- pressed in thermal units. The internal energy U ol a gas may be considered as a function of pressure and volume, of pressure and temperature, or of volume and temperature. Of these we shall only consider the case where temperature and pressure are variable. We have then p. J J The term -rr-dv, however, is equal to zero, for the av internal energy of a gas, according to Gay-Lussac's experiments, remains the same after a change in volume provided no external work is done. Equa- tion (13) becomes then dQ = %dT-^pdv. THE GASEOUS STATE. 4I If V is constant, i.e., dv = o, then dU , . . , . " This term, -p^, the increase in the internal energy caused by an increase of temperature, is, however, the specific heat for constant volume; hence dQ = -y—dT^ c^dT (for constant volume). dl If the pressure remains constant and the volume varies, i.e., if dv does not equal zero, then pdv does not disappear, and we have (14) dQ ^ c,dT ■\- pdv or dQ_ ,P_dv dT "'^ dT' dQ This term, -7™, under these conditions is the specific heat at constant pressure, i.e., ^^' '*- dT~ "^ dT- By differentiating (9) with respect to v and T we have pdv = RdT or ^dr 42 ELEMENTS OF PHYSICAL CtiEMISTRY. Substituting tliis in (15) we find (16) C, = C, + R, when Cf and C^ refer to the specific heats for one mol of gas, and R is the naolecular gas constant. If in the system no heat is absorbed or given up by conduction or radiation, i.e., dQ = O, we have an adiabatic process. From (9), by complete differentia- tion, we have pdv + vdp = RdT. If in (14) we eliminate dT, by this equation, remem- bering that C^ + R = C,, we obtain (17) dQ = ^vdp+^pdv. Since in an adiabatic process dQ = o, then 0=~^vdp+^pdv. If, then, we represent the ratio of the specific heats -?r by >^ (for air = 1.41), then vdp -\- kpdv = O or ,dv , dp k — ^ -i- = o. V ' / THE GASEOUS STATE. 43 And by integrating between the limits/, v, and/,, z/,, we have lp-\- klv= lp,-\- klv, or Iv - Iv^ (i8) or in another form (■9) MS* From (19) we see that for an adiabatic process this volume changes less for a change in pressure, or the pressure changes more for a change in volume, than is the case when the temperature remains constant (i.e., an isothermal process). In practice all changes are adiabatic unless they take place infinitely slowly, and we have p.-.p-.-.v^-.v,'. If the process is very slow and the temperature remains constant, then we have (Boyle's law) p^:p: WW,. If in (14) we eliminate pdv instead of dT {as in the derivation of (17)), then by the aid of (9) and (16) we obtain dT k — \ dp or 44 ELEMENTS OP PHYSICAL CHEMISTRY. or or (?)' = ©'"■• And finally from (19) (^■> f,=(r- The effect upon the temperature of a gas by an adiabatic compression (i.e., one which takes place in such a way that the heat generated is not removed, or that absorbed is not supplied, thus causing the temperature to vary) is shown by the following ex- ample: If air is compressed adiabatically at o" C. to 272/273 of its volume, i.e., the volume reduced by the 1/273 ps-ft, the temperature of the gas becomes, according to (21) ^=\^] =316.6, on the absolute scale, i.e., an increase of 43°. 6. By allowing a compressed gas to expand adiabati- cally the temperature of the gas is greatly reduced; this reduction can be calculated by aid of (20) (see Table I). THE GASEOUS STATE. 45 Table I. TEMPERATURES CAUSED BY ADIABATIC EXPANSION OF A GAS. mospheres). ^(initial from T). /'(final from T^t 100 71°. 5 - 201°. 5 200 58°. 5 - 213°. 5 300 52°. — 221°. 400 47° -9 225°. 500 44° -8 — 223°. a The temperatures are those of the gases as they expand. Owing to the specific heat of the vessels used, it is impossible to reach these temperatures in anything in contact with the gas, although very low temperatures may be thus obtained. From (16), i.e., as considered for molecular quantities, R is constant for all gases, and equal to the difference between the two specific heats. Why there should be a constant difference is self-evident, for the volume of a gram molecule of any gas under like conditions is a constant factor, and consequently by the law of Charles the amount of heat required for a certain expansion is the same for all gases. From (16) we can calculate the mechanical equiva- lent of heat. For air (7^=6.873 and C„ = 4.870, i.e., Cf,— C.„ = 2 cals. = R. R, however, is equal to 84800 gr.-cms. ; hence I cal. = 42400 gr.-cms. 46 ELEMENTS OF PHYSICAL CHEMISTRY. 13. Determination of the specific heat of gases. — The specific heat for constant volume, c^, cannot be determined directly with accuracy, since the vessel containing the gas absorbs so much more heat than the gas itself. Even under the most favorable condi- tions the ratio of the absorption of heat by the gas to that by the vessel is 1/55. From the specific heat for constant pressure and the ratio k = —, c^ can, however, be found indirectly. The specific heat" at constant pressure is determined by passing a certain volume of the gas, under constant pressure, and heated to a certain temperature, through the worm of a calorimeter and determining the conse- quent increase in the temperature of the water. We know then the number of calories which causes a certain temperature to exist in the known volume of the gas; and from this data it is easy to calculate the specific heat, Cp. The ratio /§ = — is determined in one of several ways. The method employing the velocity of sound is so well known that we shall not consider it here. The method which we shall consider is one involving the equations, we have just derived, and will serve to show their value. It was devised by Clement and Desormes. A glass balloon, holding about 20 liters, is provided with a brass stop-cock and a manometer. From this THE GASEOUS STATE. 47 vessel the air is partially rarefied and the pressure observed by aid of the manometer. Let this initial pressure be/„ and the atmospheric pressure be P. If the cock is now opened for half a second air will rush in until the external and internal pressures are the same. As the air goes in, however, heat will be developed, which, as it is not removed, will increase the temperature of the gas. By the process we have increased the pressure from /„ to P. If the initial specific volume is v^ and the final one is v, then k can be determined from the equation P _ a" The final specific volume is not known yet, however. To find this we wait until the flask and air have been reduced to the temperature of the surrounding air. If the pressure is p at the temperature t, then v,:v: -.p-.p,, or and 0. \ i>i k = log P — log/. log/ — log/," In one experiment with air P = 1.0036, /„ 0.9953, and/= 1.0088 atmospheres; hence k= 1.3524- 48 ELEMENTS OF PHYSICAL CHEMISTRY. 14. The second principle of thermodynamics. — By the first principle we have found that a certain amount of heat is equivalent to a certain amount of work. If we consider, however, the possible means of trans- forming heat into work we find that a certain condition must be fulfilled, i.e., the heat must go from a warmer to a colder body. In other words, heat can only pro- duce work by going from a higher to a lower tempera- ture. This condition has been expressed in other words. For example, heat of itself can never go from a colder to a warmer body zvithout work being used upon it. A perpetual motion of the second kind is impossible. A device to produce a perpetual motion of the second kind would be a machine which would run, for example, from the heat of the sea. It is related to the second just as an ordinary perpetual motion is related to the first principle. Since heat can only be transformed into work by a transference of heat from a higher to a lower tempera ture, it is important to know the relation which exists between the amount of heat transferred as heat and that transformed into work. For this investigation it is necessary that all the work done be external work, which can be readily observed, and then compared with the known amount of heat. We employ for this purpose a process which was originated in 1824 by Sadi Carnot and which is known as the cycle. The arrangement is such that the final state is identical with the initial one, i.e., THE GASEOUS STATE. 49 before and after the operation the body contains exactly the same amount of energy. Then the rela- tion between the heat transferred and the work done .'s very readily obtained. Naturally no heat must be lost by radiation or conduction, or the final result will be incorrect; for this reason the process is an ideal one and cannot be realized practically. Since there is no loss of heat by radiation or conduction, the process may go in either direction, i.e., it is reversible. This condition of reversibility is never obtained in practice, so that the relation which we find is the limit under the most favorable conditions. 15. The cycle. Entropy. — We assume the process to take place in four steps: I. Assume an ideal gas enclosed in a cylinder with a movable piston at a certain temperature and pressure. The cylinder is now placed in a heating-bath at the temperature J',, and the volume is allowed to increase under a constant pressure which is just greater than that of the atmosphere. By this expansion the gas will cool. Here, however, we assume heat to be absorbed by it to such an extent that the temperature remains constant. If the heat absorbed by the gas is Q^, its initial volume v^, and its final one v^, and the con- stant pressure and temperature respectively/ and T^, then the work done by the gas will be equal to / pdv. Since the temperature remains constant, equation (14) becomes ^<2i = o-\-pdv\ 50 ELEMENTS OF PHYSICAL CHEMISTRY. or, since / :: RT ~ ? V dQ.^ V or (22) Q.= ■ RTf-^. 2. The gas is next allowed to expand until the temperature falls to T^. For this, the new volume being v„ we have the relation <-> ?:=©'"■■ 3. Next the pressure is increased until the volume decreases to v^, heat being removed to the amount Q,, so that the temperature remains constant at T,. The work done here by the gas is — / *pdv ; we have then (24) a = rtA. 4. Finally, the gas is compressed adiabatically until the original volume v^ and the original temperature 7) are reached. This can be arranged by choosing v^ of the proper size. We have for this the relation We have thus carried the gas through a series of changes, and have finally the same state as that from THE GASEOUS STATE. $1 which we started. The amount of heat g, is absorbed at the higher temperature 7',, and a smaller amount 2, is given up at the lower temperature 7!,, and a certain amount has been transformed into work. The amount of heat which is equivalent to the work done is equal to =: (2i — Ga- The heat (2, has simply been trans- ferred from the temperature Tj to T^. The relation between Q^ and (2, is given by equa- tions (22) and (24) 2.^ ^_ By (23) and (25), however, !i = !i, i.e., /^=/!:i; henee i.e., the amounts of heat absorbed and liberated are pro- portional to the absolute temperatures. The amount of heat transformed into work is ^ = Gi — Qi' We have then and Q. - Q. T,- 7; <2. ~ T, ■ The heat transformed into work by any reversible process 52 ELEMENTS OF PHYSICAL CHEMISl^RY. is to that transferred from the higher to the lower tem- perature as the difference in temperature is to the lower absolute temperature. Or from the other standpoint, The work necessary to transfer a certain amount oj heat from one temperature to a higher one by a reversi- ble process is to the amount of heat as the temperature interval is to the final high absolute temperature. For r, = 0°, a will equal o and g, - G. = G„ i.e., at the absolute zero all heat is transformed into work. T — T For all higher temperatures ' „ — ^ is smaller than i. If the difference in temperature T^ — 7", is very small we can substitute for it dT, and as the differ- ence between the amounts of heat will then also be small, we may substitute dQ for Q, — Q^; we find then dQ_djr Q- r If now we consider the heat liberated as negative and that absorbed as positive, i.e., Q^ negative and <2, positive, then 'T' * 'T' ■I I J 2 or or <3., a, a, _- THE GASEOUS STATE. 53 where there are as many terms as there are tempera- tures. Q is the amount of heat absorbed or emitted. If the temperatures are close enough together, then the sign of summation may be replaced by that of integra- tion, and we have P r=°- This is the analytical expression of the second princi- ple for reversible processes. In words it means that for any reversible process the transformation of heat into work takes place in such a way that the sum of the amounts of heat absorbed and liberated, each divided by its corresponding temperature, is equal to zero. If T remains constant dQ = o, i.e., just as much heat is liberated as is absorbed, and no heat is trans- formed into work. The term / — , found as above, was called by Clausius the entropy. If this term is represented by s, then ds = 'J. T When dQ — o, ds = o, i.e., when a substance by a reversible change neither gives up nor absorbs heat its entropy remains constant. The two terms vary together and have the same signs. For this reason abiabatic changes are also called isentropic. i6. The factors of heat energy. — We have already found that the temperature is the intensity factor of 54 ELEMENTS OF PHYSICAL CHEMISTRY. heat energy. The capacity factor may be of two dE ^ dE ^ ^ ^ forms, c — —p (y^ dc =^ -^ , where dc and c are not di t proportional. The entropy is the ratio of the heat absorbed or Hberated at constant temperature to the absolute temperature of the process. In all cases where heat becomes latent, then, the capacity factor must ,- ^ fdQl CdE\ , ^^ be the entropy s ^= I ~ic=- I ^ I, tor the temperature then remains constant, the intensity factor being the temperature, as before. For this reason latent heat is often called entropic heat. We have in general, then, for all chemical processes T dQ = Tds. CHAPTER III. THE LIQUID STATE. 17. Distinction between liquids and gases. — Liquids are distinguished from gases by the fact that they possess a volume of their own, which, although dependent upon pressure and temperature, is not, as in gases, exclusively determined by them. The attraction between the molecules in a liquid must be greater than that in gaseous form, since the molecules are closer together, and the force of attraction is in- versely proportional to the square of the distance. In the liquid state this attraction must be greater also than the kinetic energy of the molecules, or the gaseous form would be assumed. Should this kinetic energy of the molecules be increased by heat, then the attraction will be overcome, and the liquid goes into the gaseous state, i.e., it boils. The external pressure has the same effect as the attraction between the molecules, for it prevents the liquid from going into the gaseous form. For this reason a liquid will boil at a lower temperature if the pressure is reduced. 18. The specific gravity and its determination. — The specific gravity, being the ratio of the weight of 55 S6 ELEMENTS OF PHYSICAL CHEMISTRY. the unit of volume to that of the standard, is equal to m the mass divided by the volume, i.e., — . The unit v of mass is the gram, and that of the volume is the volume occupied by one gram of water at 4° C. The specific volume, i.e., the volume occupied by one gram, is the reciprocal of the density. The molecular volume, i.e., the volume occupied by one mol, is then the molecular weight in grams divided by the density. The determination of the density in Physical Chemistry is made by aid of the pyknometer in the Sprengel-Ostwald form (Fig. 2). It consists of Fig. 2. a tube of very thin glass, with a capacity of 5-50 cc, which has fused into it two capillary tubes (a and b), through which the tube may be filled with liquid with great accuracy at any temperature. Upon the one capillary tube id) is a mark up to which the tube is filled in all experiments. The filling to the mark takes place at a certain temperature, the tube being THE LIQUID STATE. 57 weighed at any temperature, the liquid expanding or contracting according as the latter is higher or lower than the former. If /, is the weight when filled with the liquid, /,, when filled with water, and/, when empty, then /, — /, = weight of the liquid; /, — /, = weight of water. The term /, — /, at 4° C. gives also the volume of the pyknometer in cubic centimeters. Since the weight of the empty apparatus is made up of the weight of the glass plus the weight of the air con- tained minus that of the air displaced, the weight found, /,, is the true weight. This follows from the fact that the glass is very thin, so that the external volume is practically equal to the internal volume, thus causing the two corrections to cancel. In the case of the weight when filled with water or liquid this is not so, for the weight of the air con- tained disappears, and we must correct for that dis- placed. Since this causes the body to appear lighter than it is by the weight of the volume of air dis- placed, this weight of the air must be added to the apparent weight to find the true one. One cubic centimeter of air at o" and 76 cms. of Hg weighs 0.0012 gram. The true weights will then he />,-{- ^ and/, -J- J, instead of/, and/,, where ^ = (/, — /,)o.ooi2 gram 58 ELEMENTS OF PHYSICAL CHEMTSTRY. is the weight of the air displaced. The specific gravity, then, of the liquid is . ^ /3 + ^ - A A + ^ - A' 19. Connection between the gaseous and liquid states. — If at constant temperature a gas is sub- jected to a constantly increasing pressure, its state may change in one of two ways according to the con- ditions: a. This case has already been considered under gases. The volume at first changes more rapidly than the pressure, next in the same ratio, and finally more slowly. When the pressure becomes very high a further increase has but a slight effect upon the volume. b. Here the relation between pressure and volume is quite different, although the first step is the same, i.e., the volume changes more rapidly than the pres- sure. The ratio here, however, in contrast to «, in- freases continually. When a certain pressure is reached a new phenomenon is observed: the gas is no longer homogeneous; one part has separated which behaves differently from the rest, i.e., the gas is partly liquefied. For a constant temperature this liquefying pressure remains constant, while the volume decreases steadily, i.e., for one pressure we have a whole series of volumes. An increase in the external pressure has no effect upon the internal pressure THE LIQUID STATE. 59 (which still remains as it was), but it causes the volume to decrease more rapidly. Only after the whole gas has been liquefied can the pressure be increased. Then, however, we shall compress only the liquid, just as in the last stage of a we compressed the gas. The condition which causes a gas under compres- sion to follow a or 3 is the temperature. If this is above a certain point, which depends upon the nature of the gas, process a will be followed ; if below this point, process b. This was first recognized by Andrews in 1871. If, for example, we compress CO, gas, keeping the temperature under 30°. 92, the volume changes more rapidly than the pressure, and at a pressure of 35.4 atmospheres, at 0° C, the gas con- denses to a liquid. The higher the temperature, under 30°. 92, the higher the pressure must be to cause condensation. Thus : t p 31°, impossible to liquefy it. 30°. 92 = 73.6 atmospheres. 30° = 73.0 13°. I = 48.9 0° = 35.4 t / — 21° = 21.5 atmospheres. — 40° = ri.o " - 59°-4 = 4-6 - 70°.6 = 2.3 — 78° = 1.2 Andrews called this last temperature (31° for CO,), which is just above the liquefying-point, the critical temperature. Correspondingly we call the pressure which is necessary to liquefy the gas, just under the critical temperature, the critical pressure (73.6 atmos. at 30°. 9), and the volume which the gas occupies, under these two conditions, the critical volume. 6o ELEMENTS OF PHYSICAL CHEMISTRY. The best method of showing the behavior of a gas under compression is by plotting a curve in a system of coordinates where the pressures are laid out upon the axis of ordinates and the volumes upon the axis of abscissae. In these two curves the horizontal parts, which show constant pressure for varying volume, in- dicate that the gas liquefies. The vertical parts refer to the liquid state, while all others refer to the gaseous state. Fig. 3 shows the behavior of CO, and air, the ordi- nates being pressures in atmospheres. At 3i°.i CO, there is no horizontal part, i.e., no varying volume for constant pressure; the gas does not liquefy, since it is above its critical temperature. For all tempera- tures below 31°. I the horizontal part is present, i.e., the gas liquefies under a sufficient pressure. Under all these pressures air remains a gas aind behaves as CO, does when above 31° C. 20. Vapor-pressure and boiling-point. — All liquids have a tendency to go into the gaseous form. The pressure which strives to accomplish this is smaller than the attraction of the molecules in the liquid plus the atmospheric pressure which increases it, so that under these conditions the gas does not form. If we heat the liquid, this tendency to go into the gaseous form increases until the attraction and the atmospheric pressure are overcome, and the liquid boils and goes into the gaseous state. If this temperature for one liquid is lower than for another, then by reducing the THE LIQUID STATE. 6i external atmospheric pressure of the latter the same temperature is obtained. In general, then, a liquid boils when its tendency to go into the form of gas, i.e., its vapor-pressure, under these conditions, becomes equal to or greater than the Fig. 3. counter-pressure over it. In other words, the vapor- pressure is equal to that external pressure for any one temperature at which the liquid and its gas can exist together in all proportions. Every external pressure corresponds to a certain temperature, and these two 62 ELEMENTS OF PHYSICAL CHEMISTRY. factors increase and decrease together. At the criti- cal temperature the significance of vapor-pressure is naturally lost. For every pressure, however, under the critical one there is a certain temperature at which gas and liquid may exist together in all proportions: this temperature is the boiling-point at that pressure. Correspondingl)7 for every temperature under the critical one there is a certain pressure at which gas and liquid may exist in equilibrium in all proportions: this is the vapor-pressiire at that temperature. 21. The heat of evaporation. — For the transfor- mation of a liquid into its gaseous form a considerable amount of heat is necessary. There are two causes for this absorption of heat. The volume must be in- creased against atmospheric pressure, and the attrac- tion of the molecules must be overcome in order that they may be separated by the same distance as they are in the gaseous state. The former is of the least value. Its amount may be calculated irom pv z= R7 = 2 T'cals. ; * i.e., for every mol of water- vapor formed from liquid water 2 X (273 -j- 100) = 746 cals. are used for the expansion. Since the heat of evaporation of I mol (18 grams) of H,0 to steam at 100° is 9650 cals., and the amount necessary for the expansion is but 746 cals., it will be observed that this molecular attraction is very great. * Where v represents the volume of the gas, which is so large that that of the liquid may, be neglected, pv then represents the actual work of expansion against the constant atmospheric pressure/. THE LIQUID STATE. 63 The ratio of the molecular heat of evaporation (= molecular weight multiplied by the heat of evap- oration of I gram) to the absolute boiling-point has been found experimentally to be a constant for a very large number of organic liquid substances, and equal to approximately 21, i.e., Mr ^ = 21. By aid of this equation unknown heats of evaporation may be calculated (with a possible error of 5^) from the molecular weight and the boiling-point. 22. Volume relations. — In calculating the molec- vt ular volume, -7, of various substances Kopp found that for corresponding ethyl and methyl compounds there was a difference in molecular volume of 22 units. This means that- the molecular volume of CH, is 22. Further, he found by substitutions that the molecular ■volume of C is equal to twice that of H ; hence CH, = H, = C,= 22, i.e., the molecular volumes for C and H are C = II, H= 5.5. Since the molecular volume of water at 15° is 18.8, that of O in hydroxyl combinations is 18.8 - (5.5 X 2) = 7.8. For O in the carboxyl group we find a value of 12.2. The volume occupied by one mol of the compound C;«H«0/0^, 64 ELEMENTS OF PHYSICAL CHEMISTRY. where Op = carboxyl O and Oq — hydroxyl O, is then V = \\m ■\- S.5« + 12.2/ + 7.8$f. These figures are those found by Kopp. Some have since been revised, and in addition it is now possible to find the molecular volume in the state of solution as well as in the liquid state. For further details the student is referred to Ostwald (AUg. Chem., Vol. I., 2d ed.). 23. Refraction of light. — La Place in determining the velocity of light in different media found the re- lation fi^ I — = constant for all temperatures d when n is the refraction of light and d is the density of the medium. For many liquids this holds true, approximately, but Gladstone and Dale found empiri- cally a relation which holds more exactly and for many more liquids than the former. This is = — = constant. d Further, in the case of a mixture the constant is found to be equal to the sum of those for the ingredi- ents. We have then n — \ ;?, — I «, — I where / is equal to 100, and /, and /, are the per- centage by weight of the two ingredients. By aid of this it is possible to make an optical analysis. THE LIQUID STATE. 65 Example. — An unknown mixture of amyl and ethyl alcohols gives n = 1.3822, d = 0.8065. What is the percentage composition of each in the mixture ? For amyl alcohol «, = 1.4057, d^ =0.8135; for ethyl alcohol «, = 1.3606, ^, = 0.81 1 ; hence 100 X 0.4738 = A X 0.4987 + A X 0.4501 ; and since/, +A = 100, we find/, = 48.8, /, = 51.2. By direct weighing it was found that/, is 48.9 and/, is 51. 1. This formula holds also approximately for solutions, i.e., for finding the coefficient of refraction in the solid state from that of the solution and the solvent. 24. Surface-tension. — The surface-tension x of a liquid is the force tvliicJi is necessary to form a surface --'trriira' Fig. 4. of it one centimeter in length. Some liquids wet the walls of a glass tube, while others do not, and upon this depends the shape of the surface formed. Im- agine a plate of glass suspended vertically in a vessel 66 ELEMENTS OF PHYSICAL CHEMISTRY. of water, i.e., a liquid which wets the glass. The glass will then be wet as high as a (Fig. 4) and the surface abc will decrease in size by assuming the shape afic. In doing this a certain weight of liquid will be raised. When the weight P of this portion raised is equal to the surface-tension for that length of surface equilibrium will be established, i.e., Ix =^ P or P If now instead of using a plate we use a capillary tube, with a radius equal to r, I will become 2nr and the weight raised will be P = 2nrx. On the other hand we have P = nr'hs, where h is the height of the liquid in the tube and s is the specific gravity of the liquid. We have then 2 7trx = Ttr'As or X = \/2hrs, i.e., the surface-tension of a liquid can be obtained from its specific gravity aud the height of a capillary column with the radius r. THE LIQUID STATE. 6j Although this method is difficult to carry out, it has given accurate results, and illustrates the principle of surface-tension better than the others. At the critical temperature the surface-tension becomes zero. From this fact it is possible to calcu- late from the surface-tension of a liquid its critical temperature. Empirically the surface-tension has been found to vary with the temperature as follows: where x and x„ are the surface-tensions at t and o°, and /? is the temperature coefficient of the surface- tension, i.e., its change for a change of i°. Let X be the surface-tension at the critical tempera- ture (i.e., X = o) and x„ that at any observed tem- perature. If we know /3 and x„ by experiment it will be possible to find i, which will then be the difference in degrees between the critical temperature and that of observation. If jr„ referred to the surface-tension at 0° C, then t would give directly the critical tem- perature in centigrade degrees. For example, for PCI, we have x; (at 75°.4) = 3.017, B = 0.0137; hence o = 3.017 — 0.0137?: 3.017 t = = 220. 0.0137 68 ELEMENTS OF PHYSICAL CHEMISTRY. The critical temperature is then A few results as obtained in this way are compared in Table II with the critical temperatures determined directly. Table II. Substance. t^ obs. i . calc. Methyl acetate 238 240 Ethyl acetate 275 257 Methyl propionate 281 263 Ethyl propionate 296 281 Propyl propionate 320 305 Isobutyl propionate 324 319 Phosphorus trichloride 286 295 This agreement is astounding when we consider the difficulty in deterrriining the surface-tension and its temperature coefficient, as well as the empirical nature of the original equation. CHAPTER IV. THE SOLID STATE. 25. Remarks. — A solid is a substance which pos- sesses a form of its own, i.e., its shape is not depend- ent upon that of the vessel in which it is. In Physical Chemistry the properties of solids are of less importance than those of the other two states of aggregation, so that here we shall not consider them to any great extent. 26. Atomic heat. Law of Dulong and Petit. — The atoms of all elements have the same capacity for heat. A few have been found to give different values, but in general for all elements we find that atomic heat = atomic weight X specific heat = 6.34. This is the law of Dulong and Petit. By the aid of it it is possible to determine the atomic weight from the experimentally determined specific heat for one gram. We have . , 6.34 atomic weight = t^ — -. — - — ? . ^ specific heat of i gram 27. Changes in the state of aggregation. — The most important process concerning solids, for our pur- 69 70 ELEMENTS OF PHYSICAL CHEMISTRY. pose, is their formation from the liquid state, or the formation of the liquid state from them. If a liquid is cooled very carefully it is possible to reduce its temperature below the solidifying-point and yet have no solid formed. If the vessel is jarred, or a crystal of the substance thrown in, crystallization takes place immediately, and the temperature rises rapidly to the true solidifying-point. Besides this change in temperature we have also one in volume, since the specific volume in the one state differs from that in the other. When a gas condenses its form to the liquid state heat is given off. In the same way heat is also given off when a solid is formed from a liquid, and this is the cause of the rise in the temperature spoken of above. The heat which is liberated by a liquid solidi- fying (or absorbed by a solid liquefying) is called, for one gram of substance, the latent heat of solidification {of fusion). This heat, referring to one mol of sub- stance (heat for i gram X mol weight), is called the molecular heat of solidification {fusion). The effect of pressure upon the solidifying-point may be derived by the consideration of equation (6). This becomes, then, for the equilibrium between heat and volume energy sdT = cdi = vdp. Or, since s can only be determined as a difference between two states, (j, — s^dT = (v, — v^yiT, THE SOLID STATE. 7 1 where j, is larger than j, and refers to the state which absorbs heat. Since the entropy s varies with the amount of heat, we may write (t - %)^^ = (''■ - ^")^^- For all processes of solidification this term ~,' — ^' w becomes ^, where w is the latent heat of fusion of one gram of the liquid. We have then (26) T{v, - Z.J dT or (27) dT _ Tiv, - v^) dp w Since z/, refers to the state which, when formed, absorbs heat, it will represent the volume of liquid in any change from the liquid to the solid state. If the specific volume of the solid is greater than that of the liquid, i.e., if the solid floats upon the liquid (ice in water), the freezing-point will be depressed by pres- sure. If, on the other hand, the specific volume of the solid is less than that of the liquid, i.e., if the solid sinks, the solidifying-point will be raised by in- creased pressure. In the former case the pressure will tend to keep the substance in the liquid form, since that has the smaller volume, and a lower temperature will be neces- 72 ELEMENTS OF PHYSICAL CffEMlSTJ^Y. sary to produce the solid. In the latter case the formation of the solid will be aided by the pressure, and consequently the temperature need not be lowered to such an extent. The actual effect of pressure upon the solidifying- point is, however, not large. Thus an increase of pressure up to 8.1 atmospheres depresses the freezing- point of water but 0.059° C. Since at the solidifying- or freezing-point the solid and liquid exist in contact in all proportions, the vapor-pressure of the solid must be equal to that of the liquid, both at the same temperature. That this must be so follows from the following process: Imagine in the annular-shaped vessel Fig. 5 water Fig. 5. at b, ice at a, and the vapor of the two at c: all being at the temperature of 0° C. If the vapor-pres- sure over the water at b is greater than that over the ice at a a distillation from ^ to « must take place. By the evaporation at b heat will be absorbed, and by the condensation of this vapor to ice at a, under THE SOLID STATE. 73 the diminished pressure, heat will be liberated. A layer of ice will thus be formed at b, while the heat liberated will melt an amount of ice at a. We would have, then, an exchange of heat at the same tem- perature. If this were possible perpetual motion would be possible; since, however, the latter is im- possible, the vapor-pressure of the solid must be the same as that of the liquid, both having the same temperature. CHAPTER V. SOLUTIONS. 28. Definition of a solution. — A solution is a homogeneous mixture which cannot be separated into its constituents by mechanical means. The power to form solutions varies with the state of aggregation. Thus for gases it is unconditioned, while for solids, the opposite extreme, it is small, though still present. 29. Gases in liquids. — A true solution here is one in which there is no chemical reaction between the liquid and the gas, i.e., the gas may be expelled by heat. Henry's law, which was verified by Bunsen, enables us to find the amount of gas which will be absorbed. When a gas is absorbed in a liquid the amount dissolved is proportional to the pressure of the gas. Since, however, pressure and volume (at con- stant temperature) are inversely proportional, the law may also be expressed as follows : a given amount of liquid absorbs, at any pressure, the same volume of gas. If now we introduce the idea of concentration (i.e., the amount of substance divided by the total volume), instead of the pressure, we obtain also another form 74 SOLUTIONS. 75 for the law : The amount of gas dissolved by a liquid is such that there is always a constant ratio between the concentration of the gas in the liquid and in the space above it. This ratio does not depend upon the actual concentrations, but only upon the ratio of the two. The size of this ratio is dependent only upon the nature of the liquid and its temperature. If, instead of a single gas, a mixture of gases is dis- solved this law also holds, in that each constituent is absorbed from the mixture to the same extent as if it were present alone. The condition of the molecules of gas in the liquid is of interest. Dalton offered the first theory for this by assuming that the gas-molecules simply fill up the space between those of the liquid. This, however, is not in accord with the facts, for an increase in tem- perature increases the separation of the molecules of the liquid, and yet decreases the amount of gas absorbed. What probably does take place is this: There is a certain attraction between the molecules of the gas and those of the liquid. The molecules of the liquid on the surface, then, attract those of the gas until the ratio of concentration is reached. This layer of the surface is, however, no longer in equilibrium with the rest of the liquid, so that the layer sinks, dividing the gas, with which it is saturated, with the other layers with which it comes in contact. The new surface layer thus formed absorbs the gas (although to a smaller 7^ ELEMENTS OF PHYSICAL CHEMISTRY. extent, for the pressure is reduced) just as the original layer did, and goes through the same process, until finally the mixture is homogeneous and in equilibrium with the gas. If the pressure is increased more gas dissolves, for the pressure acts in the same direction as the attraction of the molecules, and increases it. If the temperature is increased, however, the liquid molecules have a greater tendency to go into the gaseous state, and the mutual attraction is diminished. The gas absorbed in this way is not to be considered as liquefied; its molecules simply become constituents of the liquid, just as those of the latter are constitu- ents. When a gas is absorbed in a liquid there is always a change of volume in the latter. As a general rule the less compressible a gas is the greater is the in- crease of volume caused by it. The chief difficulty experienced in compressing a gas, it will be remem- bered, is caused by the volume occupied by the mole- cules themselves, i.e., b in the equations of Natterer and Van der Waals. We should expect, then, that the increase in volume would be proportional to the term b for the gas in question. That this is so is shown by Table III. / Table III. Gas. Az- b O O.OOII5 . 000890 N 0.00145 0.001359 H 0.00106 0.000887 SOLUTIONS. 77 This shows that the volume of gas absorbed is reduced almost to that occupied by the molecules in the gaseous form. The pressure exerted by the attraction of the molecules of gas and liquid must be fully 3000 atmospheres in order to reduce the volume of the gas to such an extent. 30. Liquids in liquids. — We have in this case three possibilities: 1. No solubility, i.e., the formation of two layers and no homogeneous mixture. 2. A mutual solubility, but not in all proportions (as water in ether = 3^ and ether in water = 10^). 3. The same relation as with gas, i.e., a solubility in all proportions (as water and ethyl alcohol). The first and second possibilities are difficult to dis- tinguish, for an increase in temperature will cause the one constituent to vaporize, and this vapor will dis- solve in the other liquid, thus removing the mixture from the first to the second class. The most important point for us to consider here concerning these mixtures is the manner in which they evaporate. Since the three cases, as far as this prop- erty is concerned, reduce to two (i.e., since 2 is identical with 3 when only one layer is present, and with I when there are two layers) we shall consider first («) cases where two layers are present, and second {b) those for which the mixture is hoino- geneous. a. A liquid which consists of two constituents, 78 ELEMENTS OF PHYSICAL CHEMISTRY. present in two layers, has for its vapor-pressure the sum of those of the constituents. The boiling-point of such a mixture, then, is that temperature at which the vapor-pressure becomes equal to the pressure of the atmosphere; it lies still lower, therefore, than the boiling-point of the lowest boiling constituent. This is difficult to observe experimerttally by direct heat- ing, owing to the bumping. If, however, the vapor of another boiling liquid is passed through the mix- ture the whole is heated evenly and to the same degree, and the mixture distils below the temperature at which either constituent alone will do so. Assume the vapor-pressures of the constituents to be/i and/,; the volume of each in the vapor which distils over will be proportional to these pressures. The weight of the vapor of each is equal to its den- sity multiplied by its volume, or by what is propor- tional to it, its vapor-pressure. We have, then, the weights ($■, and q^ of the constituents in the vapor from the proportion If the vapor-pressures of the two constituents and their weights in the vapor are known it is possible to determine the vapor-density of the one in terms of the other. Naumann was the first to do this, using the equation Pi A SOLUTIONS. 79 All the above holds true for all mixtures which form two layers. If the two layers each contain a certain amount of the other the vapor-pressure is still equal to the sum of those of the two layers, i.e., the mixtures are to be treated just as though they were the pure constituents. With regard to the vapor-pres- sures of the two layers formed when the two constit- uents are mutually soluble to a slight degree, Kono- walow has shown that the liquid B when saturated with A has the same vapor-pressure as the liquid A wheri saturated with B, both being at the same temperature. From this it is clear that equal amounts of each layer must distil over at any one temperature, so that it is impossible to separate the two constituents by distil- lation so long as two layers are present. If, however, at first a much smaller amount of one layer is present than of the other the two layers will distil over in equal amounts, until finally one layer will be left behind. This naturally will then behave as any other homogeneous mixture (see b). b. When the liquids mix in all proportions, or in general when there is only one layer of liquid, then it is possible to make a complete separation of the constituents by a fractional distillation, provided the vapor-pressures of the two differ. The vapor given off by a homogeneous mixture has a different composition from that of the liquid, and its vapor- pressure is no longer equal to the sum of those of the constituents; it depends upon the action of the So ELEMENTS OF PHYSICAL CHEMISTRY. liquids upon one another and upon the vapors given off. If we use a system of coordinates in which the vapor- pressures are laid out upon the axis of ordinates, and the percentage composition of the liquid upon the axis of abscissae, we find that the vapor-pressure never reaches the sum of those of the constituents, but at times goes even below the smaller of these. Fig. 6 Fig. 6. gives three types of curves thus obtained, which will illustrate the principle involved. We find in I a maximum of the vapor-pressure corresponding to a certain strength of solution. In II this maximum has disappeared, while in III the opposite extreme is observed, i.e., a minimum of the vapor-pressure which corresponds to a certain composition. I. Mixtures which correspond to this type possess a maximum of the vapor-pressure for a certain com- position, i.e., the lowest boiling liquid is a solution of a certain strength. An example of this is a 75^ solu- tion of propyl alcohol in water. This "J^j, solution will distil at the lower temperature from any propyl- alcohol solution until the constituent which was pres- ent at first to the smaller extent has disappeared, when the other will remain behind in the almost pure SOLUTIONS. 8 1 state. Thus if we start with a 50^ solution of propyl alcohol a 75^ solution will be distilled until all the alcohol is removed and water is left in the flask. If we start with a 90^ solution of the alcohol the 75$^ solution will be given off until all the water has been used up and pure propyl alcohol remains behind. II. For mixtures of this type we can make a com- plete separation by distillation, i.e., obtain both products in the pure state. The vapor-pressure of all mixtures lies between those of the two constituents; consequently the one with the higher vapor-pressure will be given off in the almost pure state, leaving the other behind. Solutions of ethyl and methyl alcohol in water belong to this class. III. Here we have a minimum of the vapor-pres- sure corresponding to a certain composition, A solution of this composition will, then, always be the last to be distilled, since its boiling-point is the highest of all possible mixtures. This is just the opposite of case I, where the solution of a ce1:ain composition is given off first. All solutions, for example, of formic acid and water will distil off an amount of one con- stituent until the solution remaining contains 70^ of formic acid. A 90^ solution will give off almost pure formic acid until just enough remains to form a 70^ solution. A 50^ solution will give off water until the 70^ solution is left behind. 31. Solids in liquids. — When a soluble solid comes in contact with a liquid it dissolves. When no more 82 ELEMENTS OF PHYSICAL CHEMISTRY. , substance is taken up by the liquid at any one tem- perature the solution is said to be saturated at that temperature. There are two general methods of making a saturated solution; but in either it is always to be remembered that in order for a solution to he saturated the solid substance must always be in contact with it. I. The substance, in the form of a powder, is brought into the liquid, and agitated at a certain tem- perature until no more is dissolved; or, II. The solution is made as above, but at a higher temperature than the desired one, to which it is reduced later. Both these methods give the same result if properly carried out, although the latter is quicker and so to be preferred. 32. Osmotic pressure. — If in a tall jar we place a layer of pure water over a colored solution we observe after a time that the whole liquid is colored. The colored molecules have diffused through the liquid until a homogeneous mixture results. This" action shows that there must be a certain tendency for the molecules to get as far apart as possible, which causes them to go from the solution into the water. This deduction was first made by Pfeffer, who called the pressure thus exerted by the molecules in a solution the osmotic pressure. If a vessel is provided with a partition of such a nature that it allows free passage to the molecules of the solvent, but not to those of SOL UTIONS. 83 the solute, and we have water on one side and a solu- tion on the other, the molecules of the solute will exert a pressure upon it; this is the osmotic pressure of that solution. Semipermeable partitions of this sort were found by Pfeffer, so that he was able to actually measure the different osmotic pressures in terms of a column of mercury. An apparatus by which the principle of Pfeffer's measurements may be understood is shown in Fig. 7. The cylinder A is Fig. 7. made of porous clay, and is designed to support the semipermeable film. To produce this the pores of the cylinder are filled with a solution of copper sul- fate. After the excess of this is removed the cup is filled with a 3^ solution of potassium ferrocyanide, and allowed to stand for a day in a solution of copper sulfate. In this way the pores are filled with a film of copper ferrocyanide, which is permeable to the molecules of water, but not to those of many salts. The process of measuring is as follows: The now semi- 84 ELEMENTS OF PHYSICAL CHEMISTRY. permeable cylinder is washed out and filled with the solution to be measured. The rubber stopper CC which is attached to a long, narrow glass tube, is now inserted jn such a way that the solution rises a short distance in the tube, i.e., the cylinder is entirely filled with solution without air. The cork BB is now fas- tened in the vessel DD, which is filled with water. The liquid rises slowly in the tube until equilibrium is reached, i.e., until the weight of the column of liquid is just equal to the osmotic pressure. The height of the column thus represents the osmotic pressure, pro- vided the specific gravity of the liquid is known. The molecules of the solute in the cylinder strive to get through the walls into the water; since, however, the film is not permeable to them, they exert a pres- sure upon it. Equilibrium can also be reached, how- ever, by water going into the vessel and reducing the concentration of the solution. Since in nature all tensions between bodies are equalized, the above course is the only one which can be followed. Water then goes into the semipermeable cell until the weight of the column of liquid is just equal to the osmotic pressure. The force with which the water goes in is, of course, equal to that with which the molecules of solute would go out, for in each case the final result would be the same, i.e., a homogeneous mixture. It is quite evident that it is not the water itself which exerts the pressure upon a semipermeable dia- phragm, for its molecules have a free passage through SOLUTIONS. .85 the film. Further, equal amounts of solute in equal volumes of different solvents give the same osmotic pressure, i.e., the osmotic pressure depends upon the solute and is independent of the nature of the solvent. The nature of the semipermeable partition has also no action upon the value of the osmotic pressure.. That this must be true can be proven by the following process ; Imagine A and B (Fig. 8) to be two unlike Fig. 8. semipermeable partitions placed in a cylinder of glass. Between these assume the solution and lower the tube in a horizontal position in a vessel of water. If the pressure P is exerted by the partition A, and /, (a smaller one) by B, then water will go through A until the internal pressure P is reached. Since B allows only the pressure/, however, this pressure P can never be reached, so that we should have a steady current of water going from A to B. This, however, would be a perpetual motion, which is impossible; hence the partition A must give rise to the same osmotic pressure as the partition B. Osmotic pressure is, then, independent of the nature of the solvent and the partition, the latter simply being a means of making it visible. 86 ELEMENTS OF PHYSICAL CHEMISTRY. Pfeffer measured, by means of an apparatus of the type of the one described, the osmotic pressures which exist in a large number of solutions and found them in many cases to be very great. A few results for sugar solutions of varying concentrations at 15" C. are given in Table IV. Table IV. c P Pic c P Pic I 53-8 53-8 4 208.2 52.1 I 53-2 53-^ 6 307-5 51-3 2 101.6 50.8 I 53-5 53-5 2.74 151.8 55-4- where c is the percentage composition of the solution, p is the osmotic pressure, in cms. of mercury, and p/c is the ratio of pressure to concentration. Considering the difificulties in measuring and the imperfections in the semipermeable film, the ttrm p/c is to be considered as constant; hence the osmotic pressure is proportional to the concentration of the solute. Further results at different temperatures showed that the osmotic pressure is proportional to the absolute temperature . Van't Hoff in 1887 from this much knowledge derived, by the aid of thermodynamics, an analogy between the behavior of gases and substances in solu- tion. Here, however, we shall not go into his reason- ing, but shall consider how his results may be arrived at in a more simple manner. As we have seen by Pfeffer's results, the term//f remains constant in solutions where / is the osmotic S0LU2-I0NS. 8; pressure and c is the concentration. \/c, however, is equal to v, the volume ; we have then p/c = pv. Since now the osmotic pressure is proportional to the absolute temperature, as is also the term pjc, we have pv = constant X T: This equation is so much like the one for gases that it immediately suggests that there is some kind of a connection between them and substances in solution. If now the value of the constant is determined, then by comparing it with the gas constant it is possible to find this connection. Since we are to use the molec- ular gas constant in the comparison, it will be neces- sary also to find this constant for i mol. A i^ solu- tion of sugar at o° gives an osmotic pressure equal to 49-3 X 13-59 = 671 grams per sq. cm. Since the molecular weight of sugar (Ci,HjjO„) is 342, the volume in which i mol is dissolved (to give a i^ solution) is 34200 cc. We have then p^v. 671 X 34200 constant — ^^^ = ^r = 84200 gr.-cms., T 273 while the molecular gas constant is R = 84800 gr.-cms. The constant is the same, then (within the experi- mental error), for the gaseous state and the state of a 88 ELEMENTS OP PHYSICAL CHEMISTRY. substance in solution; consequently for each state, considering i mol, we have pv = RT. Van't Hoff summed up his results in the following form : the osmotic pressure of a substance in solution is the same pressure which that substance would exert were it in gaseous form at the same temperature and occupying the same volume. Since i mol of gas at o° and "j^ cms. of Hg occu- pies the volume of 22.4 liters, at the volume of i liter it will have the pressure of 22.4 atmospheres (Boyle's law). This pressure, 22.4 atmospheres, is then the osmotic pressure which is exerted by i mol of solute in I liter of solution, i.e. , by a normal solution. From this osmotic pressure, 22.4 atmospheres, it is very easy to find that for any other concentration given in terms normal. Thus a n/\o solution gives an osmotic pressure of 2.24 atmospheres. If the osmotic pressure of a solution is determined it is then possible to find from it the molecular weight. A 2^ solution of sugar gives, at 15° C, an osmotic pressure equal to 101.6 cms. of Hg. If this were a normal solution it would give a pressure equal to 22.4 X 76 = 1702.4 cms. of Hg. The 2^ solution contains 20 grams in the liter; hence 20: M: : 101.6: 1702.4; M= 335. SOLUTIONS. 89 This result is accurate enough to show that in water the molecule of sugar is of the same size as is given by the formula. This holds true for most of the methods for molecular weight determination which we shall consider; they show the size of the molecule, i.e., whether it is single, double, etc. The determination of the osmotic pressure directly by the use of PfeiTer's apparatus is not very satisfac- tory, owing to the difficulty in preparing the semi- permeable film and to the fact that this latter is easily ruined. An indirect method is given later, under the vapor-pressure of solutions, which is of great use. A simple method of observing the effect of osmotic pressure is given by Tammen.* If in a moderately strong solution of copper sulfate we place a drop of a strong solution of potassium ferrocyanide it is im- mediately surrounded by a semipermeable film of copper ferrocyanide. Since the ferrocyanide solution is more concentrated than that of the copper sulfate, water goes through the film, from the copper salt to the ferrocyanide, and dark streaks are observed to sink from the bubble. These streaks are of stronger copper-sulfate solution, which is formed from the other by the loss of water, and sink on account of their increased specific gravity. If the ferrocyanide solution is weak and other substances are added to it until no streaks are observed to sink from a bubble, then there must be an equal number of molecules in * Weid. Ann. XXXIV., 229 (i888). go ELEMENTS OF PHYSICAL CHEMISTRY. the two solutions, i.e., in the copper solution and the ferrocyanide plus salt. If the ferrocyanide solution is weaker than the one of copper sulfate (in terms normal), then water will go out of the bubble, which will decrease in size. Consider a cylinder, as shown in Fig. 9, filled with a solution, and provided with a semipermeable piston Solution Salt Fig. 9. {a). If by a certain weight this piston is lowered water will go through it, and we shall have separated an amount of solvent from the solution. If the amount of water separated previously contained i mol of solute the osmotic work is given by the equation pv = RT, where v is the change in volume under the constant osmotic pressure /. It is possible to determine the molecular weight of any substance in solution by the separation of a certain amount of the solvent. For each mol the work is 2 7" cals., so that the number of mols in the definite amount of solvent may be calcu- lated and from this the molecular weight of the solute. SOLUTIONS. 91 The separation of the solvent may take place in one of a number of ways, as, for example, freezing, evap- orating, etc. 33. Electrolytic dissociation or Ionization. — The molecular weights of most organic substances are found correctly from the osmotic pressure. For in- organic salts, however, varying results are obtained. The osmotic pressure is here always too large and consequently the molecular weight is too small. This reminds one immediately of the abnormal densities of gases. In the case of the gas the volume increases, and the weight of unit volume is decreased ; in that of the solution, however, the volume remains constant, while the pressure changes. Both of these abnormal results can only be accounted for b)' the assumption of an increased number of molecules or of particles acting as molecules. Let us for a moment consider the idea which was held in 1887 for the conduction of electricity by elec- trolytes. Faraday had proved that electricity can only move in electrolytes upon certain small component parts of the substance. These particles he called ions, and assumed that they absorb electricity from one electrode and surrender it up to the other. The Grotthus theory assumed these ions to be formed from the molecules by the current, which they afterward carried from electrode to electrode. Arrhenius in attempting to find an explanation for the abnormal osmotic pressures found by experiment 92 ELEMENTS OF PHYSICAL CHEMISTRY. that those substances, and only those, which give abnormal osmotic pressures in solution are capable of conducting the electric current, and if they are dissolved in other solvents in which they behave normally they lose this power. Considering this law and the theory which then existed for the conduction of electricity in electrolytes, it is quite evident that the ions which conduct the current must always be present, i.e., they are not formed by the current. The ions of course act as molecules and so increase the osmotic pressure. Since the molecular electrical conductivity* in- creases with the dilution (i.e., the number of liters in which I mol is dissolved), Arrhenius found it necessary to divide the particles present in a solution into two classes: the molecules or inactive particles, and the ions or active particles. Increasing dilution increases the number of active particles or ions at the expense of the inactive ones. At infinite dilution, then, the inactive part would practically disappear, and we should have only ions. If a is the degree of dissocia- tion or ionization, i.e., the fraction of the total sub- stance which is present as ions, then for any dilution where yu„ is the molecular conductivity for the volume in question, and )x^ is that at infinite dilution. * See Chapter X. SOLUTIONS. 93 The ions which are formed from a substance must be charged very heavily with electricity; otherwise they would not conduct the current. We have, for example, in a solution of sodium chloride undis- sociated molecules of NaCl and ions of Na and CI. Since the ions are charged with electricity, they do not behave as they would in the molecular state, i.e., Na ions do not decompose water and CI ions are not given off as a gas. Further, some ions are always charged with positive electricity, while others are charged with negative; but no ion is known which is one time positive and Another time negative. For this reason an element going into solution in water, for example, cannot dis- sociate into ions, for it is always necessary for equal amounts of positive and negative ions to be formed. When a substance dissociates it is necessary always that there be a certain relation between the ions and the undissociated portion, Ostwald proved this to be true for all the organic acids which are slightly dis- sociated. The equation of equilibrium is the same as for dissociating gases, i.e., (28) Kc — c^c.,, where K is the dissociation or ionization constant, t, and c, are the concentrations (in mols per liter) of the two ions, and c the concentration of the undissociated por- tion. Here, just as with gases, the dissociation decreases 94 ELEMENTS OF PHYSICAL CHEMISTRY. by the addition of one of the products of dissociation. If, for example, we add to a sodium-chloride solution a cer- tain amount of a solution of potassium chloride, then the product c^c^ (i.e., Na X CI) is increased until it is larger than Kc; it cannot exist in this state, however, so that equal amounts of Na and CI ions (i.e., c^c^ unite to form NaCl (i.e., c) until the equation is ful- filled. We may also express this formula in another form in which we keep separate the c, (CI) ions of the two substances. We have then (29) Kc' = c/{c: + c,"), where c" is the concentration of CI ions in the form of the potassium salt added to the sodium chloride; + c,' is the concentration of Na ions and c,' that of the CI ions of the sodium chloride which are left free and, of course, equal to c/ ; c is the concentration of the undissociated portion (NaCl). Contrasting (28) and (29), we find K the same in both, so that, c' > c, c^' < <:„ ^/ < c,. Examples of dissociation are given later all through our work, and the methods by which it is determined, as well as the use to which it is put, are discussed at length. 34. Solution-pressure. — We are now in position to understand what takes place when a solid goes into solution. If undissociated in the solution the mole- SOLUTIONS. 95 cules are given off into the solvent, in the same way as in sublimation they are given off into the gaseous space, until a certain pressure is reached. The pres- sure which a solid exerts to get into solution is called the solution-pressure. If ions are formed in the solu- tion, then the solid will dissolve until each ion, as well as the undissociated portion, reaches a certain concen- tration. If a substance is dissolved in its own solu- tion, naturally less solid will dissolve. In the same way a substance which dissociates is less soluble in a solution containing an ion in common with it than in pure water. The solution-pressure is equal to the osmotic pres- sure of a saturated solution. This can be measured as follows: if a weight is placed upon the piston (Fig. 9) which is greater than the osmotic pressure the piston will sink; if smaller, the piston will rise; but if just the same as the osmotic pressure the piston will remain stationary wherever it is placed. Liquids dissolving in others also have a certain solu- tion-pressure and they can be treated just as solids are. 35. Vapor-pressures of solutions. — It has been known for many years that the vapor-pressure of a solution is lower than that of the pure solvent. Babo (1848) and Wiillner (1856) found, further, that the depression of the vapor-pressure is proportional to the amount of solute present; and for the same solution the depression for any temperature is the same fraction of the vapor-pressure of the pure solvent. ELEMENTS OF PHYSICAL CHEMISTRY. If / is the vapor-pressure of the pure solvent, p' that of the solution, and w the percentage of solute, then P where k is the relative depression, — : — , for a i^ P solution. Raoult in 1887 found that a more general relation is obtained if the relative depression is determined for one mol of substance in a certain amount of solvent. By experiment he found that all solutions containing one mol of substance in a certain weight of solvent possess the same vapor-pressure; i.e., the MOLECULAR depression of the vapor-pressure is constant for all sub- stances in the same solvent. Further, he found that solu- tions of equimolecular amounts of substance in equal weights of different solvents to give relative depres- sions of the vapor-pressure which are proportional to the molecular weights of the solvents. This shows that the vapor-pressure depends upon the relative number of the molecules of the solute and solvent. Raoult summed up his results as follows: One mol of any substance dissolved in 100 mols of any solvent causes a constant relative depression of the vapor- pressure. This law may also be expressed in another form: The vapor -pressure of a solution is to that of the pure solvent as the number of mols of the solvent is to the SOLUTIONS. 97 total number of mols present in the solution, i.e., of solvent plus solute. For very dilute solutions the number of mols of the solute is so small in comparison with that of the solvent that the ratio becomes one, i.e., the vapor- pressure of the solution is practically the same as that of the solvent. For any solution we shall have /_ A 'p~ N-\- n or P — p' _ n where N is the number of mols of the solvent and n is the number of the solute. This equation will only hold when the solute is non- volatile, a condition which is practically fulfilled when its boiling-point is 140° above that of the solvent. If this condition is not fulfilled it is still possible to cal- culate the depression of the vapor-pressure. Here, however, it would take us too long to consider it.* Since the relative depression of the vapor-pressure of a solution depends upon the number of mols of solute and solvent plus solute, it is possible by its aid to determine the molecular weight of the solute if that of the solvent is known. Since the ions act exactly as molecules in this respect, the results are only correct when the substance is undissociated. If, *See Nernst, Zeit. f. phys. Chem., VIII., 116 (1891). 98 ELEMENTS OF PHYSICAL CHEMISTRY. however, dissociation does take place and the molec- ular weight is known, then it is possible to determine by it a, the degree of dissociation. For the determination of either of these two terms it is more convenient to use an altered form of equa- tion (30), i.e., p— p' _ n (31) p' N w Since « = — , where w is the weight of solute and W m its molecular weight, and i\^ = ^, in the same way we have or (32) iff' p — p' _ wJf ~~p' ~ mW wM p' m = W p-p' Example. — 2.47 grams of ethyl benzoate dissolved in 100 grams of benzene gives a solution with a vapor- pressure of 742.6 mm. of Hg, while pure benzene has one of 751.86 mm., both at 80° C. Find the molec- ular weight of ethyl benzoate. w = 2.47, M = 78, W = 100, / = 742.6, p = 751.86, p — p' = 9.26; hence m = 154, while from the chemical formula we find C,H,COOC,H, = 150. SOLUTIONS. 99 If water is used as a solvent, and the substance dis- sociates, the dissociation may be found approximately, as is described later. For the organic solvents cor- rect molecular weights are obtained, for in them the dissociation is so small that it may be neglected. 36. The relation between osmotic pressure and the depression of the vapor-pressure. — The formula used above was found empirically, but it is possible to derive it theoretically by the aid of the conception of osmotic pressure. Imagine an apparatus in the form of Fig. 10. The tube h, which contains a solu- 1% Fig. 10. tion, has at its lower end a semipermeable partition, which allows passage to the molecules of the solvent, but not to those of the solute. This tube is placed in the vessel F, which contains the pure solvent. Assume now that the whole apparatus is covered with a bell jar, from which the air is exhausted. lOO ELEMENTS OF PHYSICAL CHEMISTRY. The two liquids will be in equilibrium when the weight exerted by the column hG is equal to the osmotic pressure of the solution. Both liquids will evaporate, one at h and the other at G. The vapor- pressure of the solution at h must then be the same as that of the solvent at the same place. If this were not true liquid would either condense or evaporate at h, and this would disturb the equilibrium between the height of the column and the osmotic pressure in such a way that water would go through the partition. This would cause, however, a continuous cycle, from which we might obtain work without any change in temperature, i.e., a perpetual motion; hence the vapor-pressure of the two liquids must be the same at h. The actual pressure which the pure solvent has at h is equal to its vapor-pressure minus the weight of the column of vapor hG. This is, then, the vapor-pressure of the solution. Let there be N mols of the solvent and n mols of the solute. The osmotic pressure, i.e., the, pressure which the substance would exert in gas form, under the like conditions, from (9), for i mol is P=^, or for n mols (33) /> = ^ SOLUTIONS. lOI N mols of the solvent, however, will weigh NM grams, where M is the molecular weight ; hence MN v = —, where s is the specific gravity of the solvent. Substi- tuting this in (33), we obtain (34) P=im- P, the osmotic pressure, is also equal to the weight of the column of solution, i.e., (35) P = ks', where / is the specific gravity of the solution. Com- bining (34) and (35), we find nRTs As long as the solution is dilute s will not differ greatly from s' ; hence (36) h = -^^. The weight of the column of vapor hG is equal to the difference between the vapor-pressure of the solvent, p, and that of the solution, /', i.e., ha = / — p' , 102 ELEMENTS OF PHYSICAL CHEMISTRY. where a is the specific gravity of the vapor. In case the column is of a considerable height the density of it will depend upon the pressure, i.e., adh = dp or (37) a = df_ dh' , M , The density a, however, is equal to — ; hence Mf> a -^ RT or when combined with (37) dp _ Mp dk ~RT'' i.e., dpRT = Mpdh or and finally by Integration between the limits /c=/ and/t =/ Combining this with (36), we find nRT _RT p MN ~ ~Wp' SOLUTIONS. 103 or ,P_-± >' ~ N' P This term, /-j, can be written in the form and this when developed in a series gives '(■+^)=^--M^r+... Here, however, the second term may be dropped, since p — p' is small, and we find or which is the equation found experimentally by Raoult. n Whenever -r^ < o. i we may use, as before mentioned, the simpler equation ' P- P' _^ p N' Since the osmotic pressure is given by P = hs'. L P' n N P -P' n p N-^rn 104 ELEMENTS OF PHYSICAL CHEMISTRY. and h by (38) is given as we find P = hs' — -j^s'l^ gr.-cms. {R = 84800 gr.-cms.). To obtain the value of P in atmospheres it is neces- sary to divide this result by 1033 gr.-cms., the pres- sure of one atmosphere. We have then / N r^ RT ,,p RT ,p—p' (39) P = ?> ^1^1 — iTJ- s — 17^ atmos. ^^^' ioi2>M p' 10S3M p' Example. — 2.47 grams of ethyl benzoate in ben- zene gives /' = 742.6, while /= 751.86 (at 80°). Find the osmotic pressure. R^^ 84800, 7^=273-)- 80 = 353°, M— 78, / = 0.8149,/' —/>' = 9.26; hence /'= 3.6 atmospheres at 80° C. 37. Increase of the boiling-point. — Since the vapor- pressure of a solvent is depressed b,y the solution of a substance, the boiling-point must increase, so that from this term it is also possible to determine the molecular weight of the substance dissolved. We have only to find for this purpose the relation between the number of mols of solute to the corresponding increase in the boiling-temperature. This relation can be found by the aid of the second principle of thermo- dynamics, and this we shall do later. First, however, SOLUTIONS. 105 it will be well to consider the law as found empirically, from the fact that the vapor-pressure of a solution is lower than that of the pure solvent. One mol of any substance dissolved in 100 grams of solvent must always cause a certain definite increase in the boiling-point of that solvent, since it causes a definite depression of the vapor-pressure. If A is the increase due to a i^ solution, then MA is that due to I mol in 100 grams of solvent. We have then K = MA or M = ^, where K is the molecular increase of the boiling-point, i.e., that due to the solution of i mol of substance in 100 grams of solvent, which must be constant for all substances in the same solvent. If g grams of substance are dissolved in G grams of solvent and the increase of the boiling-point is ^, then 100/ and we have for the molecular weight This term K can be found for any solvent by ascer- taining the increase in the boiling-point due to a cer- lo6 ELEMENTS OF PHYSICAL CHEMISTRY. tain amount of a substance whose molecular weight is known, and solving the equation for K. Thus I mol of any substance dissolved in lOO grams of ether increases the boiling-point of the latter 2i°.i. This is calculated from the solution of a smaller amount of substance, so that the increase is smaller and more accurately determined. The term K may also be found from the theoretical equation already mentioned, giving the relation be- tween the number of mols added and the consequent increase in the boiling-temperature. This is derived as follows: Assume in lOO grams of a solvent whose molecular weight is ^and whose boiling-point, under atmospheric pressure p, is T that there are n mols of solute. Under the pressure / the solution boils at the temperature T -\- dT. At the temperature T the vapor-pressure of the solvent is /, and consequently at the temperature T-{- dT it will be / + dp. The difference in vapor-pressure between the solvent and the solution at the temperature T-\- dT is p-\-dp — p = dp. The variation of the pressure with the temperature in a change of the state of aggregation has already been given, equation (26). We have then dp w dT T{v, - v,y where v, is the volume occupied by the substance in SOLUTIONS. 107 gas form, which is so large compared with v^ that the latter may be neglected. We have then dp w For I mol RT p Substituting this value in the previous equation, and changing w, the latent heat for i gram, into /, the latent heat for i mol, we have dp _ Ip df~2T' or (40) 'i=s^^- The relative depression of the vapor-pressure caused by the solution of n mols in 100 grams of solvent is equal to dp or, since dp is small as compared to p, dp_ T' According to Raoult's law of the relative depression Io8 ELEMENTS OF PHYSICAL CHEMISTRY. dp . 1 ,. « of the vapor-pressure this term — is equal to ^, P — P _ n_ , j^ _ 1^ -^e liave then ^/>_ « _nM _ldT p ~ N ~ lOO 2r'' Since, however, /, the molecular heat of evaporation of the solvent, is equal to Mw, where w is the heat for I gram, it follows that nM _ MwdT loo~" 2^" or ,^ 0.02^' dT= n, w which for « = I reduces to w i.e., the increase in the boiling-point caused by the addition of i mol of substance to lOO grams of solvent. Some of the values of K, determined in both ways, for they give the same results, are: benzene, 26.70; chloroform, 36.60; carbon disulfid, 23.70 ; water, 5.20. The theoretical part of the above was first deduced by Arrhenius, and the empirical part, previously, by SOLUTIONS. 109 Beckmann. The apparatus which is used for the practical determination was also devised by Beckmann and is shown in Fig. 1 1. About 20 cc. of the solvent Fig. II. are placed in the vapor-bath B, which causes the boil- ing-tube A to be heated evenly. The boiling-tube has a piece of platinum wire fused in its bottom, and no ELEMENTS OF PHYSICAL CHEMISTRY. this is covered with small garnets or beads to prevent all bumping and to cause the boiling to take place gently. On top of these beads a weighed amount of the solvent is placed and the thermometer so fixed that the bulb is covered with the liquid. The heat- ing is to be done carefully until the liquid boils, the evaporating part being condensed in the spiral tubes. After a short time the temperature becomes constant and the reading of the thermometer is taken. This gives the boiling-point of the solvent, the amount of which present is known. A weighed amount of the substance is now introduced into the tube and the temperature of boiling again observed. This is the boiling-point of the solution. The thermometer is one which was invented by Beckmann* and is divided into hundredths of a degree. Example. — Beckmann found for a solution of 2.0579 grams of iodine in 30. 14 grams of ether an increase in the boiling-point of the ether equal to o°.566. What is the molecular weight of iodine ? ^=2.0579, <^= 30.14, J = 0.566, K=2\.\o; hence 2.0579 m = 21.10 X 100 — ,, = 254. 0.566x30.14 r corresponds to 254; hence in a solution of ether iodine exists in diatomic molecules. * For details see Ostwald, Handbook of Physico Chemical Measurements, Macmillan. SOL UTIONS. 1 1 1 38. Depression of the freezing-point. — More than one hundred years ago Blagden found by experiment that the freezing-point of a solvent is depressed by the addition of any substance to it. Raoult found further (1887) that one mol of any un- dissociated substance dissolved in 100 grams of any one solvent causes a constant depression of t lie freezing-point. If one mol of any substance dissolved in 100 grams of any one solvent causes a depression equal to K, then if A is the depression caused by a 1% solution MA = K, where M is the molecular weight of the solute, or If g grams of substance are dissolved in G grams of solvent, and the depression is A, then A = ^ \oog or M^ .ook£, where K may be found experimentally, as in the case of the boiling-point, when a substance of known mo- lecular weight is used. It is also possible to find an equation by which the value of JC can be determined for any solvent. This equation is derived as follows : Assume a very large 112 ELEMENTS OF PHYSICAL CHEMISTRY. amount of a solution, containing Pit of solute, in a cylinder which is provided with a semipermeable pis- ton (Fig. 9). Let the freezing-point of the pure solvent be T, and its latent heat of solidification, i.e., for I gram, be w, further, assume T — A° X.o be the freezing-point of the solution. Cause an amount of the solvent to be separated from the solution at T° by a pressure upon the piston. If the amount of solvent separated previously contained i mol of sub- stance in solution, and if the amount of solution is so great that this has not caused an appreciable increase in the concentration, then the osmotic pressure, /, will remain unchanged, and we shall have to do the osmotic work pv = RT ^ 2T cals. Now allow this volume of solvent to freeze at T. ■r, 1 • , • lOoMw By this we obtam — p — cals. at T, smce the weight looM of this volume is — p— , where M is the molecular weight of the solute. Next reduce the temperature of both ice and solu- tion A°, i.e., to r— A°, and allow the soHd to melt in ^, , . , , . , lOoMw the solution, by which — -= — cals. are absorbed at T — A°. Finally, raise the temperature of the whole system again to T. The two amounts of heat, i.e., the amount liberated by the cooling of the system A° and that absorbed by SOLUTIONS. 113 the heating J°, cancel. By this reversible process we have done the work pv = 2 7'cals., 1 00 Mw and by it have transferred „ — cals. from T — A to T, for heat has been absorbed at T' — A and lib- erated at T. By the second principle of thermo- dynamics we have then 2T A looMw "" T P or 0.02 T AM w P ' This term — p-, however, is the molecular depression of the freezing-point, i.e., that due to a solution of i mol in 100 grams of solvent. If we call this t, then w The value of K varies naturally with the different solvents, but is constant for any one. Some of the values are: water, 18.9; acetic acid, 38.8; benzene, 49.0; phenol, 75.0. From the fact that the vapor-pressure of a solution is lower than that of the pure solvent it is necessary that the freezing-point of the solvent be depressed by the addition of any substance. Since the ice which separates is always the pure solvent, and the freez- 114 ELEMENTS OF PHYSICAL CHEMISTRY. ing-point is that temperature at which both solid and liquid may exist together in eqiiilibrium in all propor- tions, the vapor-pressure of ice and liquid must be the same. This has already been proven, and if it were not true a perpetual motion would be possible. In Fig. 12 WW is the vapor-pressure curve for water, ss Fig. 12. that for a solution, and ii that for ice. At the point ? = o ice and water have the same vapor-pressure, and so are in equilibrium. The solution and ice, however, will only be in equilibrium at the tempera- ture corresponding to the intersection of the two curves, i.e., the freezing-point of the solution must lie below that of pure water. The more substance in the solution the lower the vapor-pressirre will be and the lower the intersecting point will lie, i.e., the lower the freezing-point will be. This is the same as the em- pirical law already used by which the depression is proportional to the amourit of substance dissolved. When water is used as a solvent, and the substance SOLUTIONS. 115 is dissociated in it, the molecular weight found will be •too small. If X is the true molecular weight and x' is that found, then X -.x' •.•.i:\; i.e., the ratio of the true molecular weight to that found is the same as that of the total number of molecules present to the number present provided no dissociation takes place. This follows from the fact that the greater the number of molecules present in a certain volume the smaller must be the molecular weight. i, the total number of molecules present, is given by the equation i =■{} — a) + 2a or i = I — a; a = t — I for binary electrolytes; where i — a represents the fraction of the molecules which are left in the undis- sociated state, and 2a gives the amount of ions formed from each decomposing molecule, a being the degree of dissociation. The experimental value of i is readily determined from the true molecular depres- sion and that found, i.e., for water molecular depression found 18.9 This term 18.9 is the value of K for the solvent used. An example of the method as used to determine Il6 ELEMENTS OF PHYSICAL CHEMISTRY. molecular weight and the dissociation will make the calculation clear. A solution of 0.681 gram of acetic acid (which is very slightly dissociated) in lOO grams of water causes a depression of o°.2i68. What is the molecular weight of acetic acid ? K= 18.9, ^=0.681, G=ioo, J = o°.2i68; hence 0-68I i(f = 18.Q X 100 ^5— ■ = 59.4, ^ 0.2168 X 100 ^^^ while CH,COOH = 60. A 0.0107 normal solution of KOH gives a depres- sion of o°.0388. Find the degree of dissociation. Since i mol in 100 grams of water causes a depres- sion of 18°. 9, I mol in 1000 grams would cause one of i°.89. Our molecular depression is then 0.0388 ^ . , . 3.6261 — - — = 3.6261, and I = — 5 — = 1.919. 0.0107 •^ ' 1.89 ^ ^ a := i — I = 1. 919 — I = 0.919, or the KOH is 91.9^ dissociated. The apparatus for the determination of the freezing- point was devised by Beckmann and is shown in Fig. 13. In the freezing-tube^ the solvent (15-30 cc.) is introduced and a freezing-mixture (ice and salt) is placed in C. The temperature is allowed to fall until the liquid is overcooled i or 2 degrees; then it is stirred, ice forms, and the thermometer rises to the true freezing-point and remains constant. The SOLUTIONS. 117 cause of the rise in temperature is the sudden forma- tion of ice, which gives up heat to the liquid. ^ is a tube which acts as an air-bath and causes the cooling to take place more evenly. After the freezing-point of the pure solvent is determined the solution is intro- FlG. 13. duced and the process repeated. Or the solution may be made in the tube by dropping a weighed amount of solute into the known amount of the solvent. The result will only be correct when the ice which separates is the pure solvent. Il8 ELEMENTS OF PHYSICAL CHEMISTRY. One point to be kept in mind is that the freezing- point thus determined is of a solution which is more concentrated than the one started with, for some of the solvent has been removed by the freezing. This follows from the fact that the freezing-point is that temperature at which ice and solution exist together in equilibrium. The concentration of the solution due to the separation of ice may be calculated from the overcooling, for it has been found for water that for each degree of overcooling 12.5 grams of water separate from each liter. 39. Division of a substance between two non- miscible solvents. Depressed solubility. — If a water solution of succinic acid is shaken with ether, the acid is divided between the two solvents in such a way that there is always a certain ratio between the two amounts, independent of the relative amounts of the two solvents. Table V shows this. Table V. H,0 (i). Ether (2). In HjO (3). In Ether (4). Coefficient (5), 70 cc. 30 cc. 43-4 7-1 6 almost 49 cc. 49 cc. 43.8 7.4 6 ■' 28 cc. 55-5 cc. 47-4 7-9 6 exact Columns 3 and 4 give the number of cc. of a Ba(OH), solution which is necessary to neutralize 100 cc. of the solutions. The coefficient of partition depends in absolute value upon the temperature and the dilution of the water solution. For decreased temperature and concentration the constant decreases. SOLUTIONS. 119 If there are two or more substances in solution the coefficients of partition are the same as if each substance were present alone. The solvent behaves here with respect to the sub- stance dissolved just as it would to a gas, i.e., between the two parts (liquid and solution) there is a certain coefficient of absorption — the concentration of substance acting the same as the pressure of the gas, and the temperature having the same influence in both cases. If one solvent is soluble in another the amount of the one dissolved will depend upon its state, i.e., whether it is pure or contains a substance in solution. Just as with solids, we have for liquids a certain solu- tion-pressure, i.e., a force which causes it to dissolve. In the same way that the vapor-pressure is depressed by the addition of a substance, so is the solubility of one liquid in another. This follows from the analogy between gases and substances in solution. If / is the solubility of .<4, i.e., the amount oi A in the unit of volume of B, and /' is the solubility of the same after an addition of substance to A, then in analogy to we have where N is the number of mols of solvent A and n the number of solute in A. p P' n p' ~ N I T v n I20 ELEMENTS OF PHYSICAL CHEMISTRY. This was first deduced by Nernst, who used the formula as a method for the determination of the molecular weight. If an excess of ether is agitated with water and the freezing-point of the mixture determined it is found to be lower than that for pure water. If the ether contains an amount of substance which is insoluble in water, then its solubility in water will be decreased, and the freezing-point of the mixture will be higher than before. If t is the freezing-point of the water solution of pure ether, and t' is that of a saturated solution of ether plus substance in water, then t and t' can be substituted for / and /'. We have then t' n w/f, f ~ N ■ W/M = t — f wM t' W or where m is the molecular weight of the substance dis- solved in ether, w is the weight dissolved, M is the molecular weight of ether, and Wis its actual weight before mixing with water. Since the solution of ether in water is saturated, ether as well as ice will separate, so that the solution still remains saturated. There is, however, a differ- ence in solubility of the ether at ( and i'. This Nernst* corrects in the t — t' oi his results by the * Zeit. f. phys. Chem., VI., 29 (i8go). SOLUTIONS. 121 aid of a formula. Table VI gives a few results of this method. The benzene and naphthalene are the sub- stances, which in each case are dissolved in the ether. Table VI. substance. w t-t' t - t' (corr.) m (cal.) m (found) Benzene 2.04 0.078 0.080 78 77 5.87 0.208 0.219 78 82 Naphthalene.. 3.42 0.080 0.082 128 128 t in all cases is equal to — 3°. 85, i.e., the freezing- point of a saturated solution of ether in water. The solution of substance in ether here is slightly increased in concentration, owing to the separation of the ether with the ice, but, as very strong ethereal solutions are rarely used, it has but little influence. CHAPTER VI. THE ROLE OF THE IONS IN ANALYTICAL CHEMISTRY. 40. Ions. — With a very few exceptions all the reactions of analytical chemistry take place between the ions. In order to detect chlorine in a substance, for example, it is necessary that it exist as an ion, and that an ion of silver be brought in contact with it. Then and only then is solid silver chloride precipitated. If the chlorine is present merelj' as a constituent of a complex ion no silver chloride will be formed. Thus potassium chlorate does not give with silver nitrate a precipitate of silver chloride, for its ions are K and ClO^. The ions, then, are the active constituents of all solutions. 41. The color of solutions. — The color of a solu- tion depends always upon the condition of the solute in the solvent. If a substance is not at all dissociated or but slightly so, then any color it may possess is due to the undecomposed molecules. In case the substance is completely ionized the color of the solu- THE ROLE OF THE IONS. I23 tion will be the result of the mixture of the colors of the ions; or if only one is colored that color will be the color of the liquid. When partly dissociated a substance in solution will have the color due to the mixture of the colors of the ions and molecules. There is always a chance of error here if the color of the solid is assumed to be the color of the molecule in solution. The color of a crystal is, for example, very often different from that of the substance in the fornl of powder, and, further, it is possible for a dis- sociation to take place in the water of crystallization. In this latter case the solid would exhibit the same color as the colored ion does. The only correct way to find the color of the undissociated molecule in so- lution is to use a solvent in which the substance is not dissociated to any extent; then the two colors will be the same. This is not difificult to carry out, for all solvents have a different dissociating power, and either alcohol, ether, benzene, chloroform, or acetone will be found to serve the purpose. The ions of most acids are colorless; consequently all salts of a metal in very dilute solutions will have the same color, i.e., the color of the metallic ion. In more concentrated solutions this is not true, for many molecules are colored and, as they are now present to a greater amount, the color of the solution is the result of the mixture of these and the ions. An example of this is given by solutions of cuprous chloride, where the color of the molecule is yellow. The copper ion 124 ELEMENTS OF PHYSICAL CH-EM'ISTRY. is blue, so that the color of solution of CuCl, may be either yellow, green, or blue- according as it is undis- sociated or ionized to a lesser or greater degree. All copper solutions when very dilute, provided the nega- tive ion is colorless, are of a blue color. If in any way the colored ion of a solution is removed the color naturally disappears. All dilute cobalt solutions show the pink color of the cobalt ions. If it is boiled, however, with a solution of cyanide of potassium the color disappears. This is due to the formation of the complex ion CoCN, , which is colorless, i.e., the salt KjCoClr, is formed, + which dissociates into the two colorless ions 3K, and CoCN.. It is possible in this way from color observations to gain an idea of the ions into which a substance is dis- sociated in solution. An example of this is an inves- tigation carried out by Ostwald.* He noted that the color of potassium bichromate in solution is reddish yellow, while that of the chromate is yellow. The ions of the former are ' 2K and Cr,0„ while those of the latter are + 2K and CrO,. The so-called chromic acid (H,CrO,), however, possesses a reddish-yellow color similar to that of the * Zeit. f. phys. Chem., II., 78 (1888). THE ROLE OF THE IONS. I25 bichromate. This would lead one to assume the presence of Cr,0, ions instead of those of CrO,. Further investigation, indeed, proved that this is true, and instead of the ion CrO^ we have Cr,0,. Except in neutral or alkaline solutions, when CrO, comes in + . contact with H ions it goes into Cr,0,. This method becomes quite delicate when, instead of observing the color directly, we observe the absorp- tion spectrum. 42. The action of indicators. — An indicator is a substance which possesses a different color in an alka- line solution from what it does in acid. The indi- cators are themselves slightly dissociated acids or bases, and the change in color is due simply to the rise or disappearance of the colored ion. Instead of attempting to find a general rule as to their behavior, we shall consider a few typical cases which will make the principle clear. Phenol-phthalein is a weak acid', i.e., in water it is ionized to a small extent into ions of H and those of the colored negative radicle. In the molecular state, i.e., in an alcoholic solution, it is colorless, while the negative ions are red. If to a colorless alcoholic water solution potassium hydrate is added the following reaction takes place: The indicator is but slightly dissociated into H ions, but when these come into contact with ions of OH they unite with them to form undissociated water. This is due to the fact that 126 ELEMENTS OF PHYSICAL CHEMISTRY. + only an infinitesimally small amount of H and OH ions can exist together without forming undissociated water. This removal of H ions, however, destroys the equilibrium which exists between the ions and the undissociated portion of the indicator (by equa- tion 28); consequently more of the indicator dissoci- ates. Again, the H ions are removed and the process is repeated until in the solution we have ions of K and the negative colored radicle, and the solution is red. It is a general rule that all weak acids are very much less dissociated than their sodium or potassium salts. In few words, then, the process consists in the removal of the H ions by those of OH, and the formation of the potassium salt of the indicator, which is so much dissociated that the red color of the nega- tive ion is visible. If acid is added to this salt, then ions of H come in contact with those of the negative radicle, and since they cannot exist with them to the extent which those of K can, they unite again to form the almost undis- sociated indicator, and the color disappears. Ammonium hydrate is a very weak base, i.e., its dissociation is very small (about 1.5^ in a k/io solu- tion), and its number of OH ions is so small that an acid salt is formed, or, in other words, only part of the indicator is changed into the salt, and the change of color is not sharp and decisive. All acids give very satisfactory results, because the THE KOLE OF THE IONS. 1 27 number of H ions in the indicator is very small and therefore easily influenced. In general, then, with phenol-phthalein only the stronger alkalies can be used, but both strong as well as weak acids may be readily determined (contrast with methyl orange). Methyl orange* is a medium strong acid, i.e., it is dis- sociated to'a larger extent than phenol phthalein. Its molecule is red and its colored negative ion yellow. Addition of a strong acid, by the action of its H ions, causes the dissociation to decrease and consequently the red color of the molecule appears. If an alcoholic solution which is red is mixed with a base, H and OH ions unite to form H,0, and a dissociated salt is left, which shows the yellow color of the negative ion. A very weak acid will have no effect upon this indicator, for the number of its H ions will not be great enough to influence those of the indicator (example, H^COj). Methyl orange is, then, adapted to all bases, weak as well as strong, but only to strong acids (contrast with phenol-phthalein). Paranitrophenol is slightly dissociated in water into + ions of H and C.H^NO,. The molecule is colorless, while the negative ion is yellow. If to a slightly dis- sociated solution of this, i.e., one which shows only a faint tinge of yellow, KOH is added, H,0 and the potassium salt are formed, and the intense yellow color of the negative ion appears. * Experiment seems to cast some doubt upon this behavior. See Waddell, Jour, of Phys. Chem., Mar. i8g8. 128 ELEMENTS OF PHYSICAL CHEMISTRY. Cyanine forms a hydrate with water and dissociates into OH ions and those of the positive radicle. The molecule is blue and the ions colorless. If acid is added to the blue solution the H and OH ions unite to form water, and the salt which is formed has no color, since the ions which compose it are colorless. If a base is added to the colorless solution the hydrate is again formed, which, being less dissociated than the salt, causes the blue color of the molecules to pre- dominate. Litmus is an acid indicator and is slightly dissociated + in water into H ions and the blue negative ion; the color of the molecule — red — however, predominates. If a base is added H^O is formed and also a salt, which causes the blue color of the negative ion to appear. Acid causes the formation of the acid again, which is red. Cochineal when dissolved in water is very slightly dissociated and the yellow-red color of the molecule + is visible. The ions are H and a negative violet one. An alkali causes the salt to be formed and the violet color of the negative ion appears. To make the change as marked as possible alcohol is added, so that the color changes from yellowish red to violet. In all cases it is to be remembered that a certain portion of the acid or base used is employed in caus- ing the indicator to change, so that the amount of the latter should be as small as possible. TH£ ROLE OF THE IONS. 1 29 The action of the acid in all cases is to drive back . + the dissociation by its H ions or else to remove by them the OH ions. The base drives back OH ions + or removes H ions. 43. The solubility product. — All substances are soluble to a certain extent. Those which we call in- soluble — BaSO,, AgCl, etc. — are in reality soluble to a certain extent, but this is so small that in analytical chemistry it may be neglected. In the theoretical study of the subject, however, this solubility becomes of vast importance. A saturated solution of any salt is one which con- tains the maximum amount of substance. Nernst * found in such a solution that the undissociated portion always remains constant. Consider equation (28): Kc = c^c^. If c remains constant, then, since A' is also a constant, the product c,f, must also be constant in a saturated solution, i.e., (41) f,c, =■ constant = s. This product, equal to s, was called by Nernst the solubility product, and upon it all our work in analyti- cal chemistry is based. The results of Nernst's work are summed up in the two following laws: * Zeit. f. phys. Chem., V., 372 (1889). I30 ELEMENTS OF PHYSICAL CHEMISTRY. I. In a saturated solution of a partly dissociated salt the mass of the undissociated portion reinains constant, even when a second substance is added. II. The product of the masses of the ions formed by the dissociation also remains constant in a saturated solution. This solubility product means in words the amount which the product of the concentrations of the ions (in grams per liter) can reach and still remain ions. By the addition of another substance with an ion in common this product is exceeded, and an undis- sociated substance is formed, which, since the solu- tion is saturated, separates out in solid form. It is necessary, then, for a precipitate always and only to be formed when its solubility product is exceeded. This is shown by the fact that a solution of a solu- ble bromate added to a saturated solution of silver bromate, causes solid silver bromate to separate out. If the solution is not saturated at first, then the un- dissociated portion formed saturates the liquid and after that separates out in solid form. Since the larger the concentration of one ion the smaller need that of the other be to reach the solu- bility product, one ion can be entirely separated by the addition of an infinite amount of the other. In practice a large excess of the precipitants causes all of the ions in question, except a negligible amount, to be separated. Thus in precipitating BaSO, from BaClj by H,SO, an excess of the latter will make the THE ROLE OF THE IONS. 1 31 precipitate even more insoluble. This increased in- solubility of the solid is retained by washing the pre- cipitate with a water solution of H^SO,. In the solu- ++ tion of BaCl, we have ions of Ba and 2CI, and in the + acid we have ions of 2H and SO,. The product of the ++ concentrations of the Ba and SO, ions exceeds the solubility product of BaSO, ; consequently this sepa- rates out, first in undissociated form, then as a solid. The greater the number of free SO, ions the smaller ++ is the number of those of Ba which remain free; hence the necessity of adding an excess of the precipitant. If we use equivalent amounts of BaCl, and H,SO,, ++ then a considerable amount of Ba and SO, ions will remain uncombined. The decreased solubility of a substance in a solu- tion which contains an ion in common with it can also be understood by the aid of the conception of the solu- bility product. An example will show the process in the clearest manner. Imagine AgBrO, going into solution in a solution of KBrO,. In pure water ions + of Ag and BrO, would be given off until the solubility product of AgBrO, is exceeded sufficiently to saturate the solution with AgBrO, in the undissociated form ; after that the product would be just reached. In the case of the solution the ions of AgBrO, are given off as before ; but, since we have already a large excess of 132 ELEMENTS OF PHYSICAL CHEMISTRY. Br'O, ions present, the solubility product will be reached with a smaller amount of AgBrO,, hence less will dissolve. Another process which is governed by this principle, though seemingly not, is the increased solubility of an insoluble salt in a solution in which it forms a com- plex salt. A complex salt is one which is formed of two or more simple ones, and which has different ions from these. Thus a KCN solution dissolves AgCN; + the ions, however, are to a very large extent K and AgCN,, although Ag in the ionic form is present to a very slight extent and CN ions to a much greater extent. In a water solution of AgCN when saturated we have a certain solubility product. In its solution in KCN the solubility product between the Kg and CN ions still exists, i.e., is the same as in a water solution. Since, however, the CN ions are greater in + + numbers than those of Ag, the Ag ions must have a smaller concentration than they do in a water solution of AgCN, which has been proven experimentally. The process in detail is as follows : Ions of Ag and CN are given off until the solubility product of the AgCN is reached. Since, however, the number of CN ions is large, undissociated AgCN is formed, which unites with CN ions to form the complex ion AgCN,. This, THE ROLE OF THE JONS. 133 however, causes more AgCN to dissolve and dissociate, and the process continues until all the AgCN is dis- solved or until the solution is saturated. In the solution just formed we shall have, then, just enough + ions of Ag to reach, with those of CN, the solubility product of the AgCN,. From this process it is possible to find the condi- tions which must be fulfilled in order that any salt may dissolve to form a complex salt. If the number of ions of Ag given off are too small to exceed with the CN ions the solubility product of the AgCN, then no AgCN is formed, and so no more substance dis- solves. If the number is great enough to exceed this product, then the substance will dissolve readily. The solubility then depends only upon the relation between the Ag ions in a water solution of the salt and those in the complex salt solution which would be produced if the salt dissolved. All silver salts which in water solution give off fewer ions of Ag than exist in a water solution of the complex salt are insoluble in a salt solution which would cause the formation of this salt. In general, then, we have the law: When the concen- tration of the metal ion in a water solution of salt is greater than that of a water solution of a complex salt, then the simple salt ivill dissolve in any solvent which will produce this complex salt. If the concentration is smaller the solid will not dissolve to any greater extent than it does in pure water. 134 ELEMENTS OF PHYSICAL CHEMISTRY. This law is of great importance for determining the relative solubility products of different salts. Thus it has been observed that all silver salts except the sulfid are soluble in KCN solution, i.e., Ag,S is the most insoluble salt of silver, and in a water solution contains fewer ions of Ag than exist in a solution of KAgCN^. The concentration of Ag ions in a n/20 solution of KAgCu, is 3.5 X 15;* consequently a saturated solu- tion of Ag,S must contain but an infinitesimally small portion of substance in the dissociated state. When one remembers the extreme dilution of the saturated solutions of these difficultly soluble salts it is quite evident that all but a very small fraction must be present in the ionic state, since the dissociation in- creases with the dilution, so that if the ions are driven back the part left in solution must be infinitesimally small. 44. General analytical reactions. — Here we shall not treat the analytical scheme in detail, but only consider those reactions which, viewed from the stand- point of the theory of dissociation, possess particular interest. f METALS OF THE SECOND GROUP. Calcium oxalate is a comparatively insoluble sub- stance, i.e., it possesses a small solubility product. Oxalic acid is not dissociated enough, however, to *The author, Zeit. f. phys. Chem., XVIII., 513-535 (1895). f For details see Scientific Aspects of Analytical Chemistry, Ostwald. THE ROLE OF THE IONS. 135 precipitate CaO completely from the salts of strong acids. This is due to the fact that the reaction as it progresses gives rise to ions of H and of the negative radicle, i.e., to a strong acid, and the H ions of this drive back the dissociation of the oxalic acid to such an extent that the solubility product of CaO is no longer exceeded, even with the larger excess of Ca ions present. In contrast to this all calcium salts are precipitated by ammonium oxalate, since that is dis- sociated to a larger degree, and the acid formed has no effect upon its dissociation. If, however, NaCl is added to CaClj more CaO is precipitated by H^O, since the CI ions of the NaCl drive back the dissociation of the HCl formed during the reaction, and its influence upon the dissociation of the H,0 is decreased. Ammonia is too weak a base (1.5^ ionized in a «/io solution) to form the hydrate of calcium, since the number of its OH ions is too small to cause the solu- bility product of the Ca(OH), to be exceeded, even ++ with the excess of Ca ions. The hydrates of sodium and potassium, however, are dissociated to a greater degree, and the precipitate is formed. Ca(OH), is soluble in 500 parts of water, but an excess of the precipitant will reduce this to such an extent that it may be used to detect calcium. Strontium salts may be precipitated as the sulfate, but as this is quite soluble it is necessary to add 13^ MLEMEtfTS OP PHYSICAL CHEMISTRY. alcohol and an excess of the precipitant to decrease the dissociation and consequently the solubility. Since H,SO, is ionized to a lesser degree than HCl and HNO3, the sulfate is soluble to an extent in them. The process here is as follows: The SrSO^ gives off ++ ions of Sr and SO^; these latter, however, owing to the large number of H ions of the acid, unite to form undissociated H,SO,. This loss of SO, ions causes more SrSO, to dissolve and dissociate, so that the SrSO, dissolves more than it does in pure water. If a large excess of a solution of a soluble carboi^ate is poured upon dry SrSO, the latter is transformed into SrCO,. This is due to the fact that SrCO, has a much smaller solubility product than SrSO.. In the ++ -- + solution we have ions of Sr, SO,, 2Na, and CO,; the ++ Sr ions unite with those of CO, to form SrCOj, which saturates the solution and then separates out as a solid. This causes more SrSO, to dissolve and dis- sociate, which in its turn is transformed into the car- bonate, until finally we have solid SrCOj in a solution of NaSO.. The solubility product of SrCOj is so much smaller than that of the SrSO, that SrSO, is transformed completely by a mixture of equal amounts of car- bonate and sulfate, for the decrease of the dissocia- tion of the SrSO, by the SO, ions is not enough to prevent the formation of the SrCOj. If the solubility THE ROLE Of THE IONS. 137 product of a sulfate differs less from that of the car- bonate, as in the case of barium, then the sulfate when treated with a mixture of equal amounts of a carbonate and sulfate will not be changed. This is true because the SO, ions added decrease the dissocia- tion to such an extent that the solubility product of the carbonate is not reached. We have so many SO, ions in the solution that the metal ions with those of the COs could only reach the solubility product of the carbonate if it were considerably smaller than that of the sulfate. This is a method of separating Sr from Ba, for the SrCOj formed is soluble in HCl, while the unchanged BaSO, is not. Strontium sulfate is characterized by its extreme slowness of saturation, so that BaSO,, which is formed immediately, can be separated from the SrSO,, which is formed only after a time. Magnesium. — The hydrate of this metal is very slightly soluble, but an excess of OH ions reduces the solubility very largely, so that it may be used as a + + quantitative precipitate. The fact that Mg and OH ions can exist to an extent together without uniting explains why Mg(OH)j is not precipitated by NH,OH, when it is by NaOH and KOH. The NH,OH con- tains just enough OH ions to exceed the value of the solubility product of Mg(0H)3, but the ammonium salt which is formed by the reaction has a decreasing effect upon the dissociation of the NH,OH, so that 138 ELEMENTS OF PHYSICAL CHEMISTRY. the product can no longer be exceeded. In this way an ammonium salt added to one of magnesium will prevent the precipitation of the latter as Mg(OH), by NH.OH,. METALS OF THE THIRD AND FOURTH GROUPS. These groups of metals are characterized by the fact that their sulfids have such large solubility products that they are soluble in dilute acids. A general law as to the solubility of the sulfids in acids is as follows, its derivation being self-evident: If the solution of H^S gas in acid, such as would be formed if the sulfid dissolved in acid of a certain strength, contains a smaller number of S ions than there are in a water solution of the sulfid, then the sulfid is soluble in acid. If a larger number, then the sulfid is insoluble. If the sulfid contains in H,0 a larger number of S ions than HjS in acid solution, then when it is placed in acid the solubility product of the H,S is overstepped and + H and S ions unite, saturate the solution, and finally escape as H,S gas. If the sulfid contains a smaller number of S ions in water, then it will be impossible for the solubility product to be overstepped and no HjS will be formed. Iron. — This metal has two ionic forms — one with ++ two charges of electricity, Fe, and the other with +++ three, Fe. .The solubility product of FeS is so large that it is not formed by H^S in neutral salt solutions, THE ROLE OF THE IONS. 139 for the acid formed drives baclc the dissociation of the H,S (which is ionized to a very small extent into H and S), so that the S ions become too few to exceed the solubility product of the FeS. The salts of the strong acids are hydrolytically dissociated. This is a process which usually takes place to a certain extent in all dilute weak acid solu- tions. It is a union of the undissociated portion of the salt with the water to form free acid and base. If M is the positive and S the negative radicle of the salt, then in the solution we shall have ions of M and S and also molecules of undissociated MS. This reacts with the water according to the scheme MS + H,0 = HS+MOH, and forms the acid and base. For the case of the iron salts of weak acids in dilute solutions this is almost complete. On account of this we have difificulty in obtaining clear solutions except in the presence of acids, for the hydrate formed, Fej(OH)„, is in the colloid state and separates out readily. The so-called basic-acetate separation of iron is based upon this dissociation. The iron salt is diluted and nearly neutralized with a carbonate, and then acetic acid and an acetate added, and the whole solution boiled and filtered hot. This treatment causes the acetate of iron to be formed, the dissocia- tion of which is decreased by the presence of the acetate added, so that we have a very dilute solution I46 ELEMENTS OP PHYSICAL CHEMlSTRV. of acetate which is but very slightly ionized. By heat the hydrolytic dissociation increases, and the hydrate of iron separates out completely. In this way it is possible to precipitate iron as a hydrate without mak- ing the solution alkaline. The ferro and ferri ions have different colors, the former being a greenish black, the latter a yellowish brown. H,S gas reduces the ferri to ferro ions, with the liberation of sulfur. By boiling with KCN the ferro- and ferricyanides are formed, which dissociate into a complex ion, so that they give no reaction for iron. The volumetric method of estimation of iron by permanganate of potash depends upon the change from the ferro to ferri ions. Thus 2KMnO. + loFeSO, -f 8H,S0, = K,SO, + 2MnS0. + 5Fe,(S0.), + 8H,0, or, since it is an ionic reaction, loFcSO, = 5Fe,(S0.)., ++ +++ loFe = 5Fe,. The end of the reaction is the point at which the color of the undecomposed KMnO, is observed. +++ Aluminium has only trivalent ions, Al. All salts react acid, showing hydrolytic dissociation, which with the weaker acids in dilute solutions is complete as is the case for iron. THE ROLE OF THE IONS. I4I The behavior of Al(OH), with ammonia, caustic soda, and potash is just the opposite to that of Mg(OH),. The two latter dissolve the hydrate, while the former does not. This is due to the fact that with the stronger alkalies, i.e., those which contain a large +++ number of OH ions, the Al ion takes the place of the hydrogen in the OH, and we have the complex ion AlO . This causes more of the Al(OFI), to dissolve and dissociate, etc. With ammonia the concentration of OH ions is too small to cause the ion to be formed to any extent; hence the Al(OH), is not so soluble. Cobalt and nickel. — The sulfids of these two metals show a marked peculiarity. They are not formed in dilute acid solutions, and yet when once formed are not dissolved by acid of this strength. One explana- tion offered for this is that there are two forms of each sulfid, one of which is formed in alkaline solutions and is insoluble, while the other is formed in acid solutions and is soluble. Since, however, NiS and CoS are formed in the presence of H ions, as, for example, with acetic acid and small amounts of HCl, it would seem that this behavior is a direct conse- quence of the process already sketched for the pre- cipitation of metals by H,S. All metals which in a water solution of their sulfids contain a greater con- centration of S ions than is present in a saturated water solution of H,S in presence of acid, are soluble 142 ELEMENTS OF PHYSICAL CHEMISTRY. in acid. All metals which contain a smaller concen- tration are insoluble. It is natural, then, to inquire what would be the action if the two concentrations were the same or nearly so, for there must be some one metal which is between the two extremes of solu- bility, i.e., the very soluble and the very insoluble. The rapidity of solution will depend upon the differ- ence between the two concentrations, so that if they are nearly alike the reaction will be infinitely slow, i.e., will not take place. If now NiS and CoS fulfil this condition, then no sulfid will be formed from the salt in presence of acid, and the sulfid when once formed will be insoluble in the acid solution. In other words, if for NiS and CoS this condition is true, then the product which is already formed will not change practically, for the change would be in- finitely slow. Of course if more acid is added the concentration of the S ions in the H^S will become smaller than that of the S ions in a water solution of this sulfid, and the latter will dissolve.* Zinc forms with the alkalies a complex ion, ZnO, just as aluminium. The sulfid of this metal is as in- soluble as any of this group, for neutral salt solutions are almost completely precipitated by H,S, notwith- standing the H ions of the acid formed during the reaction. The acetate in presence of a neutral acetate is completely precipitated by H,S. * This subject is at present under investigation in the labora- tory, and a report may be expected in a short time. THE ROLE OF THE IONS. 143 METALS OF THE FIFTH AND SIXTH GROUPS. The sulfids of the metals of these groups possess a concentration of S ions in a saturated water solution which is smaller than that of H,S under the same con- dition, even in the presence of dilute acids. Conse- quently they are precipitated in acid solutions by H,S gas. Cadmium. — The salts of this metal are characterized by their slight ionization. The effect of this upon the solubility is marked, for when no dissociation takes place the precipitate is usually soluble in an excess of the precipitant. As a rule, complex ions containing Cd are not stable, i.e., they dissociate into a larger number of Cd ions than the simple salts, so that these latter are formed from the complex salt. For this reason CdS is formed from K,Cd CN, by H^S gas. Copper can be estimated from the cuprous iodide. The reaction is Cu + 2l =: Cul + I. In order to make the Cul as insoluble as possible it is necessary to have ions of I present. An addition of HjSOj serves this purpose, in that it forms from the undissociated I, HI, which is dissociated into H and I ions. Copper sulfid is even more soluble than cadmium sulfid, and Jtoo much acid will prevent entirely the pre- 144 ELEMENTS OF PHYSICAL CHEMISTRY. cipitation. An excess of water, however, causes it to form. Mercury salts are not largely dissociated, and con- sequently many of the reactions take place between the molecules. The cyanide of mercury does not conduct the electric current, i.e., is not at all ionized, and strange to say is not poisonous. This latter property seems to a very large degree to depend upon the dissociation of a body, i.e., upon the ions. As hydrolysis takes place to a very large degree, an addition of acid is necessary to obtain a clear solution of a mercury salt. Bismuth is hydrolytically dissociated in its salt solu- tions to a greater degree than any other metal, so that this property is used to separate it from other metals. If a salt solution in acid is diluted to a larger extent the hydrate immediately precipitates. This very imperfect sketch will show the nature of the action of the ions in analytical chemistry and open up to the student an entirely new field of thought. We have considered only the more important and characteristic reactions, and for further examples the student must be referred to the special work upon the subject which has already been mentioned. CHAPTER VII. THERMOCHEMISTRY. 45. Definition. — Thermochemistry is the subject which treats of the connection between chemical and thermal processes. Since almost every chemical process is accompanied by temperature changes, and is more or less influenced by them, this subject becomes of great importance. If a substance is changed from one state into another in such a way that all difference in energy appears in one form as heat, for example, then this amount, when determined, is proportional to the dif- ference of chemical energy in the two states. 46. Applications of the principle of the conserva- tion of energy. — Hess was the first to apply this prin- ciple to thermochemistry. In 1840 he announced the law that the amount of heat generated by a chemical reaction is the same whether it takes place all at once or in steps. In other words, all transformations froin the same original state to the same final state liberate the same amount of heat., irrespective of the process by which the final state is reached. The heat liberated in a 145 146 ELEMENTS OF PHYSICAL CHEMISTRY. reaction, then, depends only upon the final and initial states. This was entirely the result of experiment on the part of Hess, who did not consider it as self- evident. Upon this law the whole subject of thermochemistry is based, for by it it is possible to find indirectly the heat liberated or absorbed by any reaction. This is true even though it is not possible to carry out the reaction in practice. We have simply to consider the process as a part or a sum of others which have been measured; then by the addition of the equations all undesired substances may be caused to disappear, and finally the desired reaction is found. Thus, for ex- ample, to find the heat of combustion of carbpn with oxygen to form carbon monoxide, a reaction which cannot be measured directly, we use the two known reactions c-f 2(9= co,-^()^oK and CO-\-0=CO,^ 6?.oK. Here the same final product is obtained by each, so that by changing the sign of the second and adding the two we find C + (9 = CC -f 290;^; i.e., 290 Ostwald calories {K) are liberated. This term K is the heat which is necessary to raise the THERMO CHEMIS TRY. 147 temperature of i cc. of water from o to ioo° C. It is related to the other two calories in the following way: K— loo cals. (i.e., i cc. H,0 i° C); \oK= I Cal. (i.e., i kg. of H,0 i° C). In all reactions we shall distinguish the state of aggregation according to the system proposed by Ostwald, in which for liquids light characters are used, for solids heavy ones, and for gases italics. In considering substances in solution it is necessary to know the heat which is liberated by this process. This is called the heat of solution. For different amounts of water of course this will vary, so that for uniformity the heat of solution is always understood to be the heat (positive or negative) which is liberated by the solution of i mol in so much water that an addition of more water will give no additional heat effect. This is designated by the addition of the abbreviation Aq (aqua) to the symbol of the sub- stance. It is always to be remembered, however, that an isolated Aq in an equation will not cancel with an H,0, which refers to i8 grams of water. Another example of the determination, by the proc- ess of elimination, of the heat liberated by a reaction is taken from the work of Thomsen. His object was to find the heat of formation of SeO,Aq, i.e., the heat generated when SeO, is formed from its elements and dissolved in a large amount of water. He found 148 ELEMENTS OF PHYSICAL CHEMISTRY. SeO, + 2HClAq + 2NaHSAq = 2NaClAq + 2H,0 + Se + 2S + 734^^; 2NaOHAq + 2HClAq = 2NaClAq + ■2^%K{-) ■ 2NaOHAq + 2lI,S = 2NaHSAq + 250^^; 4II+ 2O = 2Hfi + 1267K{-); 4H+2S =2H,S + g4K. By adding these equations, with the sign changed as indicated, it follows that Se + 2(9 + Aq = SeO,Aq + 564/f, The process consists, then, in combining a number of measured reactions in such a way that the final one is obtained. 47. The heat of formation. — If the heats of fortna- tion of the substances from their elements are known, then it is simpler to substitute these in a reaction and solve for the unknown term. This saves the trouble of eliminating from a large number of equations. If in a reaction we imagine all substances to be decomposed into their elements before the reaction begins, then the final result of the reaction, after both sides are in the final state, will naturally be the differ- ence in the sums of the heats of formation on the two sides. We have, then, the rule: To find the heat liber- ated by a reaction it is ^simply necessary to subtract the sum of the heats of formation of the original substances from that of those of the final substances, the heat of formation of elements being counted as zerQ^ THERMOCHEMISTRY. 149 Since, in tiie reaction, Pb + 2/= Pbl, + 398;r, o = Pbl, + 398Jr, or Pbl, = - i^ZK, we can substitute for the chemical symbols in an equa- tion the negative values of the heats of formation and solve for the unknown term. Examples. — Find the change in heat energy caused by the decomposition of MgCI, by Na. We have MgCI, + 2Na = 2NaCI + Mg + xK. The heat of formation of MgCI, = 1 5 \oK, and that of NaCl = 1954.^, or 2NaCl = 39o8i^; hence — 1 5 loisT = — 3908 + X ; X = 2398Z'. What is the heat of formation of KMnO.?" We have the reaction 2KMnO,Aq + 5SnCl,HClAq = 5SnCl,Aq + 2KC]Aq + 2MnCl,Aq + 8H,0 + 3867^, or, substituting the negative values of the heats of formation, we obtain 2X — 4057 — 6290 = - 7859 - 2023 - 2500 - 5469 + 3867 ; hence X = i8i8i«r. ISO ELEMENTS OF PHYSICAL CHEMISTRY. To do this by the method of elimination would take a large number of equations and not be nearly as easily obtained. The heat of formation of organic compounds can be found from the heats of combustion in oxygen. If the elements remain behind in the uncombined state the negative value of the one would give the other. The elements, however, unite to form H^O, COj, etc., so that the heats of formation of these must be subtracted from the heat measured to obtain the true value of the formation from the elements.* 48. Chemical changes at a constant volume.— By the first principle of thermodynamics, equation (13), if the volume remains constant the term pdv disappears, and we have dU = dQ (constant v). By integration this becomes where U^ refers to the internal energy in the initial state and {7, to that in the final state. We have then i.e., the intrinsic energy in one state is equal to that in the other plus the amount of heat generated by the change. The chemical symbols which we use in such * For further information upon any of the above points and for data see Ostwald, Lehrbuch der allgemeinen Chemie, Vol. II., Part I. THERMOCHEMISTRY. 15' cases express, in addition to the ordinary chemical meaning, the intrinsic energy of i mol. 49. Chemical changes at a constant pressure.— If the volume changes during the reaction, i.e., if a gas is formed, then we may no longer neglect the term pdv in equation (13), for the external work of expan- sion absorbs heat. This term may be neglected in all cases where no gas is formed, simply because the correction is very small as compared with the heat of the reaction. Thus, for example, we will calculate the correction to be used for an increase of volume equal to i cc. Here dv = I and/ = 1033, grams and/^'w = 1033 gr.-cms., which is equal to — 0.02439 cal. or 0.0002439^^. 42355 If I mol of base is mixed with i mol of acid 137K are generated, and the volume increase is equal to 20 cc. The correction to be applied here is 20 X o.ooo2439isr = 0.004878^^', which is so small compared to IS7K that it may be neglected. When a gas is formed the increase of volume may become very large; consequently it is necessary to use the correction. We have, equation (13), dU = dQ— pdv, or integrated 152 ELEMENTS OF PHYSICAL CHEMISTRY. or For I mol of gas, however, pv ^ RT = 2T cals. or 0.02 TK; hence U,^U,+ Q,-o.o2TK; i.e., in any reaction, for each mol of gas formed, at the absolute temperature T under constant pressure, the in- trinsic energy of the substance is decreased by 0.02TK. In the case of the absorption of the gas the intrinsic energy is increased by this amount. Thus if I mol of gas is formed by a reaction at 18° C. the amount of heat used for its formation is 0.02 X (273° + 18) = 5.82.^. At any one temperature this is independent of the pressure p. Under constant pressure the symbols, in addition to the chemical significance, mean also the intrinsic energy plus the term pv for i mol. This term/z; may be either positive or negative according as the gas is absorbed or formed. 50. Relation between results for constant vol- ume and constant pressure. — We have for constant volume and for constant pressure Q^ = {U,-U:)Jr0.02TK; THERMOCHEMISTRY. 153 hence Qp= Q.-0.02TK and Q.= Qp + o.02TK. In this way it is possible to make all our determina- tions for constant volume and then calculate the result to constant pressure. This latter is the more useful term, for all our reactions take place in that way under atmospheric pressure. The former, however, is that usually determined in the bomb-calorimeter. An example of the calculation from one condition to the other is given below. Find from the reaction H,-\-0= H,0 -f 674.84^ at 18° const, vol., the heat generated if the reaction takes place under a constant pressure. By the reaction i mol of hydrogen disappears with 1/2 of a mol of oxygen; consequently the heat under constant pressure will be larger than that for constant volume by the amount 0.02 TK for each mol which disappears. We have 1.5(0.02 X 2<^i)K= 8.73, which is to be added to the result above. We obtain 674.84 -f- 8.73 = 683.57isr for const, pressure at 18° C. 51. Effect of temperature. — The variation of the heat energy with the temperature can be derived in the following manner: li M \s the' molecular weight of the substance in, whose specific heat is c, then »^(<") = ?«(/', + Mc{t" - i'). 154 ELEMENTS OF PHYSICAL CHEMISTRY. If represents any chemical reaction where m' and m" are the two reacting substances and ;« is the product, we can find Q for any other temperature by the aid of the former equation. Thus at the temperature t" we shall have m\t,') - lifc\t" - t') + m'\t") - M"c"{t" - f) or w'(,»)+»«'V') = m^,,)-\^Q,MM'c'+M"c"+Mc){t"-t'); i.e., since the terms M'c', M"c", and Mc are the molecular heats, the heat energy for any temperature t" can be obtained from that observed at t' , by adding to it the product of the difference of the molecular heats before and after the reaction into the difference in temperature. For this purpose we have assumed the specific heat to remain constant for the interval of temperature. This is not strictly true, but if the average specific heat is used the error will be very slight. For great accuracy it would be necessary to introduce the specific heat as a function of the temperature, i.e., to have terms such as M I ^ c^dt in place of those of the form Mc(t!' - f). 52. The ions in thermal reactions. — If two salt solutions which are nearly completely dissociated are THERMOCHEMISTRY. 155 mixed, no change of heat takes place provided that the ions do not unite to form molecules. This is the law of the thermoneutrality of salt solu- tions, which was first observed by Hess. At the time of its discovery the theory of solution had not arisen and consequently some difificulty was experienced in explaining it. According to our present conception of ions, however, it is quite simple. Since there is no union between the ions, i.e., these exist free just as they were, consequently there is no reason why a change of heat should take place. If the ions of two solutions unite to form undis- sociated molecules, then, for a definite amount of un- dissociated substance formed, we have a certain amount of heat liberated. This is the case in the neutralization of any base by any acid ; the amount of heat evolved depends upon the number of ions which are free in the acid, base, and salt formed. If the acid, base, and salt are completely dissociated, then for I mol of acid and i mol of base we observe an amount of heat liberated which is equal to 137^. This is caused by the union of a sufficient number of ions of H and OH to form i mol of water, and is the heat of association of water. We have, for example, H + C1+ K + OH = H.O + K + CI 4- 137^5: or H + OH = H,0+ 137^- 156 ELEMENTS OF PHYSICAL CHEMISTRY. If the acid and base are not completely dissociated the result will be smaller, because heat is absorbed by the further dissociation of these. In general, we shall have, then, if the salt is not completely dissociated, i.e., if more heat is liberated by its undissociated product being formed, q = x-\- w,{l — a,) - w,(l - a,) — w,(l - a,), where a, = dissociation of acid, w^ = its heat of dissociation, a, — " " base, w, = " " " a, = " " salt, w, = " " " " + X = heat of association of i mol of H ions with I mol of OH ions to form i mol of H,0; i.e., tke heat generated by the neutralization of an acid by a base is equal, for each mol of water formed, to ijjK plus the product of the heat of dissociation of the salt into the undissociated portion minus the same products for the acid and base. Naturally the negative value of the heat of associa- tion of H and OH ions is the heat of dissociation of + water, i.e., the heat necessary to form i mol of H and I mol of OH ions, from water. If a precipitate is formed by two practically com- pletely dissociated salts the negative value of the heat liberated is the heat of solution of the solid. Thus THERMO CHE MIS TRY. 157 AgAq + NO.Aq + NaAq + ClAq = AgClAq + NaAq + NO.Aq + i^ZK or A^Aq + ClAq = AgClAq + i SSAT; i.e., when i mol of AgCl goes into solution the heat + generated is equal to — 158^, or when i mol of Ag ions unite with i mol of CI ions, to form i mol of AgCl, 158^ are liberated. CHAPTER VIII. CHEMICAL CHANGE. A. Equilibrium. 53. Reversible reactions. — If we bring a number of reacting substances together in a chemical system, then after a sufficient length of time the reaction will reach an end. To represent any chemical reaction we may use the equation «.^.+ M, + n,A, + . . . = n/A/ + n,'A: + n,'A,'+.. Here «, molecules of A,, n, of A^, n, of A„ etc., unite to form n/ molecules of A/, n,' of A,', «/ of A/, etc. When these substances can remain together for an indefinite length of time without the reaction going in either direction they are in chemical equilibrium. Reactions go from left to right when we start with the substances A^, A,, etc., and from right to left when we start with A/, A,', etc., are called reversible or reciprocal reactions, provided that in each case, when we start with equivalent amounts, the final equilibrium is the same for both directions. 158 CHEMICAL CHANGE. I59 An excellent example of such a reaction is QH.OH + CH3COOH4ii^"CH3COOC,H. + H,0. Alcohol. Acetic Acid. >3J" Ethyl Acetate. Water. If we start from the right side we obtain a certain definite amount of those on left and vice versa, which is shown by the sign ^. For example, i mol (46 grams) of alcohol plus i mol (60 grams) of acetic acid, or I mol (88 grams) of ethyl acetate plus i mol (18 grams) of water, will always give the same final state, in which we have 1/3 mol alcohol + i/S mol acetic acid ■ -f- 2/3 mol ethyl acetate + 2/3 mol water. 54. The law of mass action. — The first theory of chemical action was that proposed by Bergmann in 1775- By this the value of chemical affinity can be expressed by a certain number; when the affinity of the substance A for the substance B is greater than that for the substance C, then the latter will be com- pletely separated from its companion A, according to the equation AC ArB = AB^ C. This theory was found later, by experiment, to be untrue, for it left out of consideration the important factor the mass. Subsequent work culminated in 1864 in the law of mass action as announced by Guldberg and Waage. l6o ELEMENTS OF PHYSICAL CHEMISTRY. By it all chemical action is proportional to the active masses of the substances present at any time, i.e., to the amounts present in the unit of volume. The exact meaning of this law is best grasped by considering the process of reasoning by which it was developed. If two substances which are capable of reaction are together in a system, for example, then A and B will only be transformed into A' and B' when the molecules of A and B come together at one point. The greater the number of collisions the more rapidly the molecules A and B will be formed, and the greater will be the chemical force of the reac- tion. The number of collisions, however, is propor- tional to the number of molecules; hence the chemical force is proportional to the masses of the substances. In order to obtain a mathematical expression of this law we proceed as follows: Designate the number of molecules of A and B in the unit of volume by c^ and c,; the frequency of collision of the molecules A and B will then be proportional to the product c,c^, since if one is absent the chemical force will be zero. Different substances will form different amounts of product, for each collision this influence Guldberg and Waage summed up in the factor k. For any one con- stant temperature this factor will remain constant for any one reaction. The chemical force or, what is pro- CHEMICAL CHANGE. l6l portional to it, the velocity of the change from A and B^Q A' and B' will be then When the reaction is reversible, however, a counter chemical force is present, i.e., to form A and B from A' and B' \ this is equal to v' = k'c.c,'. When equilibrium is established there is no change in either direction, or, what is the same thing, the change in one direction is equal to that in the other; hence for equilibrium we have the condition V = v' or /§<:/, = /&V// ; i.e., k where K is the constant of equilibrium. This is constant for any one reaction at any one temperature and varies with the latter, k and k' cannot be deter- mined absolutely, while K, which is simply a ratio, can be. This formula holds only for a simple reac- tion, such as A + B'^A' -\-B'. l62 ELEMENTS OF PHYSICAL CHEMISl^RY. We may generalize it, however, so that it will hold for any one. In the case of a reaction of the form n,A, + n^A, + . . . = «.M/ + «/^/ + . . . «,,«„..., represent the number of molecules which must come together at one point in order that the product be formed. It is necessary, then, for us to use these terms in c^c^ = Kc/cJ as exponents of the terms c^, c„ c/, c,', which represent the number of mols per liter of each substance. When n molecules are needed, then each one must be treated as in the case of a simple reaction. An example will show the necessity of this. If we have the reaction 2A^ -\- A, =^ A,', then it is neces- sary for two molecules of A, to come in contact with one of A,. We have then the frequency of collision to form A' proportional to c,CjC„ or KcX = c/. In general we have then Kc«c"^ = c!^i'c'"i! or ,- In,' ^ ln„' (42) K ^ -^-^^. For constant temperature K is constant for any one reaction. This is the final equation of the law of mass action CHEMICAL CHANGE. 163 as applied to the reaction when equilibrium has been established. The application of this to different reactions gives results of considerable importance. 55. Equilibrium between gases. — In gases, since the partial pressures are proportional to the number of molecules, we may use the former in (42) instead of the latter. We have, then, for the equilibrium of a gaseous chemical system K =-ri~ k' /,"'/,"" . . . ■ An example of this is given by the gaseous reaction H, + I, = 2HI. If the partial pressure of H is /„ that of I /„ and that of HI/, then P This case was investigated by Lemoine, who found by experiment that at 440° K = 0.0375. If we heat HI, then, to 440° it is possible to calcu- late from the amount of HI the amounts of H and I gas present at equilibrium. The total pressure of a mixture of gases is equal to I64 ELEMENTS OF PHYSICAL CHEMISTRY. where the terms on the right are the partial pressures. If in such a system as we have just considered, the pressure is increased so that the volume is reduced to the ;irth part of what it was, then each partial pressure is increased to x times its previous value. We have then ;iry - A - p^ ' i.e., tke state of equilibrium of H, I, and HI gases is in- dependent of the external pressure. This is general in all cases when there is no change in the number of molecules duri^ig the reaction. That this is so was proven by Lemoine, who found by experiment that the equilibrium constant was the same for all pressures from 0.2 to 4.5 atmospheres. 56. Dissociation of gases. — The general scheme according to which this takes place is A = n:A! + n^A: + . . . . Here, in contrast to the previous case, there is a change in the number of molecules; hence the equilibrium depends upon the external pressure. If/ is the par- tial pressure of A, the undissociated portion, // that of the product A^, and// that of A^, then //«.»' P when in this case ^ is the constant of dissociation. CHEMICAL CHANGE. 165 With increasing dissociation tlie number of mole- cules in the system increases, so that, as mentioned already, the external pressure has a great influence. Friedel found by experiment that for gases which are almost completely dissociated the undissociated por- tion is proportional to the external pressure. As mentioned under gaseous dissociation, the addi- tion of an indifferent gas to a dissociating gas does not alter the dissociation. This, however, is only true when no change of volume takes place by the mixture, for when the volume increases naturally the gas acts as a dilutent, and the increase of dissociation will be independent of the nature of the gas added. An example of dissociation is the decomposition of CO, by heat into oxygen and CO. We have 2CO, = O, + 2CO. If p is the partial pressure of the CO,, p^ that of the 0, and /, that of the CO, then It is quite common also for polyatomic molecules to dissociate by heat into others with fewer atoms. Riecker found, for example, that the molecules of S given off by evaporation are of the form Sj. These dissociate according to the scheme l66 ELEMENTS OF' PHYSICAL CHEMISTRY. These S, molecules when alone then dissociate into the diatomic molecules, i.e., 3S,. In the vapor of sulfur we have, then, molecules in the forms S^, S,, and S^, in which the number of the latter increases by decreased temperature and press- ure, at the cost of the former. For the two equations given we have K.=.^-f and K.^^, where /, refers to Sj, p^ to S„ and /, to S„ K^ being the constant for S, and K^ that for S^. 57. Equilibrium in liquid systems. — The reaction CH3COOH + QH.OH ^ CH,COOC,H, + HA as already observed, reaches the state of equilibrium when we have present 1/3 mol acid + i/3 rnol alcohol + 2/3 mol ester -j- 2/3 mol water, provided we start with i mol of each of the two constituents (either acid and alcohol or ester and water). This reaction goes very slowly at ordinary tempera- tures, but when it reaches the above final state it remains in it indefinitely. If we designate by v the volume of the system, and start with i mol of acid, m mols of alcohol, and « mols of ester (or water), then CffEMICAL CHANGE. 167 in the state of equilibrium, after x mols of alcohol have been decomposed, we shall have per liter; V ^ — X tt 11 V X ln-\- x\ ester _ — or V \ V 1 11 (( n + x (x\ water = or l-l " tl u hence k(i — x){m — x) = k\n -\- x)x. In the special case of equilibrium above, however, m = I, n = o, X =^ 2/3 ; hence 9 ~ 9 or ^=|- = 4. By substituting this value for K in the general equa- tion and solving it for x we find x=i/6{4m-{-i)-\-n— Vi6{m'—m-\- i)-{- ?,n{m—i)-\-n'', which becomes for « = o X = 2/i{m + I — Vm" — m-\- 1), which is the value for x at equilibrium when we start from I mol of acid and in mols of alcohol. l68 ELEMENTS OF PHYSICAL CHEMISTRY. This has been proven by experiment with very satisfactory results. It was also found that by using a large amount of acetic acid to a small amount of alcohol, or vice versa, the formation of ester and water is almost complete. In the same way a large amount of water upon a small quantity of ester causes the latter to be almost entirely transformed. Amylene in contact with acid forms an ester, ac- cording to the equation CH^COOH + QH,. ^ CH.COOCQH.,). If X is the amount of ester formed when equilibrium is established, v is the volume of the system, and i mol of acid is used for a mols of amylene, then = amount of amylene left ; V I — X V X V " acid ester formed ; tt ti hence ^ ia-x)(^-x) ^x^ V V ^ - K - ^^ TT = A - k! (a — x){\ — x)' The value for K in this case has also been determined experimentally. It was found that K = 0.001205. CHEMICAL CHANGE. • 1 69 Since the volume v is still in the equation, it is not possible to cause the reaction to go completely in either direction, as was the case with acid and alcohol. 58. Dissociation of double molecules. — Nernsthas proven that the law of mass action is followed in the case of the decomposition of double molecules in solution. If c^ is the concentration of the double molecules and c, that of the two single molecules produced from it, then Kc = c^. Or if V is the volume of liquid and 01 the degree of dissociation, then I — a , ,. . = amount of undissociateo portion ; V a V a V " one dissociation product ; " the other dissociation product ; and or I — a a K =— r V V K: (i — (x)v ' 59. Solid solutions. — Van't Hoff found that when finely powdered BaCO« and Na^SCU are shaken to- gether they react until 80^ of the BaCO, is trans- formed into BaSO,. Also when dry BaSO, and dry 170 ELEMENTS OF PHYSICAL CHEMISTRY. Na^CO, are shaken together 20^ of the BaSO, is transformed into BaCO,. This seems to be a regular state of equilibrium and the reaction is hastened by high pressure. 60. Ions and the law of mass action. — Previous to the work by Nernst upon the dissociation of double molecules into simpler ones, Ostwald proved that the law of mass action holds for dissociating organic acids. The reaction is analogous to that already given (§ 58), and we have (i — a)w' where a is the degree of ionization, v the volume in liters in which i mol of substance is dissolved, and K is the ionization constant, dissociation constant, or co- efficient of affinity. The latter name was given to isTby Ostwald from the fact that it determines the strength of the acid in that the larger the value of K the stronger is the acid. When K is known for any acid at any temperature, a, the degree of ionization, may be calculated from it by the formula / ,^ , KW Kv Kv A . ^4 2 From his results Ostwald formulated a general law as to the affinity of acids, i.e., to the size of the term or the number of H ions liberated. The addition of O, CI, Br, I, CN, etc. — in short, of negative groups — in a molecule increases the separation CHEMICAL CHANGE. i;i of ions of H ; the addition ofH, NH^, etc. , i.e., positive groups, decreases it; and the influence is the greater the nearer (in space) the substituted group is to that of carboxyl. Thus for CH.COOH \ooK = 0.00180, forCCl.COOH 100^=121. Some other values of K are given in Table VII. Table VII. IONIZATION CONSTANTS OF ORGANIC ACIDS. Propionic 1.34 Isobutyric 1.44 Capronic , i.45 Butyric 1.49 Valerianic 1.61 Acetic 1.80 Camphoric 2. 25 Anisic 3.20 Phenylacrylic 3-55 Succinic 6.65 Lactic 13.8 GlycoUic 15-2 Formic 21.4 Malic Fumaric Tartaric Salicylic Orthophthalic . . . Monochloracetic . Malonic Maleic Dichloracetic . . . irxio' 39-5 93 97 102 121 155 158 1170 5140 Oxalic loooo (+)* Trichloracetic . •+N* '(1) For largely dissociated salts and acids the law of mass action does not hold accurately by experiment, i.e. , no constant value for isTcan be obtained. Whether this is due to the inaccuracy of determination of the * These two acids are so largely dissociated that a small error in a. affects A" to a large degree. For all other acids see Zeit. f. phys. Chem., III., 411 (1889). 172 , ELEMENTS OE FHYSICAL CHEMISTRY. term a or to some factor in the law of mass action which has not been considered is as yet unknown; but it is likely that the former reason is the true one. For all organic acids, however, it has been found to hold strictly. If two solutions have the same degrees of dissocia- tion, then the addition of one to another with an ion in common will cause no change in the dissociation. Acid solutions of which this is true are called Isohydric. Naturally if two solutions contain the same absolute + amount of H ions they are also isohydric at that dilu- tion. If the degree of dissociation of a salt with an ion in common with an acid is d, and n is the number of mols of salt which are present, then the equation of equilibrium of the acid will become {nd -\- oi)a = Kv{i — a). For very weak acids we can generalize this as follows: at, the degree of dissociation of the acid, is very small in presence of the salt, so that a in comparison to i and nd may be neglected. Since d for salts is almost independent of the dilution, we have na = Kv, Kv a — — ; n i.e., the dissociation of a weak acid in presence of one CHEMICAL CHANGE. 1/3 of its salts is approximately inversely proportional to the amount of the salt. The case of the partition of a base between two acids depends to a certain extent upon their ioniza- tion constants. This partition takes place when there is not enough base present to saturate both acids. The final mixture consists of water, undissociated salt, and the dissociated and undissociated portions of the afcids. The equilibrium is the same as that which would be attained by the mixture of the salt of the one acid with the other acid. The affinity of each acid for the base will depend upon the number of free + ions of H which it possesses, i.e., the one containing the larger quantity will unite with the larger amount of the base. For weak acids it is possible to formulate a general law regarding the partition and the ionization con- stants. In this case a is so small that it may be neg- lected in I — «; hence we have Kv = ol\ a = VKv. With two acids of the same dilution we should have \/K = a, i/W = a', or a' sfW 1/4 ELEMENTS OF PHYSICAL CHEMISTRY. The coefficient of partition of two acids, then, is pro- portional to the ratio of their degrees of dissociation at the given volume, or for WEAK acids to — =i. This coefficient of partition is independent of the nature of the base and depends only upon the two acids. For the partition of an acid between two bases the coefficient will depend only upon the two bases. If a is the degree of dissociation of the one base and a' that of the other, we shall have for them when weak, just as for acids, a _ VW and the same generalization holds true. In order that a base may be divided equally between two acids it is necessary that they be iso- hydric, i.e., it is necessary that Kv = K'v'. An example of isohydric solutions, i.e., two which + contain the same amount of H ions, is acetic acid at a dilution of 8 liters and hydrochloric acid at one of 667 liters. These two solutions may be mixed in all pro- portions without any change in the dissociation result- ing. When mixed in equal proportions, if treated with a small amount of base, equal amounts of chloride and acetate will be formed. CHEMICAL CHANGE. 175 6i. Hydrolytic dissociation. — This process has already been mentioned. It arises from the reaction between the solute and the solvent, the non-ionized portion of the salt decomposing and forming acid and base with the water. If an acid, HA, and a base, BOH, are dissolved in a large amount of water the following reactions may take place : I. BA = B + A. II. HA= H + A. III. BOH = B + C5H. IV. H,0 = H + OTi. V. BA + H,0 = HA + BOH. Equations I to IV inclusive are the ordinary forms of ionization. , V is the equation of equilibrium, of hydrolytic dissociation. If K^ to K^ are the constants for these equations, and the concentrations are BA HA BOH B H A OH, C, C, C, <:, • • • > then, since the solids will also be present as gases, at least to a slight degree, their partial pres- sures will be ;r,, n^, . . . , tt/, tt,', . . . , and, by the law of mass action, we have kn,''^7t^^' . . ./,"'/,«» . . . = >^';r,'^>'7r,'^>' . . . //"''/>/""' . . . , where k and k' are the constants of the two reactions, i.e., from left to right and from right to left. Since, however, as long as solid is present the terms ffj, 7r„ . . . and n,', n^, . . . , will be the same, i.e., the gaseous space will be saturated with gas from the solid, the active mass of the solid at anyone tem- perature, as far as participation in the chemical reaction is concerned, remains constant. Hence the terms ;r„ n^, . . . and ;r/, «/, . . . , drop out of the general equation and we have only the gases as factors in the equilibrium, i.e., -^i"'//" • • • = '^'//"''A'"'' • • ■ or ^A^'A"" • • • = /.'"''A'""' • • • An example of such a complete equilibrium is given by CaCO^^CO, +CaO. Here we find Kn^ =pn^, where tt, and n^ are the partial pressures of the gases CHEMICAL CHANGE. 179 from the solids, and / is that of the CO,. Since n^ and ff, are constant, we have i.e., the equilibrium at any constant temperature depends only upon the pressure of the CO, gas. In Table VIII the pressures are given under which equilibrium exists at different temperatures. Table VIII. EQUILIBRIUM OF CaCOs ^^_^ COa + CaO. Temperature C. Press. mms. of Hg. Temperature °C. Press mms. of Hg 547 27 745 289 6io 46 8x0 678 625 56 812 753 740 255 865 1333 This means that CaCO, when heated in a vacuum to any temperature gives oil CO,, CaCO, and CaO until a certain pressure is reached. Thus at 547° a pressure of 27 mms. of Hg is produced. 65. Dissociation of a solid into more than one gas. — In this case it takes place isothermally, and so we have here also a complete equilibrium. The vapor of solid NH,HS shows by its density an almost complete dissociation into NH, and H,S, NH,HS*^NH,-\-H,S. At 25".! the gaseous pressure is equal to 501 mms. of Hg, i.e., since the partial pressures of the H,S and l8o ELEMENTS OF PHYSICAL CHEMISTRY. NHj are the same they are equal to nearly 250.5 mms. of Hg. Equal only nearly, because this pres- sure includes that of the undissociated NH,HS gas. Since, however, this is very small, the error is slight. If n is the partial pressure of the NH,HS gas, and/, and /, are those of the NH, and H,S, then, by the law of mass action. Kit = /,/,. Since, however, n is constant at any one tempera- ture, we have ^ = AA- The total pressure, P — ^o\ mms., is equal by Dalton's law to the sum of the partial pressures; we have, neglecting n, or and or This value K = J>,p, may be verified experimentally by observing the effect of the addition to the dissoci- ating system of one of the products of the dissociation, since the product of the partial pressures must always 2 " ' P.=P., K-- = P.P. _PP _ ~~ 2 2 ~ 4 K-- ~ 4 , (501)' _ 4 : 62700, CHEMICAL CHANGE. l8l remain constant for constant temperature. Table IX gives the results of experiments carried out for this purpose. Table IX. i,(NH,) A(H,S) /lA = ■S' 208 294 61152 138 458 63204 417 146 ' 60882 453 143 64779 Average = 62504 The average of which agrees quite well with the value previously found. In each case a certain amount of one of the products is added before the solid is sublimed, and the total pressure afterward determined. It is quite simple then to find the amount of solid which has dissociated and thus the total amount of each gas present. The case of the dissociation of ammonium car- bamate is quite similar. We have I.e. But hence NH.OCONH, ^ 2NH, + CO,; Kn = /,V, p p P' p A=-X- = — , A = - 3 3 9 3 pa which was found to hold true by experiment. 1 82 ELEMENTS OP PHYSICAL CHEMISTPY. 66. Non-electrolytic dissociation in solution. — When a solid goes into solution its action is analogous to its transformation into the gaseous state. A sat- urated solution, then, in contact with the solid at any temperature will still be saturated. We have, then, by the law of mass action, for any one temperature Kn = c or K= c, where c is the concentration and varies with the tem- perature. If the solid in going into solution dis- sociates into other molecules, then an addition of one of these will cause less substance to dissolve. This has been proven by Behren$for a solution of phenan- threne picrate in absolute alcohol, in which a decom- position into phenanthrene and picric acid takes place to a large extent. By the law of mass action Kc = c,c„ where c = undissociated phenanthrene picrate, c^ = free picric acid, and c, = free phenanthrene — all expressed in mols per liter. For any one temperature c must be constant, since the solution is saturated; hence c^c, = constant. The experiments consisted in the solution of phe- nanthrene in a saturated alcoholic solution of picric acid. In this way only free phenanthrene (f,) is dis- CHEMICAL CHANGE. l83 solved and since t, is known and remains constant, it is only necessary to determine c^. The product (t/,), determined in this way under varying conditions, was found to be constant. It is not possible, however, to work in the other way, i.e., to dissolve the substance in a saturated alcoholic solution of phenanthrene, for it has been observed that this latter polymerizes. If we make allowance for the number of complex molecules present, and thus find only the number of simple ones, then c,c, is found to be constant and the same as before. 67. Electrolytic dissociation or ionization and solubility. — If we dissolve a substance which forms ions in a solution already containing one of these ions the solubility of the salt is decreased. This process has already been described in Chapter VI. The effect of the added salt depends upon its dis- sociation; hence we may use this as a method for determining a. This was first done by Nernst, who deduced the following equations for the purpose. Let ;«„ be the solubility of a salt in pure water (in terms normal) and a^ be the ionization at this concen- tration; then mj^\ — a,) is the amount of the undis- sociated portion, while m^a„ is that of each ion. If the solubility in the presence of the second electro- lyte, the free ion of which has the concentration x, is w/ and its dissociation a, then, since the solubility of the undissociated portion remains constant. 1 84 ELEMENTS OF PHYSICAL CHEMISTRY. 7«„(i — «,) = m(\ — «) but Kml\ — a„) = (w/.o-.)" and Km(\ — a) = ma(jnot ■\- x) ; hence (in^a^ = ma{ma -\- x), by which m =\/<(^) + 7-i = or where m is determined by experiment we can cal- culate X by X = = ma. ma ma 68. The effect of temperature upon an incom- plete equilibrium. — In the case of complete equilib-' rium under constant pressure a change in the tempera- ture causes one of the phases to disappear entirely. In the case of incomplete equilibrium, however, the effect is quite different. A very small change in tem- perature causes only a very slight change in the equi- librium. This causes a corresponding change in the relative composition of the reacting constituents, in one direction or the other, which just compensates for the change which the reaction coefficient has suffered. If the volume energy of a system at constant vol- ume changes with the temperature the work, in a reversible process, is to the amount of heat transferred CHEMICAL CHANGE. 185 as the change in temperature is to the absolute tem- perature (second principle). The work with the volume constant is vdp, and the heat of the process we shall call q ; then vdp _ dT ~q'-T' V, however, remains constant, i.e., for one mol R7' V = or hence dp _ qdT p ~ RT i.e., the ratio of the pressures of a constituent of a system at two temperatures which do not differ greatly enough to cause q to vary appreciably with T. Imagine a system in incomplete equilibrium, for example, the three gases NH.HS^H,S + NH., no solid being present. We have then for the tem- perature r,, if/, is the partial pressure of the NH.HS gas, /), that of the H,S, and />, that of the NH„ Kp, = p,p,. At any other temperature T, we shall have then K'Pl — P^Pt- 1 86 ELEMENTS OF PHYSICAL CHEMISTRY. According to our equation, we shall have V/ ~ KS7, Tj' ,A _ _iM _ M. and hence A/= ^' A>/ \p:p^pj~ R^T, Tj' p: I.e., (43) ^^'-^^=|(^;-T,) = or, using Briggs's logarithms and solving for q, we have , , 4.56o(iogisr ^- log iir)r,r, , . (44) q = ^^ — T - T — ^^ gram-calones. This gives us for constant volume the relation between the coefficient of reaction K and the absolute temperature, q is the heat liberated or absorbed by the process during the change in temperature, and is r, — 7; found for the mean temperature — ^— -'- The results for q obtained by the use of (44) are smaller naturally than those for constant pressure. Applica- tions of this formula to a number of cases follow. CHEMICAL CHANGE. l87 Vaporization.— 1\\z condition regulating the equi- librium between a liquid and its vapor is the concen- tration of the latter, which depends upon the tem- perature. We have then ^-'- v~ RT If/, and/, refer to the two temperatures T^ and T,, then (43) becomes ^T, ^T, 2\t, Tj- Regnault found for water T, = 273, /, = 4-S4 mms. of Hg. T, - 273 + 11.54. A = I0-02 mms. of Hg. from which by aid of (44) the heat of evaporation of water at constant volume is found for i mol to be lOioo cals. By experiment it is found to be 10854 cals. when the volume increases. If we subtract from this 2 3", = 546 cals., which is the work of expansion, we obtain g = 103 18 cals. Dissociation of solids. — If a solid dissociates into gases the equilibrium is conditioned by the concen- tration of the latter. If in the dissociation NH.HS^H.S + NH, <:, and r, are the concentrations of H.S and NH,, that l88 ELEMENTS OF PHYSICAL CHEMISTRY. of the NH^HS being small and remaining constant, then c -- = -^ . -1 = ^ ' V, RTl ' ~ v^ RT: V meaning in each case the volume in which i mol is present. In this case P where P is the total pressure ; hence K = c,c. 4RT' If /" is the total pressure at any other temperature T„ then ^ '^'^ ~ ART,' hence or T, r. 2R\T, TJ- At the temperature T, - 273 + 9.5, P = 17s mms. of Hg. 7; = 273 + 25.1, P' = 501 mms. of Hg. hence g = 21550 cals. CHEMICAL CHANGE. 189 By direct experiment the molecular heat of dis- sociation is found to be 22800 cals. We must, how- ever, subtract /^T^ cals. from this, which amount has been used for the expansion. We have then 22800 — 1 132 = 21668 cals. = q. The general form of the equation for this process, where «, mols of one gas, «, of another, etc., are given off, is IK' -IK= («. + «, + .. .)(/f;^ - /f:) = ^Y^ - f J- This formula may also be applied to the dissociation of salts containing water of crystallization. Solution of solids. — In this case the equilibrium depends only upon the concentration and temperature; we have then K^ c, where c is the concentration of a saturated solution at the temperature 7",. If d is the concentration of such a solution at another temperature T^, then it is possible for us to calculate by (44) the heat of solution of the solid, the increase of volume being so small that it is practically equal to zero, van't Hoff found by experiment with succinic acid in water that for r, ~ 273, c = 2.88 mols per liter; and for r, = 273 + 8.5, c' = 4.22 mols per liter. igO ELEMENTS OF PHYSICAL CHEMISTRY. For I mol, then, i.e., by (44) q ■=■ 6900 cals., while Berthelot by direct experiment found 6700 cals. Ionization of solids in solution. — If a substance is very slightly soluble, then the solution must contain very little undissociated substance and the heat of dis- sociation must be the same as the heat of solution, i.e., equal to the negative value of the heat of precipi- tation. Thus for AgCl we have AgCl = Ag + CI. If the solubility at 7\ = c, and at T, = c' , in mols per liter, then as before '^-"=ik~Y)- For T^ = 273 + 20, c = 1. 10 X I0"5, T^ = 273 + 30, c' - 1.73 X IO-5; hence q = 15900. For the negative heat of precipitation we found (Chapter VII) 15800 cals., which is an excellent agreement. Dissociation of gaseous bodies. — When a substance A dissociates according to the scheme A = n,A, + n,A, + . . . the equation of equilibrium is Kc — t/jf,«2 . . ., CHEMICAL CHANGE. 19' where c, c,, c„ . . . aire the concentrations of A, A^, A^, . . . in mols per liter. If a is the degree of dis- sociation, then when i mol is present in the volume v I — a n^a fi,a If the mol occupies the volume v^ at T,, and the volume V, at T^, then „«, + «, + ... ^ ^»i + », + ... _g_(2 L\ the terms «,"', n"', . . . cancelling. For the dissociation of N^O, we have N,o. :: 2N0,. Here «, = 2, n, = o, and we obtain If we assume the vapor-densities to be A and A' at r, and T^, then 3-179 - ^ , 3-179 - ^' a = 2 and «, = -j, , where 3. 179 is the vapor-density of N,0, as calculated from the molecular weight (96) and based upon air as a standard. The volume occupied by i mol of N,0, under atmospheric pressure is increased by the dissociation. 192 ELEMENTS OF PHYSICAL CHEMISTRY. One mol occupies at any temperature 0.08 ig 7" liter / 22. 36^ I from 5-). The volume in which i mol is present ^. 1 7Q will be increased, then, to o.oSigT " ^ liters. This % I 7Q term ' . however, is equal to (i + a) (i.e., 3.179 : A :: i + -i' : i); hence f, = T^i + a,) and z/j = T^i -\- a,), and we obtain From the results r. = 273 + 26.1, J = 2.65, a, = 0.1986, T, = 273 + 1 1 1.3, //'= 1.65, a, = 0.9267, we find ^ = 12900 cals. After subtracting the heat equivalent to the work of expansion* we obtain from the experimental result q = 12500. By a similar calculation it is also possible to find the heat of dissociation of other gases. Dissociation in solution. — We have already consid- ered this as applied to substances which are quite in- soluble where the concentration of the undissociated portion may be neglected. When no solid is in con- tact with the solution, and the undissociated portion * The gas is supposed to increase its volume without doing work. CHEMICAL CHANGE. 193 may vary, the calculation is quite different. For this we use the same formula as in the case for gases, since substances in solution behave just as they do in gaseous form, bnt the volume of a liquid varies but slightly for small temperature differences, z/, = v,. If the dissociation at one temperature Z', is «■,, and that at Z', is «■,, then for binary electrolytes we have ^K^ ^ ^<(i - «,) R\T, tJ Arrhenius has proven this equation thoroughly by calculating q from experimentally determined values of a at different temperatures. By it he found the heat of dissociation for the different acids, bases, and salts, and thus was able to calculate what the heats of neutralization should be by the equation (Chapter VII) q^x^Wlx- ^O - W^i - «,) - Wii - «,). In Table X the values obtained from this formula are compared with those values found by experiment. Table X.* THE HEATS OF NEUTRALIZATION OF NaOH. Acid. Calculated. Observed. HCI 13700 cals. 13740 cals. HBr 13760 " 13750 " HNO, 13810 " 13680 " CjHjCOOH 13400 " 13480 " HiPOj 14910 " 14830 " P(OH) 15460 " 15160 " HFl 16120 " 16270 " * Arrhenius, Zeit. f. phys. Cheni., IV., 110(1889). 194 ELEMENTS Of PHYSICAL CHEMISTRY. This agreement is very close considering the chance of error in both results. From this it is clear that the value of the heat of neutralization of a base by an acid cannot be assumed to be proportional to the strength of the acid. The two latter acids are very weak, and yet they give the greatest heats of neutralization. The heat generated, as already mentioned, depends upon the heats of dissociation of the acid, base, and salt. B. Chemical Kinetics. 69. Application of the law of mass action. — By the law of mass action the reaction n,A, + n^A^ + . . . = n^Al + n^A^ + . . . becomes kc^-^c^'^ . . . = k'c/^cj"'. . . The speed with which a reaction will take place is pro- portional to its constant /^ in that direction. The total speed with which the reaction goes is, then, equal to the difference in the speed in the two directions. We have then V = V — v' =z kc^^^c"' ... — k'c^^-^'c^"-^ . . . If in the time dt the substance being formed is in- creased by an amount dc, then dc ^ = kc^^^c"-^ ... — /^V/«iV/«2' . . . CHEMICAL CHANGE. I9S This, however, is only true when the reaction takes place isothermally, i.e., when the heat liberated or absorbed is removed or supplied, so that no change in the temperature follows, for the constants k and k' are dependent upon the temperature. If we have four substances present at the beginning, i.e., t = o, in the concentrations «,, «„ «/, and «/ mols, and x mols of «, and a, are decomposed at the time t, then dx — = k{a, - x){a, -x) — k'{a,' + x){a,' + x). The ratio p- can be found at equilibrium (i.e., for dx -J- = o), as has already been explained. In general, however, this formula is very much sim- plified by the fact that most reactions are almost com- plete in one direction, so that the second term will be so small that it may be neglected. We have then dx -^ = k{a, - x){a, - x). 70. Reactions of the first order. — For convenience we divide all reactions into orders. Thus a reaction of the first order is one in which but one substance suffers a change in concentration. Raw sugar in water solution is transformed in the presence of acids almost completely into dextrose and laevulose; it is inverted. The speed of the reaction is 196 ELEMENTS OF PHYSICAL CHEMISTRY. very small and increases with the amount of acid added. The progress of the reaction may be observed by aid of the polariscope. The uninverted portion revolves the plane of polarization to the right, while the two products revolve it to the left. If a^ is the positive angle of revolution at the time ^ = O, i.e., that which is due to the uninverted sugar alone, the amount of which is « mols; a/ is the negative angle after com- plete inversion; and a is the angle at the time t — then, since the revolution is proportional to the concentra- tions, X, the amount inverted, is found from a„ — a X ^= a For the time ^ = 0° a = «„, i.e., x = o; for the time t = 00 , i.e., after complete inversion, a — — a^' or X =^ a. This process was first measured by Wilhelmy (1850), and it has played an important role in the history of chemical mechanics. The process follows the scheme C.,H,,0., + H,0 = 2QH,,0., whether acid is used or not, for the concentration of the latter does not change during the reaction. According to the law of mass action, the speed is pro- portional to the amounts of sugar and water. The latter, however, is present in such an excess that its CHEMICAL CHANGE. 1 97 action is constant. The speed of reaction, then, is proportional to the amount of sugar present and we have a reaction of the first order. The formula is then j, = k{a~ X), where, for if = o, ;ir = o. ^ is the inversion constant, which depends only upon the temperature. By in- tegration this becomes — l{a — x) — kt -\- constant or, since, for t = o, x = o, /(«) = the constant ; i.e., t a — X t a -(- «/ The meaning of the constant k is in words as follows: its reciprocal value multiplied by the natural logarithm of 2 gives the time in minutes which is necessary for the transformation of one half the total amount of substance, a as is shown by the substitution of — for x. Further, for all reactions of the first order this constant k is found to be independent of the original concentration of the substance. Table XI gives the values for k for a 20^ sugar solution in presence of a 0.5 N. solution of lactic acid at 25° C. 198 ELEMENTS OF PHYSICAL CHEMISTRY: Table XI. INVERSION OF SUGAR. t (miautes) a !,„,» _^ 34°. 5 1435 3i°.i 0.2348 4315 25°. 0.2359 7070 20°. 16 0.2343 1 1 360 J3°-98 0.2310 14170 10°. 01 0.2301 16935 7°-57 0.2316 198 1 5 5° .08 0.2291 29925 - T'.65 0.2330 00 - 10°. 77 Since only the constancy of the term — / is t a — X to be shown, we use the more convenient system of logarithms, i.e., that of Briggs. 71. Catalytic action of hydrogen ions. — The in- version of sugar, as well as all other reactions of the first order which are hastened by the action of acids, + is due to the H ions which are present. This action is known as a catalytic action of the acid present, which retains throughout the reaction its original con- + centration. The reason why H ions act in this way is unknown; but that they do have an accelerating effect is an established fact. The more concentrated the acid used the more rapidly the sugar is inverted, without, however, any exact proportionality. The inverting action increases CHEMICAL CHANGE. I99 more rapidly than the concentration. In presence of a neutral salt of the acid its inverting power is in- creased by about \o<^ for the stronger acids, but decreased for the weaker ones. Since the acceleration of the speed of the reaction is caused only by acids, and since they are distin- + guished by the presence of H ions, the action must be due to these latter. Further than this, a series of acids arranged in the order of inverting power is found to be in the same order in which they stand in refer- + ence to ionization. Thus it is plain that the H ions have this effect, although the inverting power is not strictly proportional to the ionization. For example, a 0.5 .N solution of HCl inverts 6.07 times as rapidly as one of a o. i .N solution, while it contains only + 4.64 times as many ions of H. There must be, then, another factor to be considered, and this Arrhenius found to be the presence of other ions which increase the catalytic action of those of H. This theory ex- plains very simply the anomalous behavior already mentioned. The speed of reaction increases more rapidly than the concentration of H ions, owing to the fact that the negative ions are also, present to an equal amount and so increase the inverting action of the H ions. In the presence of a neutral salt we have two sep- arate actions upon the acid; the first causes a decrease 200 ELEMENTS OF PHYSICAL CHEMISTRY. in the concentration of H ions of the acid; the second accelerates the action of the H ions left. The stronger acids are less influenced by the former than the weaker ones, because they are dissociated more nearly to the extent to which their salts are. Consequently the second action predominates, and we have the greatest inverting power for the strong acids when a salt is present. For the weaker acids, however, the second action is the smaller; hence the inverting action is less for weak acids in presence of a salt. Another reaction of the first order which depends + for its speed upon the catalytic action of the H ions is the formation of alcohol and acid from an ester. For example, CH.COOQH. + H,0 = CH.COOH -I- QH.OH. The equation is the same as before, i.e., follows the formula t a — X where a is the amount of ester in mols at the time ;f = o, and x is the amount transformed at the time t. X is equal also to the amount of acid or alcohol formed during the reaction for the time t. The progress of this reaction is observed by titrating the amount of free acetic acid present. 72. Reactions of the second order. — Here two molecules suffer a change in concentration during the CHEMICAL CHANGE. 201 reaction, i.e., the constant depends upon the concen- tration of two substances. We have then dx -^ = k{a - x)ib - x) ■ or I t[/(^ — x) — l{a — x)] = kt -\- constant. a — b For / = o X =^ o, and the constant is I a i.e., -fb-la); k = I ^ {a - x)b (a — b)t {b — x)a' If we use equivalent amounts of the two substances, then a = b, and we have ^ = k(a- X) or k = \-l t (a — x)d Here, as before, the reciprocal value of k multiplied by the natural logarithm of 2 gives the time in minutes which is necessary to transform one half of the original substance ; but here k is proportional inversely to the original concentration. An example of a reaction of the second order is C,H.C00C,H3 + NaOH = CH,COONa + C,H,OH. 202 ELEMENTS OF PHYSICAL CHEMISTRY. Table XII gives the value of this constant as calculated for different lengths of time at the temperature of io°C. Table XII. / (minutes) k . ... 489 2.36 1037 2.38 2818 2.33 The question as to how the speed varies with the nature of the base used has been investigated quite thoroughly. Thus Reicher found for strong bases (those which are much dissociated) approximately equal values for k; for weak ones, however, he found very much smaller values. Ostwald observed that the weak bases ammonia and allylamine fail to give a con- stant value for k. Thus for ethyl acetate and am- monia he found the values given in Table XIII. Table XIII. / (minutes) k 60 1.64 240 1.04 1470 0.484 This failure to give a constant he recognized, how- ever, as being due to the effect of the neutral salt formed (ammonium acetate). When a large amount of this is present a constant is obtained, as is shown by Table XIV. CHEMICAL CHANGE. Table XIV. (minutes) k o .... 994 0.138 6874 0.120 15404 0.119 203 This shows still a slight effect of the neutral salt formed, for it is impossible to keep it perfectly con- stant. Arrhenius found that the base acts simply in pro- portion to the number of OH ions it contains, we have QH3OAH. + OH + Nt = C,H30, + Nt + C,H.OH. According to this, all bases containing the same amount of OH ions should give the same constant. We can re-arrange our equation to the form dx — = k'a.((i — x)(b — x), where a is the degree of dissociation of the base. Since the strong bases are but slightly influenced by the addition of salts with an ion in common, the con- stant of KOH can be taken, with very little error, to be that due to a certain concentration of OH ions. From this it is then possible to calculate the constant for NH^OH, with or without an ammonium salt present A n/\o solution of KOH at 24^.7 C. gives a constant equal to 6.41. Such a solution of NH^OH 204 ELEMENTS OF PHYSICAL CHEMISTRY. is but 2.69^ dissociated, while that of KOH is 97.2^ dissociated. The constant, then, for NH,OH in the absence of salt is , 0.0269. k = -6.41 = 0.177. 0.972 ^ ' ' The effect of the neutral salt upon the NH4OH can be readily calculated from the dissociation of the two; we have, then, the constant for NH,OH in presence of a neutral salt, k = 0.41, 0.972 ^ where a is the concentration of OH ions in the NH^OH in presence of the amount of the neutral salts. Table XV compares the values thus obtained with those actually observed by experiment. Table XV. SPEED OF REACTION CAUSED BY NH40H. s a k (calculated) k (observed) 2.69 % 0.177 0.156 0.00125 I. 21 0.08 0.062 0.005 U.7I 0.047 0.039 0.0175 O.II8 0.0078 0.0081 0.025 0.082 0.0054 0.0062 0.05 0.042 0.0028 0.0033 This is a very good agreement, considering the pos- sible experimental errors, so that we may conclude that t/te velocity of saponification is proportional to the number of OH ions present, and independent of the radical combined with them. CHEMICAL CHANGE. 20$ 73. Reactions of the third order. — If the three molecules are present in equimolecular amounts we have or _ I .x{2a — x') ~ t 2a^{a — xY One reaction of this order has been studied by Noyes. The reaction is 2FeCl, + SnCI, = aFeCl, + SnCl,, +++ ++ ++ ++++ 2Fe -|- Sn = 2Fe + Sn. When SnCl, = FeCl, = 0.0625 inol ^ value of ^ = 85 was found. For such a reaction the reciprocal value of k multiplied by the natural logarithm 0/2 also gives the time in minutes which is necessary for the trans- formation of one half the substance present ; but here it is inversely proportional to the square of the original concentration. In general, the time is equal to 7IU2 for any reaction, but for one of the «th order it is proportional for different concentrations to the (« — i)st power of the original concentration. The practical value of the reaction velocity is to find the number of molecules which take part in a reaction. Thus if we find ^ to be a constant, as calculated for a reaction of the first order, the reac- 206 ELEMENTS OF PHYSICAL CHEMISTRY. tion in question must be one of the first order. In this way, by use of the different equations, it is pos- sible to find the number of molecules changing their concentrations in any reaction. 74. Incomplete reactions. — Thus far we have con- sidered the speed of reactions which are complete, i.e., those which only cease when all the original substance has been transformed. In the case where this is not true it is necessary to use the equation in its original form. Such a case is the one already studied, C,H,OH + CH3COOH ^ CH3C00C,H, -f H.O, which goes with a certain speed until two thirds of the acid and alcohol are decomposed. When the amount of ester is equal to x, then dx when we begin with i mol of each acid and alcohol and have no water of ester present, k^ and k^ are the speed components in the two directions. We found earlier that Ar = | = 4. By observing the change for any time we find /^t 2 — IX k From these two values 7^ and k, — k. we can find k,. CHEMICAL CHANGE. 20/ The reaction so measured, however, does not give a constant value for k^. This was accounted for by Knoblauch. When alcohol and acetic acid form ester and water the reaction goes faster than it should, according to the theory. This is due to the + catalytic action of the H ions present when the reac- tion goes in the one direction. If, then, the con- + centration of H ions is retained the same throughout the reaction, k^ should be constant, which has been found by experiment to be the case. This shows the importance of secondary reactions, which may give entirely false results unless accounted for. 75. Reactions between solids and liquids. — The solution of a substance in an acid depends for its speed upon the surface of contact between the two and upon the strength of the acid; but many second- ary reactions take place, so that our results are only approximate. If we assume the surface to be so large that it changes but slightly during the reaction we may con- sider it as constant; we have then, if 5 is the surface, dx where x is the amount dissolved in the time t. By integration we find t a — X 2o8 ELEMENTS OF PHYSICAL CHEMISTRY. The formula in this form was found by Boguski to give constant results, within the experimental error, for the solution of marble in acid. 76. Speed of reaction and temperature. — Empiri- cally it has been found that the speed of a reaction always increases by an increase in the temperature. The increase in many cases is very great. Thus a rise in temperature of 30° causes the speed of sugar inversion to be five times as great as before. Owing to the internal friction and its variation with the tem- perature, it is impossible to gain any assistance from the thermodynamics; so that we have no theory which will account for such an enormous increase in the speed. CHAPTER IX. PHASES. 77. The phase rule. — This law, which was deduced by Gibbs, gives the relation between the number of phases and substances in a chemical system at equilib- rium. Imagine a complete equilibrium consisting of y phases of n substances. Consider one phase alone. This phase will contain, then, a certain amount of all the n substances. It may be a gas or a liquid, for all substances go into the gaseous form and into solutiort, at least to a slight extent. In this phase which we are considering let the con- centrations of the n substances be c^c^ . . . c„. Since the equilibrium is complete, the composition of this phase will be altered by a change of concentration, temperature, or pressure of the system. A change in one of these will cause a corresponding change in the other two. This we may express by the equation F{c^c^ . . .c„, p, T) = o, where F is any function of the variables. 209 2IO ELEMENTS OF PHYSICAL CHEMISTRY. The composition of one phase, however, determines that of all others which are in equilibrium with it; for all phases which are in equilibrium with one must be in equilibrium with each other, and this is only possible for a certain ratio of concentration between the constituents. Thus for a certain liquid phase we have a certain gaseous one in equilibrium with it and perhaps a solid. It follows, then, that the com- position of all the phases is a certain determined func- tion of the same variables. For each phase, then, we have an equation of the form Since there are y phases, we have, then, y equations of this form^ There are, however, n -\- 2 variables in each equation, so that li y = n-\- 2, i.e., if we have two more phases than substances, we may find a cer- tain definite value for each unknown quantity, since we have the same number of equations as of unknown quantities. 'In this case there is only one value for each c^c^ ...€„, p, and T at which the system may exist in equilibrium. When n substances are present in n -\- ^ phases we have equilibrium only for a certain temperature, a certain pressure, and a certain ratio of concentration of the single phases, i.e., n -\- 2 phases of n substances can only exist at a certain point {transition point) in m system of coordinates. If one of these values is changed, then one phase disappears entirely, and we have n-\- \ phases of n PHASES. 211 substances. In this case there will he n -{- 2 variables (unknown quantities) in n -\- i equations ; hence it is not possible to find an absolute value for any one. We find, however, relative values of the n -\- 2 varia- bles c,c, . . . c„, p, T, i.e., for each value of 7" we have only a certain value of / and of each of the terms c,c, . . . t„ at which a complete equilibrium can exist. // is necessary to have at least n substances present in order that a system containing n-\- i phases may exist in complete equilibrium. Gibbs deduced these laws mathematically in a differ- ent manner from this, although the above derivation will represent the idea quite as clearly and with greater simplicity. An example of the use of this law is given by the dissociation of CaCO,, i.e., three phases. We must have for complete equilibrium at least two phases, CaO and CO^. Naturally the system might contain more than two substances, but two is the lowest limit, i.e., the minimum number of phases. Where we have n -\- i phases of n substances, and the conditions are altered, one phase disappears and we have n phases (equations) and n-\-2 variables (unknown quantities), i.e., the composition of the phases is uncertain and we have an incomplete equilib- rium. Such a one is a mixture of water and alcohol, i.e., two substances in two phases, liquid and gas. At a low temperature we shall have a complete equi- 212 ELEMENTS OF PHYSICAL CHEMISTRY. librium, for ice is formed and two substances are present in three phases. 78. The equilibrium of water in its phases. — Fig. 14 gives the curves for water when plotted in a rvr SP 1 wv Fig. 14. system of coordinates of which the abscissa are tem- peratures and the ordinates are pressures — in other words, the vapor-pressure curves of water, ice, and steam. Along the curve WV liquid and gas are in complete equilibrium. The curve IV is made up of the values at which gas and solid can exist in perfect equilibrium, i.e., it is the vapor-pressure curve for ice. The curve IW represents the conditions for the coexistence of water and ice. As the freezing-point is but slightly influenced by pressure, this curve forms only a slight angle to the axis of pressures (i atmos. = depression of 0.00752°). The point in which the three curves intersect is the transition-point, i.e., the triple point in which all three phases can exist in equilibrium. If we start with the system containing water and vapor, i.6., down the curve WV, in a closed vessel from PHASES. 213 which heat can be absorbed the point of intersection is reached, water freezes, and we have three phases of one substance, i.e., the transition-point. If the tem- perature is still decreased either the liquid or the gaseous phase will disappear. Which of these de- pends upon the ratio of concentration. If the volume of vapor is great enough, all the liquid will freeze and the curve IV will be followed. On the other hand if the volume of liquid is large all gas will disappear and the curve IW will be produced, for the amount of ice which separates will increase the pressure and thus cause all the gas to condense. In a system of coordinates, then, an incomplete equi- librium is represented by a surface, i.e., W, V, and I; a complete equilibrium by a line, as WV, IW, and IV; and an equilibrium of n substances in n -\- 2 phases by a point, as 0. The equation already developed, i.e., Q = T^{v, - z-O, can be used for the change from one phase to another when the difference in volume in the two states is known. This holds, as has already been shown, for the case of ice and water, as well as for that of water and vapor and for different allotropic states of the same element. By the aid of the phase rule it is always possible to detect from the relation of pressure and temperature the formation of a new phase. 214 ELEMENTS OF PHYSICAL CHEMISTRY. Whenever in an investigation a multiple-point is found in a series of coordinates it shows that a new phase has arisen. Upon this fact is based the dis covery of a number of new hydrates, etc.* * For further information upon this subject the reader is re- ferred to Bancroft's Phase Rule. CHAPTER X. ELECTROCHEMISTRY. A. The Migration of the Ions. 79. Electrical units. — The unit of the resistance offered to the electric current is the ohm, i.e., the resistance at a temperature of 0° C. of a column of mercury 106.3 cms. long, with a cross-section equal to I sq. mm. The unit of current strength is that strength which will separate 0.328 mg. of copper from a solution in one second, and is called the ampere. The unit of electromotive force is the volt, which is of such a value that a Daniell cell is equal to i.io volts.. The unit of the amount of electricity is the coulomb; I gram of H ions carries with it 96537 coulombs. The intensity factor of electrical energy is the volt, while the capacity factor is the coulomb, i.e., E = en, where e is the amount of current in coulombs and n is the electromotive force in volts. To find the heat equivalent of electrical energy, 215 2l6 ELEMENTS OF PHYSICAL CHEMISTRY. since the latter is equal to i volt X i coulomb, we have only to measure the heat generated when i coulomb is driven through a circuit with an electro- motive force of I volt, or, when i coulomb has its potential lowered from i volt to zero. We find that I volt X I coulomb = 0.236 cal. or 4.24 volts X I coulomb = i cal. The mechanical equivalent of electricity is, then (since I cal. = 43280 gr.-cms.), I volt X I coulomb = 10210 gr.-cms. = 10210 X 93i-iS — 10017541.5 = 10' ergs. If all electrical energy is transformed into heat, then ne = kA, where A is the amount of heat and k is the heat equivalent of electricity. 80. Faraday's law. — This, with the theory of solu- tion and electrolytic dissociation, is the basis of all our work in electrochemistry. The law may be expressed as follows: All movement of electricity in electrolytes takes place only by the concurrent inovement of the tons in such a way that equal amounts of electricity cause chemically equivalent amoimis of the different ions to move. One gram equivalent of ions carries with it 96537 coulombs of electricity, as has been determined by ELECTROCHEMISTRY. 217 experiment. One coulomb causes, then, 0.0,1036 gram of hydrogen ions to move ; hence it will cause 0.0,1036 X a grams of any other element to move, where a is the equivalent weight of the element. 81. The migration of the ions. — All electrolytes, as has already been stated, contain ions which are charged with enormous amounts of electricity. The electrodes become charged by the current which passes to them and attract or repel electrostatically the charged ions in the solution, since like charges repel and unlike attract one another. The ions, then, are separated in this way and gather around the electrode. As soon as the difference of potential of the two elec- trodes becomes great enough, the ions give up their charges and assume the ordinary state of elements. In the case of the conduction of a current by a metal, it is necessary for equal amounts of positive and nega- tive electricity to go in opposite directions. In elec- trolytes this was found by Hittorf to be unnecessary. It is always necessary, however, for the solution to contain equal amounts of positive and negative ions, and as long as this condition is fulfilled the conduc- tion may take place by the movement of equal or unequal amounts of the positive and negative ions. If the unit quantity of electricity is passed through the solution, then 1/2 may be transported by negative ions and 1/2 by positive ones, or 3/4 by negative and 1/4 by positive, etc. In other words, it has been found that the ions are moved with varying velocities 2l8 ELEMENTS OF PHYSICAL CHEMISTRY. by the current. The above relations are shown by Fig. 15, where the white circles (negative ions) move B A O O O o o o V=4 t — I — J — 1 000 0000000 OOOuOOOQOOO V-2 Fig. 15. twice as rapidly (and consequently twice as far) as the black ones (positive ions).* The distances ii and v represent the number of ions of each kind which have been moved. The relation between the two numbers, i.e., the loss and gain of the ions on each side, will give them the relative velocity of migration of the two ions. In .this way we can find the relative velocity of migration of any ion. The following consideration will make this quite clear: Assume the vessel in Fig, A c D B Fig. 16. 16 to be divided into three portions, AC, CD, and DB, and filled with a solution containing 30 gram equiv- * Here six equivalents of tlie substance have been separated upon the electrodes. Of these four are missing from the left side, while only two are missing from the right side. ELECTROCHEMISTR Y. 2 1 9 alents of HCl. We have, then, 10 gram equivalents in each division. If 96537 coulombs of electricity are passed through the cell from B 'lo A \ gram equiv- alent of H ions and i gram equivalent of CI ions will be separated upon the electrodes A and B. These gases we assume to be removed as they are formed. These 96537 coulombs pass through the whole solu- tion and have a certain effect upon the equilibrium of the ions. First we will imagine the H and CI ions to move with the same velocity and then with differing velocities, and find the relation between the change in concentration and the relative velocities. I. If the velocity for each ion is the same, then 1/2 gram equivalent of CI ions, charged with 48268 _cou- lombs, will migrate from BD through DC to CA ; and -I- 1/2 gram equivalent of H ions, with the same amount of electricity, will go from AC through CD to DB, Altogether i gram equivalent, charged with 96537 coulombs, has passed through the section CD. Since -I- I gram equivalent of H ions has been removed by decomposition from .^Cand 1/2 gram equivalent has migrated to it, we have left g\ gram equivalents of H ions and 9J gram equivalents of CI ions, since but 1/2 gram equivalent of this has migrated from it. Con- sequently we have in AC g^ gram equivalents of HCl. In BD we have the same number, since i gram equivalent of CI ions has disappeared and 1/2 gram 220 ELEMENTS OF PHYSICAL CHEMISTRY. equivalent has migrated to it and 1/2 gram equivalent + of H ions has migrated from it. In the section DC the concentration is unaltered, i.e., just as many ions have left as have been carried to it. The concentration of solution left upon the two sides is, then, the same for solutions containing two ions which possess equal velocities of migration. + II. Assume the velocity of the H ions to be five times that of those of CI. In this case after i gram equivalent of H and i gram equivalent of CI have separated in the gaseous state the whole system will have suffered a change. 5/6 + of & gram equivalent of H ions, charged with ^(96537) coulombs, will migrate from BD through DC to AC, and 1/6 of a gram equivalent of CI ions, with ^(96537) coulombs, will go from AC through CD to DB. Altogether, as before, i gram equivalent of ions will go through the section CD, carrying with it 96537 coulombs of electricity. The original composition of the solution in CD is again unchanged. In^C" we have lost i gram equiv- + alent of H ions in the form of gas, and gained 5/6 of a gram equivalent by the migration ; consequently we + have 9f gram equivalents of H ions left. 1/6 of a gram equivalent of CI ions has migrated, so that in AC viQ have 9! gram equivalents of HCl. ions ELECTROCHEMISTRY. 221 In BD we have lost 5/6 of a gram equivalent of H and I gram equivalentof CI ions as gas, but have gained 1/6 of a gram equivalent of CI by the migration ; consequently we have 9^ gram equivalents of HCl left. From these two examples the following law may be deduced: The loss on the cathode (i.e., AC) is related to that on the anode (BD) as the velocity of migration + of the anion (CI) is to that of the cation (H). In this way Hittorf determined the relative velocity of migration of the different ions. At first glance it would seem that if one ion moves more rapidly than the other one side would be posi- tive and the other negative ; but the consideration on page 220 shows that this is not so, and that both sides remain neutral. The one difficulty confronting us is to account for the separation of more ions than are sent to the electrode. This is actually the case, however, and it is probable that around the electrodes in each liquid there is a greater concentration of ions than in any other place. 82. Determination of the relative velocity of mi- gration. — The practical determination of the relative migration velocity is merely a matter of analysis. The apparatus which is used for this purpose is a decom- position-cell, so arranged that no metal can drop from one electrode to the other. The apparatus is fiilled with solution and the current passed through for a certain length of time; the electrodes are of 'the metal 222 ELEMENTS OF PHYSICAL CHEMISTRY. which is contained in the salt. After a certain time one half the liquid is drawn off and a definite amount of it analyzed. This analysis will give us the loss of metal on the one electrode, from which that on the other may be calculated. If n is the fraction of the cation which has migrated from the anode to the cathode when one gram equiva- lent has been separated, then i — « is. that fraction of the anion which has gone to the anode. These two quantities n and i — n are called the transference- numbers of the cation and anion. We have then n _ loss at anode _ u I — 11 loss at cathode v ' where u is the velocity of migration of the cation and V that of the anion. The relative velocity of migration is the distance covered by one ion divided by that covered by both, i.e., the sum of the two distances; and this is what we determine here, for the distances covered are, of course, proportioned to the amounts transported. An example will make the determination of this clear: Hittorf electrolyzed a solution of AgNOj until 1. 2591 grams of Ag were separated. A certain volume of liquid from the cathode gave before the experiment 17.46249 grams of AgCl, and after it 16.6796 grams of AgCl, i.e., a loss of 0.7828 gram of AgCl or of 0.5893 gram of Ag. If no Ag had come to the cathode by migration, the solution would have lost 1.2591 grams of Ag; it ELECTROCHEMISTRY. 223 lost, however, only 0.5893 gram; hence 1.2591- 0.5893 = 0.6698 gram Ag has come to it by the migration. If just as much of the Ag had come by migration as had been separated, the transference- number of the Ag would have been i, i.e., the NO, ions would not have migrated. Only 0.6698 gram of Ag has migrated, however; hence the transference- number for the Ag in AgNO, is found from the pro- portion 1. 2591 : 0.^698 ■.■.\:x = 0.532. The transference-number of the NO, ions is then I — 0.532 = 0.468. Constant results are only obtained here after the concentration has become small enough to prevent the influence of the undissociated molecules — then further dilution has no effect. Constant temperature is also necessary, for an in- crease seems to cause the velocities to become equal- ized. The nature of the solvent has also a great influence upon the velocities of migration. n If one ion is divalent, then the ratio gives the relation of one ion of the divalent element (two charges) to two ions of the monovalent one. A table containing a large number of results from experiments of this sort is given by Kohlrausch,* a *Wled. Ann., LV., 287 (1893). 224 ELEMENTS OF PHYSICAL CH'EMISTRY. few of which follow in Table XVI, and the student is referred to 'them for further information. Table XVI. hittorf's transference-numbers for the anions. Solutions i/io equivalent normal. Substance. v 1/2 KjSOi 0.60 1/2 CuSO» U.64 1/2 HjSO, 0.21 i/2K,COs.. 0.37 1/2 NajCOj 0.48 1/2 LijCOs 0.59 KOH 0.74 NaOH 0.84 KCl 0.507 NaCl 0.63 LiCl 0.70 Substance. NH4CI .... 1/2 BaClj . 1/2 CaClj. . 1/2 MgCU . HCl KNO3 NaNOa AgNOs 1/2 CaNOs KCIO3 0.508 U.61 0.68 0.68 0.21 0.50 o.5i 0.526 o.6r 0.46 B. The Conductivity of Electrolytes. 83. The specific conductivity. — In measuring the electrical conductivity of a solution we obtain results in two different units: one refers to the same amount of all solutions, the specific conductivity ; the other refers to equivalent molecular amounts, the molecular conductivity. The specific conductivity of a solution is the recip- rocal of the resistance in ohms of a cube of the solution with an edge equal to i centimeter at a temperature of 0°. Another form of this is also used: it is the reciprocal of the resistance of a column of solution i meter long ELECTROCHEMISTRY. 22$ with a cross-section of i square millimeter referred to that of a like column of mercury at o°. This is the Siemens unit and can be turned into ohms by dividing by 1.063. The former definition is the better for our purpose, for from the conception of it the molecular conduc- tivity can be more readily understood. The values of the specific conductivity as based upon the above definitions are not equal. The rela- tion between them can, however, be easily determined. Since the conductivity is proportional directly to the section and inversely to the length of the column, we have the proportion I .01 -: — :: 1:0.0001; i.e., the specific conductivity referred to the cube (1 cm. X I cm. X I cm.) is loooo times that referred to the column (100 cms. X .01 sq. cm.). 84. The molecular conductivity. — Since the con- ductivity of a solution depends almost exclusively upon the amount of substance dissolved, it is more convenient for us to obtain results which refer to equivalent amounts of the latter. The molecular conductivity of a substance is the recip- rocal of the resistance of the amount of solution which contains a mol of substance, the electrodes being i cm. apart and large enough to contain between them the entire solution. This can Jhen be readily calculated 226 ELEMENTS OF PHYSICAL CHEMISTRY. from the actually determined specific conductivity. If the mol of substance is dissolved in i liter, then the molecular conductivity is simply looo times the specific conductivity of the cube, for the section is lOOO times as great, the separation of the electrodes remaining the same. In the same way if the mol is present in two liters we multiply by 2000, etc. In genera], if / is the specific conductivity referred to the cube, and v is the number of the liters in which i mol is dissolved, the molecular conductivity is f reciprocal ohms IJL = lX 1000 XV or ly. \o Xv\ i. ( or Siemens units. Since the specific conductivity referred to the cube is 1000 times that referred to the column (100 cm. X I sq. cm.), the molecular conductivity is found from the conductivity of the column, /', by aid of the equation ,, , , ( reciprocal ohms M = /'XiooooX ioooXz'=/Xio Xz' i J. [ orSiemensunits. The equivalent conductivity is that referred to a solution containing i equivalent instead of i mol. For monovalent substances these two values are the same. ' 85. Determination of electrical conductivity. — Since the degree of dissociation of electrolytes can be determined by aid of the conductivity, the method is one of great importance. The apparatus used for this purpose as designed by Kohlrausch is shown in Fig. ELECTROCHEMISTRY. 22/ 17.* The arrangement is the same as that used for determining the resistance of metals, with the excep- tion that an alternating current is produced by an in- duction-coil, and a telephone-receiver is used instead of the galvanometer to show when the resistances on the two sides of the bridge are the same. The alter- nating current is used to prevent actual decomposition of the solution, which would decrease its concentration continually and cause polarization of the electrodes. ' In this way all substance which is deposited by the current in one direction is redissolved by it in the opposite direction, and all polarization effect is nulli- fied. / is an induction-coil which is worked by a battery (not shown in the figure). The current passes through the bridge-wire adb, the rheostat R, and the liquid resistance (to be measured) W which is kept at constant temperature. When the resistance R is in the same ratio to fTas ad is to db no tone is *For details see Ostwald, Handbook of Ph^^^k;;!;;^; Measurements, Macmillan & Co. yicocnemical 228 ELEMENTS OF PHYSICAL CHEMISTRY. heard in the telephone T, i.e., no alternating current goes through the circuit cTd. We have then W:R::db:ad; db W= R-j. ad The conductivity of the cell is then \ _ ad ^ W~^Jb' which, when the electrodes have a cross-section of I sq. cm. and are i cm. apart, is equal to /, the specific conductivity in reciprocal ohms or Siemens units. Its molecular conductivity is, then, equal to jj. = I X lo' X ■z'. It is not necessary, however, to have the electrodes of a particular size or distance apart, for this will simply introduce a constant factor in the equation which can be determined. If k is the factor, i.e., the one to transform our results into reciprocal ohms or Siemens units, then ad ^^ ^'Rdb or adv J 1 reciprocal ohms ^ Rdb I or Siemens units, where a is the reading on the left end of the bridge and b that on the right. If now with the electrode with which we are working we determine ad, db, R, and V for a certain solution, and substitute for }i the ELECTROCHEMISTK V. 229 value found at that temperature by Kohlrausch for a standard electrode, the equation can be solved for K. The solution generally used for this purpose is n/^o KCl, for which at 18° /^ = 112. 2 and at 25° /i = 129.7 reciprocal Siemens units. The form of the electrode is shown in Fig. 18, the con- nections being made by aid of the mercury in the glass n1 r 1 - ■ rr: ~. zr- — — - _F 1 " [ I Fig. 18. (Natural size.) tubes. The electrodes themselves are of Pt, which are coated electrolyticaliy with platinum-black, so as to do away with any difference of potential between them. 230 ELEMENTS OF PHYSICAL CHEMISTRY. 86. General rules. — Since the ions are the carriers of electricity in solution, the molecular conductivity at any dilution divided by that at infinite dilution, i.e., when the substance is present only in the form of ions, will give us the degree of dissociation. We have then jXV a ^ — . This conductivity at infinite dilution means simply that the molecular conductivity is not altered by further dilution. This maximum value for the molec- ular conductivity Kohlrausch found for a binary elec- trolyte to be equal to the sum of two single values, one of which refers to the anion and the other to the cation. This law of the independent migration of the ions shows that conductivity is an additive property. The truth of this law is shown in Table XVII. Table XVII MOLECULAR CONDUCTIVITIES AT INFINITE DILUTION.* K Na Li NH, H Ag CI 123 103 95 122 353 350 109 NO 118 98 OH 228 201 CIO3 115 CjHsO, 94 73 103 83 The differences of two corresponding numbers in the vertical rows, as well as of two in the horizontal ones, are nearly equal, which can only occur when the result is composed of two single and independent values. * At 18° C. ELECTROCHEMISTRY. 23 1 One kind of ion, then, always carries the same amount of electricity, independent of the nature of the other ion. The conductivity for any solution at infinite dilu- tion is }l = kill -\- v), where /^ is a constant which depends upon the unit chosen for u and v, the velocities of the two ions. The ratio of the velocities is known for any one solu- tion from Hittorf's results; hence we can calculate the conductivity from them. We have k{u -\- v) = IJ.„. But u _ n V 1 — n' hence vk = (l — «)/^00, and uk = njA„. If we express u and v in units of conductivity as well as jA, k will be equal to i ; hence v-{\- n)/A„ ; An example of this will show the method: The molecular conductivity of NaCl at infinite dilution /a^ is 103, « from Hittorf's results for NaCl is equal to 0.380 ; hence i — n = 0.620, and in terms of conduc- tivity we have 232 ELEMENTS OF PHYSICAL CHEMISTRY. and :<(Na) — 0.38 X 103 = 39.14 v{Q\) = 0.62 X 103 = 63.86. This 39.14 is the molecular conductivity of a solution of Na ions, while 63.86 is that for a solution of CI ions. In all solutions in the same solvent these values remain constant, so that it is possible for us to calcu- late what the conductivity should be for any solution at infinite dilution. This is of great use, for it is not always possible for us to reach this limiting value in the experiment, or if possible it is accompanied by a large error. A few of the average values of the velocities of migration, expressed in units of conductivity, as determined in several ways, are given by Kohlrausch and are reproduced in Table XVIII. Table XVIII. VELOCITIES OF MIGRATIONS EXPRESSED IN UNITS OF CON- DUCTIVITY.* K = 60 H = 290 NO, = 58 OH = 165 + Na = 40 + Ag = 52 C103 = 52 iTi = 33 Cl = 62 do. = 54 NH. = 60 i = 63 CHsOj = 31 ^;= 18° c. ELECTROCHEMISTRY. 233 From these results, then, it is always possible to find the conductivity of any solution at infinite dilution by simply adding the values for the ions, as given in the table. If the dissociation is not complete at a certain dilution, then it is possible to calculate just what the molecular conductivity will be. Since only the ions conduct the electricity, the molecular con- ductivity for any dilution will be // = a{u -\- v), where a is the degree of dissociation at that dilution, i.e., the fraction of the total amount of substance which is present in the form of ions. 87. The conductivity of organic acids. — The de- termination of the dissociation constant of organic acids, already mentioned, is very easily made by aid of the conductivity. The formula (I -a) can be easily transformed into one in which the known terms are conductivities by aid of the relation a = — . We find then The strength of the acid depends upon the size of this term K. It has been determined for all acids bv 234 ELEMENTS OE PHYSICAL CHEMISTRY. Ostwald, and some of his results will be found in Table VII, already given. 88. The relative velocity of migration from the conductivity. — The molecular conductivity can be determined from the ionic velocities, and hence from the conductivity we can determine velocities which would not be possible to find in any other way. Most neutral salts are largely dissociated, and hence we can reach experimentally the value for infinite dilution, i.e., the maximum molecular conductivity. If the velocity of one ion is known, that of the other then may be readily calculated. This has been done by Bredig.* He finds, in general, that the more complex the ion is the more slowly it migrates; for example, a polymeric ion migrates more slowly than the simple one. Further, the constitution of the ion seems to have an influence, for metameric ions do not always migrate with the same speed. A few velocities determined from the sodium salts (i.e., fjL^ — «(Na)) are given in Table XIX. Table XIX. VELOCITY OF MIGRATION OF THE ANION V IN UNITS OF CONDUCTIVITY. Formic acid v = 55.9 Acetic acid i/ = 43. i Monochloracetic acid v = 42.0 Dichloracetic acid v ^ 40.1 Trichloracetic acid J* = 37-5 Propionic acid i/ =: 39.0 Butyric acid i" = 35-4 Isobutyric acid v = 35.6 * Zeit. f. phys. Chem., XIII., 191, 288 (1894). ELECTROCHEMISTRY. 235 89. The absolute velocity of migration of the ions. — Thus far we have considered only the relative velocity of the ions, or else their velocities in terms of the amount of current which they can transport ; now, however, we shall consider the actual velocity in centimeters per second with which they go through the solution. Imagine two electrodes i cm, apart which have a difference of potential equal to i volt, and assume between them a solution containing i mol of positive and 1 mol of negative ions. In i second the amount of electricity e will go over. Each mol of ions will transport 96537 coulombs of electricity; hence the ratio -^ will give the fraction of the distance, 96537 ^ I cm., which each ion has traversed in i second, i.e., the velocity in centimeter-seconds. By Ohm's law E = \ r where E is the current strength, v is the electromotive force, and r is the resistance. Since z/ = i in this case, we have I e ^ — . r But ^ is the conductivity; hence instead of e we may use the molecular conductivity of the solution, i.e., e fx 96537 ~ 96537' 236 ELEMENTS OF PHYSICAL CHEMISTRY. If 96537 coulombs had passed over the total num- ber of ions would have covered i cm. This is possibly hard to grasp at first, but the following consideration will make it clear. Before the current is passed through, the positive and negative ions will be linked together by electrostatic attraction. The current will separate these and they will go in opposite directions, so that the sum of the two distances covered will be I cm. If the two are in the centre each will traverse 1/2 cm ; if near one electrode the one will have to go further than the other, so that the sum will still be I cm. The molecular conductivity of a o.oooi normal solution of KCl at 18° (in reciprocal ohms) is 128.9; hence the distance covered by the sum of the two ions is 128.9 , J —z — - = o.oon45 cm. per second ; 96537 ^^^ ^ i.e., two ions which are together at the beginning of the equivalent will be separated at the' end by 0.001345 cm. This total velocity is made up of the two single velocities. By Hittorf's experiments the relative velocities of K and CI ions in KCl are in the ratio of 49 to 51. Hence K has a velocity of 0.00066 cms. per second and CI has a velocity of 0.00069 " " " ELECTROCHEMISTRY. 237 in a o.oooi normal solution at 18° for a potential difference equal to i volt. Some other absolute velocities as given by Kohl- rausch are reproduced in Table XX. Table XX. ABSOLUTE VELOCITY OF THE IONS AT l8° IN CMS. PER SECOND. K = 0.00066 H = 0.00320 NH, = 0.00066 CI = L). 00069 Na = 0.00045 NOs = 0.00064 Li = 0.00036 CIO3 = 0.00057 Ag = 0.00057 OH = 0.00181 CrjO? = 0.000473 Cu =: 0.00031 From these results it is also possible to find the molecular conductivity either at infinite dilution or at any other dilution. For infinite dilution we have simply -^"— — u 4-v in cms. per second, 96537 ^ ^ or for any other dilution if a is known /i„ can be cal- culated from a( i^" I =: u -[- V in cms. per second. 196537/ It has been possible to prove the correctness of some of these results by experiment. Whetham's method for this purpose is as follows: If we consider the boundary-line of two equally dense solutions which contain a common colorless ion, but which are colored differently, and call the salts AC and BC, then when 2.38 ELEMENTS OF PHYSICAL CHEMISTRY. a current passes through the boundary-line the C ions go in one direction and the A and B ions in the other. If the A and B ions are the cations, then the color boundary will move in the direction of the current, and its velocity will be equal to the velocity of the ion which causes the change of color. In this way the velocities for Cu, Cr^O,, and CI were determined in centimeter-seconds.' Table XXI compares the values thus found with those calculated by Kohlrausch. Table XXI. COMPARISON OF ABSOLUTE IONIC VELOCITIES. T Whetham, Kohlrausch, by Experiment.* Calculated (new values). Cu 0.00026 0.00031 0.000309 CI 0.00057 0.00069 0.00059 O . 00047 CraO? u. 00048 0.000473 0.00046 Considering all the difficulties in both cases, the agreement is remarkably close, and proves that our idea of the conduction of the current is the correct one. 90. The basicity of an acid. — For all binary organic acids the formula holds strictly. For dibasic acids this is also true * Zeit. f. phys, Chem., XI., 220 (1893). ELECTROCHEMISTRY. 239 when the dissociation is not more than %o%, i.e., all dibasic organic acids under these conditions give off only one ion of H. In the case of the neutral salts of these acids, however, the dissociation into more than two ions take place at a much smaller dilution. The difference of conductivity, then, between two dilutions for a neutral salt must be greater for a polybasic acid than for a monobasic one. Ostwald has made use of this fact for a method of determining basicity. He found empirically that the Na salt of a monobasic acid gives a difference in molecular conductivity at z^ = 32 liters and v = 1024 liters of approximately 10 units, a dibasic one of 20 units, etc. Hence to find the basicity of an acid we have only to find the molecular conductivity of its Na salt at 32 and 1024 liters dilu- tion; then n, the basicity of the acid, is given by = - (see Table XXH). Table XXII. CONDUCTIVITY OF SODIUM SALTS OF ORGANIC ACIDS. Acid. A « = — 10 Formic 10.3 i Acetic g.5 I Propionic 10.2 i Benzoic 8.3 i Quininic 19.8 2 Pyridin-tricarboxylic (i, 2, 3) 31.0 3 (I, 2, 4) 29.4 3 Pyridin-tetracarboxylic 41.8 4 Pyridin-pentacarboxylic , 50.1 5 n 240 ELEMENTS OF PHYSICAL CHEMISTRY. 91. The conductivity of neutral salts. — Another empirical law allows us, from the molecular conduc- tivity of a neutral salt at one dilution, to determine it for any other, provided the salt is largely dissociated, i.e., if yM„ is not very different from fx^. The relation observed is as follows: A'co — -W» = '^l • «. • C.„ or when w, and n^ are the valences of the anion and cation respectively, and c^ is a constant for all elec- trolytes. When c^ is known for all dilutions, and also the terms jx^, w„ and «j, then we can find the value of p.^, i.e., the molecular conductivity at infinite dilution. If we designate n^ . n^ . c^ by d^, then /^o, = /*. + 4- Table XXIII gives the value of d^ for different dilutions and values of «, . 11^ at 25°. Table XXIII. Valence, wi.wa ^64 '^las <^ihi ^bia ^^1024 I II 8 6 4 3 2 21 16 12 8 6 3 30 23 17 12 8 4 42 31 23 16 10 5 53 39 29 21 13 6 60 48 36 25 16 This, as before mentioned, holds only for salts which are largely dissociated. ELECTROCHEMISTRY. 24I 92. The dissociation of water. — The ions of water are H and OH, i.e., to a very sh'ght degree it is a binary electrolyte. The reciprocal of the specific resistance (Siemens units) of a specially purified water was found by Kohlrausch to be 0.014. io-'° at 0° C. 0.040. 10- '° at 18° 0.058. io-'° at 25° 0.089.10-'° at 34° 0.176.10-'° at 50° One mm. of this water at 0° has a resistance equal to that of a copper wire of the same section and 40000000 kilometers long, which is long enough to go around the earth a thousand times. From this conductivity the degree of dissociation is + easily determined. One mol of H ions has a conduc- tivity equal to 290 units (in a previous table), while I mol of OH ions has one equal to 165; hence the maximum molecular conductivity of water should be pL^ = 290 -\- 165 =455 if completely dissociated. The above specific resistances refer to a column I meter long with a section of i sq. mm. That of a liter between electrodes i cm. apart would be at 18° 10' times as great as 0.040- io-'°, 0.04 X io-'° X 10' = 10-° X 0.04; 242 ELEMENTS OF PHYSICAL CHEMISTRY. hence 0.04 X 10 ' — ^-— = 0.9 X 10 ; 455 + which is the concentration of H and OH ions in mols per liter at 18°, or there are 17 grams of OH and i gram of H ions in'iioooooo liters of water. 93. The temperature coefificient. — According to Kohlrausch, the conductivity varies with the tempera- ture as follows: /'. = /^,s[i+A^- 18)], where /S is the temperature coefificient, or the change in conductivity for i" C. We can find /? by experi- ment from the equation ^ - i^^lt - 1 8)- Substances with small molecular conductivities usually have large temperature coefficients. Most of the strongly dissociated salts have a value for /3 equal to 0.025, the molecular conductivity changes 2\ is the osmotic pressure of the metal ions in the solution, three possibilities: 1. P = p. 2. P> p. 3- P p. This is the case with Zn. Ions of Zn will go from the electrode into the solution, taking with them positive electricity. This will take place only to a very slight degree, for it would increase the positive charge of the solution. These positive ions around the electrode will be attracted to the plate, which is negatively charged owing to the loss of the positive ions. We have, then, finally, the metal negative against the solution, and the electrode sur- rounded by a Helmholtz double layer of ions. If positive electricity is given now to the metal, the double layer is broken up and more Zn will go into solution in the ionic form, but as soon as the 256 ELEMENTS OF PHYSICAL CHEMISTRY. current is stopped the double layer will agaia be found. 3. P < p. This is true for Cu. A stick of copper in a solution of one of its salts will have ions of Cu from the solution deposited upon it, since the pressure toward the Cu is greater than that away from it. These ions take positive electricity with them, conse- quently the metal is positive against its solution. The double layer of ions will also be found here, for the positive ions on the metal will attract the nega- tive ones from the solution. In this case, however, the double layer will only be broken up when elec- tricity is conducted away from the Cu, when the Cu ions will again precipitate and continue to do so until p — P, when all action will cease. Since i gram equivalent of ions carries with it 96537 coulombs of electricity, the amount of metal going in or out of solution may be infinitesirnal and still cause the formation of a measurable amount of electricity. Experiment has shown that the metals Na, K . . ., etc., up to Zn, Cd, Co, Ni, and Fe are always negative against their solutions, i.e., P> p. The noble metals, on the contrary, are positive against their solutions, although in some few cases it is possible to get a solution in which P> p. In general, though, for the noble metals P < p. A negative element has exactly the same action except that, in general, as far as is known, P > p. Here, although P > p, the electrode is positive against ELECTROCHEMISTRY. 257 the solution, for the negative ions formed from the electrode leave positive electricity behind. In general, the electrolytic solution-pressure de- pends upon the temperature, the nature of the solvent, and the concentration of the substance in the elec- trode (see later). 100. Theoretical formula for the difference of potential between a metal and a solution of one of its salts. — It is possible from the conception of solution-pressure to derive a formula giving the value of the E.M.F. of a single electrode. This will depend upon the osmotic pressure of the metal ions working in the one direction and upon the electrolytic solution- tension working in the other. Since the ions go from one pressure to the other, a certain amount of work is done; this amount is always the same no matter in what form it appears, i.e., whether electrical or osmotic. When ions are transferred from the pressure P to the pressure /, the osmotic work done is equal to Jp P Jp P from which, by integration, we obtain P RTl~. P The corresponding electrical work, however, is ne„ where n is the difference of potential between the 258 ELEMENTS OF PHYSICAL CHEMISTRY. metal and solution, and e„ is the amount of electricity carried by i gram equivalent of ions. We have then or RT P 71 = / — . e. P From this we see that if tt = o, P =^ p, which corre- sponds to experience. 6„ for I mol of a monovalent ion is equal to 96537 coulombs. We must, however, express both sides of the equation in electrical units. The electrical equiva- lent of heat is 4.25. Since 72 = 2 cals., the right side of the equation must be multiplied by 4.24 in order to obtain our result in electrical units. We have then A.2A- i.q6 P 96537 X ;r(volts) = ^-^ — — T log -: ^ ^•'^ ^ ' 0.4343 ^ p or P n = 0.0575 log — volts at 17° C. For a divalent element we must multiply e, by 2 ; hence if n is the valence of the metal we have / -\ 0.000198 _, P , (45) Tt = ^ T log — volts. If we consider a cell composed of two metals, each present in a solution of one of its salts, the difference ELECTROCHEMISTRY. 259 of potential may arise (1) at the place of contact of the two metals, (2) at the place of contact of the two solutions, and (3) and (4) at the points of contact of the two electrodes with their solutions. The first can be entirely neglected provided the temperature re- mains constant. The second, as will be shown later, is always small, so that we shall have to consider only (3) and (4). At 17° the E.M.F. of the cell will be, then, / ^N / ^ 0-0575 , P 0.0575 , P (46) ;r = (;r, - n^ = -^ log ^ log - volts, the minus sign being used because ions are formed on the one side, and consequently must disappear upon the other, for there must always be an equal number of positive and negative ions in the solution. lOi. The Lippmann electrometer. — This instru- ment has already been mentioned, but the theory of it has not yet been considered, so that here we shall study its action. As a result of a large number of experiments Lippmann announced the following law: The surface-tension of mercury at its point of contact with dilute H^SO^ is a steady function of the E.M.F. of polarization upon that surface. Helmholtz by use of his theory of an electrical double layer explained this process more completely. If Hg is brought in contact with dilute H^SO,, a small portion of oxide, which is always present upon the surface, dissolves, so that practically we have a dilute solution of sulphate of mercury in contact with 26o ELEMENTS OF PHYSICAL CHEMISTRY. the mercury. The electrolytic solution-pressure of mercury is very small — so small, in fact, that the ions of Hg present in the solution are precipitated to a very slight extent upon the electrode. These posi- tive ions cause the mercury to become positive against its solution, and so the negative ions of the solution are attracted to the mercury, forming a Helmholtz double layer upon it. The bubble of mercury will now assume its largest surface, since the ions in the layer repel one another and become separated as far as possible. This flattening of the surface works against the surface-tension of the mercury. If now to this bubble of mercury negative electricity is supplied, its positive charge will decrease. By this the number of negative ions attracted by it will decrease, so that the surface-tension will become greater than the repelling influence of the negative ions, and the surface will become smaller. If we continue to supply negative electricity to the bubble, we shall finally neutralize all the positive charge in it. The layer of ions will then go back into the liquid, and the surface-tension will reach its maximum, and the potential difference between electrode and liquid will be reduced to zero. If now more negative electricity is supplied to the bubble, it will become charged negatively and attract positive ions to it, and the bubble will again flatten, i.e., the surface-tension is again overcome. The im- portant point for us, however, is that the potential difference is equal to zero for the maxiinuin surface-ten- ELECTROCUEMtSTRY. 261 sion. Since this is true, we shall have the maximum surface-tension when the mercury and H,SO, are con- nected together with a wire. If now the acid and mercury are placed in the tube already illustrated, Fig. 18, and the two poles connected, the meniscus will remain in the same position. If, instead of connecting the two wires together, the two ends are connected with a compensated E.M.F., the mer- cury will not change its position in the tube, for both sides have the potential of zero. If a difference of potential does exist, the mercury will move in one direction or the other, and continue to do so until by compensation the difference becomes zero. A glance at Fig. 19 will show the principle of this. When the key K is up, both sides of the electrometer are con- nected. When the key is pressed down, the two electromotive forces in process of compensation are connected to the sides of the electrometer and com- pared. It is often convenient to measure each single elec- trode alone; this is done by the aid of a normal elec- trode. One arrangement of this sort consists of a stream of Hg flowing into a solution of an electrolyte (KCl). If this is so fixed that the column breaks into drops just as it enters the electrolyte, the potential difference between the metal and solution is zero. If, for example, the mercury is positive against the solution, then each drop as it falls will carry with it 262 ELEMENTS OE PHYSICAL CHEMISTRY. positive electricity until the two portions are of the same potential. If to the Hg of such an arrangement we connect the metal of a single electrode, connecting the liquids with a siphon, the electromotive force determined will be that of the one single electrode, since that of the other is equal to zero. This piece of apparatus is large and clumsy, so that for practical purposes we use a normal electrode, which gives a certain deiinite potential difference as measured against the mercury electrode. The elec- trode in general use is made up according to the scheme HgCl/normal KCl/Hg, which gives a difference of potential of — 0.56 volt, i.e., the metal is positive against the solution. Fig. Fig. 22. 22 shows the form of the cell. The syphon tube is so arranged that it is always kept full of n/i KCl. ELECTROCHEMIS TR Y. 263 This dips into the liquid of the other electrode pro- vided they form no precipitate. In case they do a solution of n/\ KNOj is used, into which the two siphons from the electrodes dip. The advantage of using KNOj as a connection is that the velocity of migration of the K and NO, ions is nearly the same ; hence the passage of the current produces no change in the concentration at the two sides which would cause a difference of potential. An example of the use of this electrode in measur- ing the E.M.F. of any other is given by the following: For a combination in the form Zn - «ZnSO, - «KC1 HgCl - Hg, i.e., for Zn «ZnSO,, against the normal electrode we find n = 1.08 volts, the current going from Hg to Zn through the wire. The total E.M.F. is given by the equation - RT ,P RT ,P' where P and J> refer to the Zn and P' and p' to the Hg, since that is the only arrangement which would make n positive. We have then „ RT ,P , ^ 1.08 = / V- 0.56, RT P t^^l— = (1.08 — 0.56) = -t-o.S2 volt ; 264 ELEMENTS OF PHYSICAL CHEMISTHY. i.e., Zn is positive against a n/\ solution of its sulfate and gives an E.M.F. of 0.52 volt. The sign always refers to the E.M.F. of the electrolyte against the elec- trode. Thus, — I means that the solution is negative and the metal positive and vice versa. In the cases of Cu - «CuSO. - wKCl HgCl - Hg, we find n = 0.025 volt, the current in the wire going from Cu to Hg. We have here, since in the solution the current goes from Hg to Cu, RT ,P' RT jP n = /— .- 1— 6. / 26 / or , RT ,P 0.025 = -0.56-—/-, RT P /— = — 0.025 — 0.56 = — 0.585 ; i.e., the electrode of Cu is positive, with a difference of potential against the electrolyte equal to 0.585 volt. The process which takes place is shown more clearly by the diagram Cu«CuSO HgCl in KCkHG I ^-^0.56 I > »-> ' 0.025 hence the copper must be more positive than the Hg by 0.585 volt. ELECTROCttEMISTRV. 265 102. Concentration-cells. — a. In which the elec- trodes have different concentrations. — If the electrodes are made of two amalgams of the same metal in salts of this metal, the E.M.F. is given by the formula 0.000198™, P 0.000198 „ , P' , Tt ^— 1 log ^— 1 log —r volts. n ^ p n ^ p' If the two solutions have the same concentration of metal ions, then p = /', and we have ooooiqS™, P' ,^ n = ^^T log p volts, when P and P' are the electrolytic solution-pressures of the two electrodes with respect to the dissolved metal. Amalgams when dilute may be considered as solu- tions in which the mercury acts as the solvent; conse- quently the osmotic pressure of the metal in the amalgam will be proportional to the electrolytic solu- tion-pressure of the electrode. Since in all solutions the osmotic pressure is proportional to the concentra- tion, we may substitute for the electrolytic solution- pressures the proportional terms, the concentrations of the metal in the two electrodes. We obtain then 0.000198 ™ , c .^ n = 2—7 log -; volts. This formula was proven experimentally by G. Meyer, whose results for zinc amalgams in a ZnSO, solution are given in Table XXVI. 266 ELEMENTS OF PHYSICAL CHEMISTRY. Table XXVI, T c c' IT (obs.) IT (calc.) 11°. 6 0.003366 0. 0001 1305 0.0419 V. 0.0416 V. l8° 0.003366 O.OOOII305 0.0433 V. 0.0425 V. 12°. 4 0.002280 0.0000608 0.0474 V. 0.0445 V. 6o° 0.002280 0.0000608 0.0520 V. 0.0519 V. The results obtained by Meyer also proved the formula to hold for copper amalgams in CuSO^, etc. The derivation of the equation in the way given shows the value and use of the conception of the electrolytic solution-pressure, but it is possible to derive it without this idea, and possibly the action of the cell will be made clearer if we do so. Imagine a cell composed of two amalgams of Zn of different strengths in a solution of a Zn salt. The process of the cell is as follows: Zn goes from the concentrated amalgam into the solution and precipitates Zn upon the more dilute electrode. The action of the cell, then, is simply the transportation of Zn ions from the concentrated to the dilute amalgam, i.e., Zn ions from the higher osmotic pressure /, proportional to the concentration c, to the lower osmotic pressure/', which is proportional to the concentration c' . The E.M.F. generated by this process is then equal to the maxi- mum work which could be obtained by the process. The maximum osmotic work is equal to RTlu-. c ELECTROCHEMISTRY. 267 or RT , c — log -;. 0.4343 ^ c' The electrical work is « X 96537 X n, or, since for Zn « = 2, 2 X 96s s;?^- Hence and 2 X 90537'r = log- -7 -^ ^ "^ 0.4343 ^ 0.000198 _, c n = ^— r log p volts, which, when n=- 2, is identical with the equation found previously. In both these equations we have assumed the metals to dissolve in mercury in monatomic molecules, and this assumption is upheld by our results, both by this method and by others. If Zn had been present in the form of diatomic molecules in the mercury, our equation would have had a different value. For the movement of the same amount of ions as before we should have had, then, the osmotic work equal to \ RT ^ c - . log -;, 2 0,4343 ^ c' 268 MLEMENTS op physical CHEMtSTRY. since the number of particles would have been 1/2 what it was, and the work depends simply upon the number of particles. The electrical work would have been the same as before, i.e., 2 X 965371', and we should have had 1 0.000198 c n =^ - • 7 log —. volts ; 2 2 ^ c' i.e., the E.M.F. would have been one half what we have found it to be. The electromotive force is found, then, to be de- pendent only upon the concentration of the metal in the amalgam and its valence, and not upon the nature of the metal itself. The mercury has no effect so long as the metal dissolved in it gives the larger E.M.F. The same is true of an alloy of two metals: only the one which gives the higher E.M.F. has any influence unless its concentration is so much larger than that of the other that it counteracts its action. An example of this is given by an alloy of Cd and Zn in acid. The acid is first saturated with Zn and then only does the Cd go into solution. In the first part of this process we shall obtain only the E.M.F. of the Zn and afterwards that of the Cd. Another example of the electrodes ha\ing different pressures owing to differences of concentration is given by cells of the type of the Grove gas-battery in an ELECTROCHEMISTRY. 269 altered form. The electrodes are of platinized plati- num, in which the gas is absorbed under different pressures, and are placed partly in the liquid and partly in gas. Such an electrode is to be considered as a perfect reversible gas electrode,* i.e., one from which ions of the gas are given up, for the metal acts simply as a conductor, as has been shown experi- mentally by the use of different metals, the same results being always obtained. In this way reversible electrodes can be made of all gases. Oxygen as an electrode, however, gives off ions of OH, since those of do not exist, and absorbs O when the OH ions give up their charges to it. If we have two electrodes of H, under different pressures, in contact with a liquid containing H ions, we shall obtain a certain E.M.F. This may be calcu- lated in two ways, as we did in the case of amalgams. In the second way, however, the process is slightly different, since one molecule of H gas forms two monovalent ions. The osmotic work is equal to RT , P log 0.4343 ^ P" as before. The electrical work, however, which corre- sponds to this is .2e„;r; hence RT , P " = iM3^)^°^P- * The solution tension here of the gas electrode is proportional to the square root of the gaseous pressure. 270 ELEMENTS OF PHYSICAL CHEMISTRY. i.e., we have 2 in the denominator notwithstanding that the gas is monovalent. b. Different ionic concentrations. — In this type of cell we assume the electrodes to be of the same metal, but dipping into solutions which possess different ionic concentrations of the metal. An example is given by the arrangement Ag(AgNO, cone.) - (AgNO, dilute)Ag. From the general equation for silver we have P P 7t = RTl RTl'~j. P P Since P =. P' this becomes 7t - RTl^,, P where p is the osmotic pressure of the Ag ions on the concentrated side, and^' that on the dilute side. From this we see that the E.M.F. depends only upon the ratio of the concentrations, and not at all upon the actual concentrations, all of which was found to hold experimentally by Nernst, the developer of the theory. Since the osmotic pressure is proportional to the concentration, we may substitute the latter for the former; we obtain RT X n = /-. n c For example, with a o.ooi and a o.oi normal ELECTRO CHE MIS TRY. 271 solution of AgNO, with Ag electrodes we have, from the conductivity, ionic concentration in the proportion of I : 8.71 (not I : lo); hence the E.M.F. of such a cell is at i8° Tf = 0.000198 X 291 log 8.71 = 0.054 volt, while 0.055 volt was found by experiment. At 17" C. we have 0-0575 , / „ n = — — log -7 volts. If the concentration ratio of the ions is lo, then n = 0.0575 volt for a monovalent ion, or 0.02875 volt for a divalent one. This class includes all cells made up of solutions containing different concentrations of metal ions. In case the contact of the two liquids causes the forma- tion of a precipitate, another solution is used as a con- necting link, the two being in contact in small siphons. 103. Dissociation by aid of the electromotive force. — Since for concentration-cells we have the formula T = — ~ log — , volts, n X 0.4343 ^ / or for monovalent metals °^7'~ 0.000198^' we can find the ionic concentration of the metal on the one side provided that on the other is known. 272 ELEMENTS OF* PHYSICAL CHEMISTRY. An example of this is the cell Ag - ^AgNO, - KNO, - £KAgCN, - Ag, which gives an E.M.F. of 0.542 volt. If, for simplicity, AgNO, i/io normal is considered as completely dissociated, then the concentration of the Ag ions will be o. i normal, and, since 7^= 273 -\- 17 = 290°, we have c 0.542 , log -7 = ^^, , C — 3.0 X 10-" ; ^ c' 0.000198 X 290 ^ ' i.e., in the n/20 KAgCN, solution the Ag ions are present to a concentration of 3.0 X lO"" normal. This method was devised by Ostwald, and is one of the most delicate known, for the more dilute the one solution is, i.e., the smaller the number of its metal ions, the greater is the E.M.F. of the cell. Another typical cell of this kind is Ag - ;^AgNO, - KNO3 - ^KClAgCl - Ag, which was used by Goodwin to determine the solu- bility of AgCl. In a saturated water solution of AgCl we may assume without error that the dissociation is complete. The solubility product will be then Ag X CI = s, or the concentration of Ag ions is equal to Vs. Naturally if we have a certain strength (normal) solu- ELECTROCHEMISTRY. 2/3 tion of AgCl the concentration of the Ag ions is the same as that of the AgCl; hence ^s is the solubility of the AgCl. In the presence of KCl the concentra- tion of Ag ions will be diminished, but the product of the concentration of Ag ions, with that of those of CI, will always give the same solubility product s. If now we determine the E.M.F. of the cell when the concentration of CI ions is known, we find the concentration of those of Ag. The square root of the product of the two will give us, then, the solubil- ity of AgCl in pure water. Goodwin found when the concentration of AgNOj and KCl was o. i normal an average E.M.F. of 0.45 volt. If we work at 25" C, then a:(AgNO,«/io) = 82^ and a(KCl«/io) = 85^, and W^ = 5:450 ^ c' 0.000198 X 298 or c' = 1.94 X 10-; i.e., Ag ions are present with the excess of KCl to a concentration of 1.94 X lO"" mols. per liter. The product s is then found from 1.94 X 10-" X 0.085 = 5 = 1.64 X io-'° and 1/7= 1.28 X lo-"; 274 ELEMENTS OF PHYSICAL CHEMISTRY. i.e., AgCl in a saturated water solution at 25° is present to a concentration of 1.28 X lO"" normal. A cell of this sort may be arranged with bromides, iodides, etc., in place of the chlorides, which was also done by Goodwin, and the solubility of AgBr and Agl determined. Another illustration of this method of determining dissociation is furnished by Ostwald's arrangement for determining the dissociation constant for water. The scheme is a gas cell in the form H — Acid Base — H, when the two H electrodes have the same solution- pressure. The E.M.F. of such a cell, deducting the E.M.F. caused by the contact of the two solutions, is, for normal solutions at 17" C, n = 0.0575 log 4,- By experiment ^ = 0.81 volt; hence, since / = 0.8 (i.e., the normal acid is 80^ dissociated), P or /' = 0.8 X 10-'*. Since this is the concentration (in terms normal) of the H in the base, and the concentration of OH ions in contact with them is 0.8, the dissociation constant of H,0 is 0.8 X 0.8 X 10-". ELECTROCHEMISTRY. 27$ 4- Since H,0 dissociates into H and OH ions, the con- + centration of H and OH ions in pure water is the same, and is equal to |/(o.8y X IO-" = 0.8 X IO-' = K for H,0. By the conductivity at this temperature Kohlrausch found 0.9 X 10-'. It would be possible in this case to use oxygen elec- trodes, by which the same final result would be obtained, but platinum-black absorbs very little oxygen, so that the results are not as certain as with hydrogen, 104. The electromotive force between liquids. — If a concentrated acid is in contact with an unmixed layer of dilute acid or pure water, a difference of potential between the two liquids is exhibited. This difference is only apparent for a very short time and disappears by the diffusion which causes the entire liquid to become homogeneous. This action is due, according to Nernst, to the unlike velocity of migra- tion of the two ions of which the acid is composed. In the concentrated acid the osmotic pressure of the ions is greater than in the other; hence they are driven into the dilute solutions by a pressure equal to the difference between the two osmotic pressures. In this way both ions are driven by the same pres- 276 ELEMENTS OF PHYSICAL CHEMISTRY. sures, but the velocity of the H ions is much greater than that of any negative ion, so that for a short time the weaker solution will show the polarity of the faster moving ion. This difference of velocity, however, in any one layer of liquid will become equal- ized by the electrostatic attraction of ions with unlike charges, i.e., the faster ions increase the velocity of the slower ones, and the latter decrease the velocity of'the former. The difference of potential, however, will exist until the solution is homogeneous. The electromotive force due to the contact of two liquids can be calculated by the following process: Imagine two solutions, containing the same two mono- valent ions but of different concentrations, to be in contact. If n is the transference-number of the posi- tive ion, and i — n that of the negative ion, then by the passage of a current of electricity to the amount e„ (i.e., 96537 coulombs) the following change will take place: If the current enters the concentrated solution and then passes through the dilute one, n gram equivalents of positive ions will go from the concen- trated to the dilute solution, and during the same time I — n gram equivalents of negative ions will go from the dilute to the concentrated side. Let p be the concentration of positive and negative ions in the concentrated solution, and/' that in the dilute. The maximum work done is nRTl^, - (I - n)RTlt^^ = {2n - i); X RTl^„ P P P ELECTROCHEMISTRY. 277 11 or, if we substitute for n the value , where u is u Y. V the velocity of the positive ions and v that of the negative ions, (2n - i)RTl^, = ^t^RTl^, ^ ' p' u-\-v p' or , . u — V RT ,.p (47) '" — — i ^^7- ^" u-\-v e^ p' If u is greater than v, the current goes from the dilute solution to the concentrated one, i.e., the dilute side is positive. If v is greater than u (exam- ple, KOH), the concentrated solution is positive. If u — V, then we have, both by theory and experiment, no difference in potential. The formula for other cases is too complicated and of too little value to find a place here, so that the student must be referred to Nernst's original article* for further information. Formula (47) has been proven experimentally within the experimental error by Nernst, so that it un- doubtedly represents the true process taking place. 105. Solution-pressure of the metals. — It is possi- ble from the potential difference between a metal and its salt solution to calculate the solution-pressure in atmospheres. The equation is RT ,P «e„ p * Zeit. {. phys. Chem., IV., 129 (1888). 278 ELEMENTS OF PHYSICAL CHEMISTRY. or at 17" log P = -^^^ + log ;^ + 0.4343- If we use normal solutions, / is equal to 22 atmos- pheres if completely dissociated. The pressure of the ions, in case the salt is not completely dissociated, is then easily found (i.e., a X 22 atmospheres). Since n can be determined, the value for/* can be found for all metals. Table XXVII gives the values as deter- mined in atmospheres by Neumann. Table XXVII. SOLUTION-PRESSURES OF METALS. Zinc 9-9 X 10" atmospheres. Cadmium 2.7 X 10' " Tiiallium 7.7X10' Iron 1.2 X 10* " Cobalt 1.9 X lo" " -Niclcel...- 1.3X10" " Lead i.iXio"' Hydrogen g.g X io~'* " Copper 4.8 X l0-'» " Mercury l.lXio~" " Silver '^t. 2.3 X 10~" Palladium i.5Xlo~'* " The influence of the dilution on each side of an element depends upon the solution-pressure. If the latter is great, then dilute solutions are necessary for maximum action ; if small, then concentrated solutions are to be preferred. 106. Cells with inert electrodes. — Elements" of this type have electrodes in solutions which do not ELECl'ROCHEMISTRY. 279 contain the ions of the electrodes. Such cells are also known as oxidation- and reduction-cells. An example is given by the arrangement plat.Pt - FeCl,sol. . . . .".SnCl,sol. — plat.Pt. ++ ,++++ The Sn ions go over into Sn ions, taking up the elec- +++ tricity set free by the transformation of Fe ions into ++ those of Fe. The Fe side is, then, positive and the Sn side negative. The origin of the current in these cells is due to the oxidation on the one side and the reduction on the other. An oxidizing substance (i.e., one which is reduced) is any substance from which negative ions are formed or upon which positive ions give up their charges. A reducing substance is one in which nega- tive ions are given up or positive ions formed. 107. Processes taking place in^the cells in com- mon use. — All cells give an E.M.F. which is equal to the difference of the E.M.F's. given by the tw o - ai - ng fe electrodes; but in some few cases this is accompanied by other processes. In the following the action of depolarizers is given, as well as the change in the cell during action. " '' Clark cell. — This cell, as already mentioned, is used to give a constant E.M.F. as a standard. The scheme is ' Hg - Hg.SO, - ZnSO, - Zn. 28o ELEMEN7S OP PHYSICAL CHEMISTRY, The HgjSO, is quite insoluble, but a small number of Hg ions does exist in the solution. The Zn ions with a high solution-pressure are driven from the electrode into the ZnSO, solution,' and drive before them to the electrode the ions of Hg, since the solution can con- tain only the same number of positive and negative ions. The Hg ions give up their charges to the elec- trode, which becomes positive, the Zn being negative from the loss of positive ions. The current goes, then, externally through the wire from the Hg to the Zn, and internally from the Zn to the Hg. If the resist- ance placed against the cell is great enough, only an infinitesimal portion of the Hg ions is driven from the solution, and the cell remains constant. If the current is passed through a small resistance, however, all the Hg ions are soon precipitated upon the elec- trode and the E.M.F. is reduced, i.e., the cell is polarized. If allowed to stand, however, more Hg,SO, dissolves and the original E.M.F. is obtained. The Leclanchd cell consists of a solution of ammonium chloride, in which we have two electrodes, Zn and C + MnOj. The action of the MnO, is to prevent polarization, the processes taking place without it and with it being as follows: In the cases without MnO, the Zn with its high solution-pressure goes into solution, driving before it the other positive ions, i.e., those of NH,. These ions force those of H from the water out of the solu- tion, and they give up their charges to the electrode. ELECTROCHEMISTRY. 28l The bubbles of H collect upon the electrode and are absorbed and so given off to the air, but, as this process is slow, other ions are prevented from giving up their charges and consequently the E.M.F. decreases. To get rid of this action the MnO, is used. In contact with water we have, then, to a small extent, a solution of MnO,, which dissociates according to the scheme ++++ MnO, + 2H,0 =: Mn + 4OH. These tetravalent ions of Mn have the tendency to go into the divalent state by giving up two equivalents ++ of electricity, i.e., to form the ions Mn. In conse- ++ quence of this the Mn ions with those of NH, are driven to the electrode by the Zn ions, and, since they give up two equivalents of electricity more readily than any other ion gives up its entire charge, the elec- tricity is given up without any substance which might ++ cause polarization. We have, then, MnCl, (i.e., Mn ions) formed in the solution. The process continues so long as the solution is unsaturated with Zn ions; when saturated the action ceases. Bichromate cell. — This cell is arranged according to the scheme Zn - H,SO. + K,Cr,0, - C. The action of the two substances in solution forms bichromic acid (H,Cr,0,). This dissociates into 282 ELEMENTS OF PHYSICAL CHEMISTRY. to a lg.rge extent, and to a smaller degree into ++++++ _ H,Cr,0, + 5H,0 = 2Cr + 12 . OH. These hexavalent Cr ions have the tendency to give up three equivalents of electricity and to go into the +++ trivalent state (i.e., Cr). Accordingly the Zn ions which are forced from the electrode drive before them ++++++ the 2Cr ions, which give up three of their equivalents +++ and become Cr, which remain in the solution as ions in equilibrium with SO, ions (i.e., as Cr,(S0j3). We have, then, finally, a solution of Cr,(SOJ, left in the jar. This change in the number of electrical equiva- lents by a change of valence always takes place more readily than the change from the ionic to the elemental state and is of great value as a means of preventing polarization. Accumulators. — The action of the lead accumulator or storage-cell also depends upon a change of valence. Any reversible cell will act as a storage-cell, after it is used up, by the passage of a current through it in the direction opposite to that in which it goes of itself. The lead cell, however, is generally used on account of its high E.M.F. This consists before charging of two lead plates, one of which is coated with litharge (PbO) in a 20 Gas 268, 269 , Helmholtz 250 , Leclanch6 280 , One-volt 250 , Osmotic theory of the 257 , Storage 282 293 294 INDEX. PAGE Charles, Law of 4 Chemical change 158 at constant pressure 151 volume 150 kinetics 194 Coefficient of affinity 170 Color of solutions 122-125 Conductivity, Determination of the 226-229 of neutral salts 240 organic acids 233 , Solubility by aid of the 243 , Temperature coefficient of 242 , The molecular 224-226 , The specific 224 Constant, Dissociation or ionization 93 of a reaction of the ist order, The 197 2d " " 200 3d " " 205 hydrolytic dissociation, The 175, 176 Coulomb 215 Cycle, The 49 Dalton's law 14 Decomposition of HjO, Primary 289 val ues 283-286 Dielectric constant and dissociating power 243 Dissociation and external pressure 164, 169 . Constant of 27, 164, 170 , Degree of 29, 92, 115, 116 , Electrolytic 91-94 from E.M.F 271-275 , Gaseous 26, 164 , Heat of, of solids 187-189 gases igo, 191 , Hydrolytic 141, 175, 176 in normal solutions 245 solution. Non-electrolytic 182, 183 of double molecules in solution 169 ' HjO 241, 242, 2741 polyatomic molecules 165, 166 INDEX. 295 PACE Dissociating power and dielectric constant 243 Distillation, Fractional 79-82 Division of a substance between two solvents 118, iig Dyne 2 Electrical conductivity, see Conductivity. Electricity, Mechanical equivalent of 216 , Thermal equivalent of 216 Electrochemistry 215 Electrodes, Concentration of 265-269 , Normal 261-264 Electrolysis 283 Electrolytic solution-pressure 254 Electrometer, Lippmann 247, 259, 260 Electromotive force 246 Energy, Electrical 5 , Factors of 3 , Factors of heat 53, 54 , Intrinsic 3 , Kinetic 3, 4 , Mechanical 3 , Potential 3 , Volume 4 Entropy 53, 54 Equilibrium 3, 153, 194 and external pressure 164, 169 between gases and solids 177 , Complete 177, 210 , Constant of 161 , Effect of temperature upon 184-194 , Physical 177 Erg 2 Faraday, Law of 216 Film, Semipermeable 83 Force - 2 , Chemical 160 Freezing-point and external pressure 71 vapor-pressure 113, 114 , Depression of the 111-118 296 INDEX. PAGE Freezing-point, Determination of the 116-118 , Molecular depression of the 111-113 Gas constant. Molecular 13 , Specific 13 , Definition of a 10 , Equation of state for a 13 laws 10 Gases and liquids 55 in liquids 74 , Kinetic theory of 30 , Specific gravity of 15-17 , Determination of specific gravity of 18-25 Heat, Atomic 69 , Capacity for 36 , Determination of specific 46 , Latent 70, 71 of association 155 dissociation 187-189, 190-193 evaporation 62, 63 formation 148 neutralization 155, 156, 193, 194 precipitation 156 solution 147 vaporization 187 , Specific 36 Hess, Law of 145 van't Hoff, Law of 85-88 Hydrolytic dissociation, Constant of 175, 176 Indicators, Action of 125-129 Intensity factor 3 Ionic concentrations 270 Ionization 91-94 and solubility 183-184 constant 170— 171 Ions 91-95 and analytical chemistry 122-146 in thermal reactions .154, 155 , Migration of the 221-217 INDEX. 297 PAGB Isohydric solutions iy2 Kohlrausch, Law of 230 Liquids in liquids 77-82 , Specific gravity of 55-58 Mass actioa, Law of 159-163 Migration of the ions, Absolute velocity of 235-238 in terms of conductivity 232-234 , Velocity of 221, 222 Mol 13 Molecular weight from depression of freezing-point 116 depressed solubility 109-121 depression of vapor-pressure 98 increase of the boiling-point no Natterer, Equation of 34. Nernst, Laws of 129, 130 Non-homogeneous systems 176, 177 Ohm 215 Osmotic pressure and molecular weight 88, 89, 98 vapor-pressure 99-104 , Simple demonstration of 89, 90 theory of the cell 257 . -rpetual motion of the ist kind , 48 2d " 48 Phases 176, 209-214 Physical chemistry i Pyknometer 56 Point, Transition 212 , Triple 212 Polarization, Theory of 286 Pressure, Critical 59 , Electrolytic solution 254 , Molecular depression of the vapor 96 , Osmotic 82-91, 104 , Relative depression of the vapor 96 . Solution 94> 95 , Vapor "62 Raoult, Law of 96, in 298 INDEX. PAGE Reactions between solids and liquids 207 , Incomplete 206, 207 of the ist order ig7 2d " 200 3d " 205 , Reciprocal and reversible 158 Separation of metals by graded E.M. F. 's 291 Solid state. The 69-73 solutions 169, 170 Solids in liquids 87 Solution, Heat of 147 Solution pressure of metals 277 Solutions 74-121 , Color of 122-125 , Isohydric 172-174 , Saturated 82 , Solid 169, 170 , Vapor-pressure of 95-104 Solubility, Depressed iig-121, 131, 132 , Increased 132 from conductivity 243 E.M.F 271-273 , Law of 135, 136 of sulfids, Law of 140 Speed of reaction and temperature 208 Sugar, Inversion of 196-19S Surface-tension 65-68 and critical temperature 67, 68 Temperature, Absolute 11 , Critical 59 Thermochemistry 145-157 , Definition of 145 Thermodynamics, ist principle of 36-45 2d " " 48-54 Thermoneutrality of salt solutions 155 Units, Electrical 215, 216 Van der Waals, Equation of 35 Vapor-densities, Abnormal 23 INDEX. 299 PAGE Vapor-pressure, see Pressure. 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