3635
THERMODYNAMICS OF ELECTROLYTE EQUILIBRIA
The Thermodynamics of Electrolyte Equilibria in Media of Variable Water Concentration by R. A. Matheson’ Chemistry Department, Victoria University of Wellington, Wellington, New Zealand
(Received October ‘7, 1968)
The standard states commonly used for electrolyte solutions are discussed with particular reference to the implications of these standard states in regard to the effects of pressure and solvent changes upon the standard chemical potentials of both ions and neutral solutes. Consideration is given to both normal standard chemical potentials and those which result when aqueous solutions are considered t o consist of hydrated solutes and free water molecules. The equations obtained for the variation of conventionalequilibriumconstants and solubility products with pressure are found not to justify a previous proposal for the formulation of pressure-independent complete equilibrium constants and complete solubility products. The thermodynamics of solutions of electrolytes in mixed solvents consisting of water and a second component which does not solvate ions are considered. Experimental data for solutions of hydrochloric acid in dioxane-water mixtures and of acetic acid in the same solvent mixtures are shown t o be inconsistent with the assumption that the standard chemical potentials of the relevant hydrated solutes are independent of solvent composition. The variation of conventional equilibrium constants with solvent composition and a previous proposal for the formulation of a complete equilibrium constant supposed to be independent of solvent composition are discussed.
Introduction At constant temperature the equilibrium condition for the chemical reaction
aA
+ bB +cC + dD
apA
+
is bpB = Cpc
+
dpD
(1)
pi, the chemical potential of the species i, is a state function which depends on temperature, pressure and, in the case of a dissolved substance, solution composition. Commonly a particular state is chosen as standard state and the chemical potential of any other state related to the chemical potential of the standard state (pot) by means of the equation p6 = pol
+ RT In ai
(2)
where ai is the activity of i in the state for which the chemical potential of this substance is pi. By definition a, = 1 in the standard state, From eq 1 and 2 one obtains the equilibrium condition
Cd’c
+
dpoD -
apoA -
bpoB =
- RT In
ac‘aDd aAaagb (3)
I n the case of a gaseous equilibrium it is customary to choose standard states such that each of the standard chemical potentials is a function of temperature only2 and the equilibrium constant (ao) (aD)‘/( u ~* () u ~ )is strictly constant under isothermal conditions. However, the position is less simple when nongaseous substances are involved. For an equilibrium in solution, the determination of the equilibrium constant normally involves the study of a series of solutions in which the
concentrations of A, B , C, and D in a given solvent are varied while the temperature and pressure are held constant. Standard states are chosen so that, under these conditions, the various standard chemical potentials and the equilibrium constant are constant. However, a change in conditions may alter the values of the standard chemical potentials and cause the equilibrium constant to vary. It is usually understood that all quantities in eq 2 must refer to the same temperature and that standard chemical potentials are temperature dependent. However, for liquids and solids two conv e n t i o n ~are ~ in use in regard to pressure. These may be illustrated by considering possible standard states for a liquid. First, we may choose, for any pressure, the pure liquid ut that pressure as the standard state so that the activity of the pure liquid is unity at all pressures. Since the chemical potential of a liquid is a function of pressure, this procedure implies a pressure-dependent standard chemical potential and, if eq 2 is used, both p, and pot must refer to the same pressure. Alternatively, the pure liquid at 1 atm may be chosen as the standard state so that the standard chemical potential is independent of pressure. However, the activity of the pure liquid is not now unity a t pressures other than 1 atm. Obviously the differences between the two conventions are important in high pressure studies. For dissolved substances one may choose the pure (1) Send correspondence to author a t the Chemistry Department, Otago University, Dunedin, New Zealand. (2) See, e.g., J. G. Kirkwood and I. Oppenheim, “Chemical Thermodynamics,” McGraw-Hill Book Co., Inc., New York, N. Y . , 1961, p 104. (3) See, e.g., K. G. Denbigh, “Principles of Chemical Equilibrium,” 2nd ed, Cambridge University Press, Cambridge, 1966, p 287.
Volume 73, Number 11 November 1968
R. A. MATHESON
3636 liquid form of the substance as the standard state and thus obtain a standard chemical potential which is not altered when the solvent is changed. However, the standard state is sometimes specified by requiring that the activity coefficient of the substance shall approach unity as its concentration tends t o zero. This second procedure gives a standard chemical potential the value of which varies from solvent to s ~ l v e n t . ~ It is clear from the foregoing that, for reactions involving nongaseous substances, the equilibrium constant is not necessarily a function of temperature only. Depending on the standard states chosen for the various substances, the equilibrium constant may or may not be pressure dependent or solvent dependent. A further point t o be considered is the possibility that the solvent may be involved in the equilibrium. Commonly the dissociation of an aqueous electrolyte M X is represented thus MX
---t
M+
+ x-
so that at equilibrium poM+
+ pox- - poMx = -RT
In K
(4)
where the quantity K , which we shall call the conventional equilibrium constant, equals aM +ax-/aMx. However if the species M+, X-, and MX are hydrated, the dissociation presumably takes place thus AIX(H,O),
+ nHzO +M+(HzO), + X-(H,O),
For brevity we shall represent this reaction as hlX(aq)
+ nHzO +M+(aq) + X-(aq)
(5) where i(aq) refers to the hydrated species i, e.g., M+(aq) refers t o M+(H20),, and n = m x - u is the change in hydration number for the reaction. At equilibrium
+
poM+(aq)
+
poX-(aq)
-
~POHZO
-
pOMX(as,=
- RT In KO (6)
= -aM + ( e q ) a X - ( s )
(7)
where
KO
a M X ( a q ) (aH20)”
Following Quist and Marshall5awe shall call KO the complete equilibrium constant. For dilute solutions in pure water a t a fixed pressure, U H ~ Ois virtually constant and the distinction between the two equilibrium constants unimportant. Indeed if pure water a t the pressure in question is taken as the standard state for HzO, they are equal. However, when a H z O is not constant it is important t o distinguish between the two equilibrium constants, a point which has been stressed by Quist and Marshall in their study of the changes in conventional equilibrium constants brought about by the pressurization of aqueous electrolyte solutions or by the addition to such solutions of an “inert” The Journal of Physical Chemistry
solvent (one which does not solvate the substances involved in the equilibrium). They propose5b“That the activity (or concentration) of solvent molecules indeed be included, in molar concentration units, as very important variables in the complete equilibrium constant KO” and add that “when this is done it is found that the isothermal KO for a particular equilibrium is independent of changes in dielectric constant, whether these are made by varying the density of the solution by pressurization or by adding an ‘inert’ solvent.” SubsequentIy6 they remark that “This KO is independent of density (pressure) a t constant temperature, and therefore meets one of the requirements of a true thermodynamic equilibrium constant.” Elsewhere’ they say that “The complete equilibrium in ion-pair formation thus includes solvation of ions and ion pairs, as well as the interaction between the solvated ions to form the solvated ion pair. With this concept the correct constant for the equilibrium should indeed be a constant (at cdnstant temperature) and would correspond to the KO in (their) equation (1) where (CH,,)” could be replaced by C” (reactive solvent) for some other reactive solvent.’’ Their eq 1 is
KO
K/(~H,o)” (8) This equation follows from eq 7 if it is assumed that the activity of water is equal to its molar concentration (CH~O) and if sat+(,,) ax-(aq)/aarx(aq) is equated to the conventional equilibrium constant K . From eq 8 =
log K = log KO
+ n log C H ~ O
Quist and Marshall quote numerous instances where a is found. linear relation between log K and log CII~O These examples include cases where C H ~ Ois varied by applying pressure to the solution as well as those where the variation is effected by the addition of an inert solvent. However, they do not give a detailed thermodynamic proof that either eq 8 or its antecedent KO = K/(aHzo)” correctly relate the hydration number change and the conventional equilibrium constant for the reaction to a complete equilibrium constant KO which is a function of temperature only. We believe that an examination of the possibility of such a proof is required in order to elucidate the folloming matters. (1) The status of the hydration number changes calculated from variations of K with C H a 0 . 8 While the (4) See e.g., K. G. Denbigh ”Principles of Chemical Equilibrium,” 2nd ed, Cambridge University Press, Cambridge, 1966, Chapter 9, especially eq 9.5. Ki in 9.5is solvent dependent. (6) (a) A. S. Quist and W. L. Marshall, J. P h y s . Chern., 72, 1536 (1968); (b) ibid., see p 1537. (6) Reference 5a,p 1538. (7) W. L.Marshall and A. 5. Quist, PTOC.Nat. Acad. Sci. U.S., 58, 902 (1967). (8) Reference 5a, Tables I and 11.
THERMODYNAMKS OF ELECTROLYTE EQUILIBRIA
3637
previously mentioned linear relation between log K and O that a number x can be found such that log C H ~shows K/(CH,O)”is constant under isothermal conditions, it where 8,’refers to the hydrated salt and R,’ refers to does not prove this number to be the hydration number free water. Assuming that there is sufficient water to change for the reaction in question. hydrate all the ions present and that a fixed number of (2) The correct formulation of “true isothermal molecules of water (h) are required to hydrate nl cations equilibrium constants” (equilibrium constants which and nz anions, assumptions which we shall make are a function of temperature only) for reactions in throughout this paper electrolyte solutions. n8 = n,‘ (3) Whether the linear relation between log K and log C H t O is in any fundamental sense an explanation of nw‘ = n, - hn, the changes in conventional equilibrium constants Now the addition of dn moles of water to the solution brought about by the application of pressure or by the will increase the number of moles of free water by dn addition of an “inert” solvent. and leave the number of moles of hydrated salt unEquations 6 and 7 follow directly from the application changed. Thus of the general equilibrium condition, eq 1,to the reaction represented by eq 5 . However, if KO is to be a “true iso= X,’ (11) thermal equilibrium constant,’’ standard states must be ) ) since the value of X is independent of any assumpchosen so that ( P O M +(as) Pox-(aq)- nomzo- M ’ ’ M x ( ~ ~ and tion made about the molecular state of the constituents is a function of temperature only. While this requirement can be met, at least in principle, it is not obvious X,’= hx, (12) that it is satisfied by the choice of standard states upon It follows from eq 11 that, at least when the pure which conventionalequilibrium constants are based, or by the standard state for water implicit in the relation U H ~ O liquid is chosen as standard state for water, the standard chemical potential and the activity of free water are the = CHZo. We therefore present a discussion of common same as the ordinary standard chemical potential and standard states for electrolyte solutions with a view to activity of water, respectively. I n contrast (cf. eq 12) establishing the behavior of the corresponding standard the standard chemical potential, activity and activity chemical potentials when the solvent or the pressure is coefficient of a hydrated salt differ from the ordinary changed, a point which is of considerable significance in values of the standard chemical potential, activity, and view of the current interest in studying the properties of activity coefficient of the salt, i.e., the values of these electrolyte solutions over extended ranges of pressure. quantities which are obtained when the components of We shall discuss both the thermodynamic functions for the solution are considered to be unhydrated salt and solute and solvent as commonly obtained and those water. However, examination of Robinson and Stokes, which result when one makes explicit recognition of the discussion of activity coefficients1°shows that in the case phenomenon of hydration. of the activity and activity coefficient the difference is very small when the solute is sufficiently dilute for hn, to Thermodynamic Functions for Hydrated Solutes9 be small compared with n, and the water activity to be Consider a solution prepared from n, moles of water close to unity. and n, moles of some anhydrous salt, each molecule of Standard States for Electrolyte Solutions which ionizes to form nl cations and n2 anions. OrdiThe standard states for the ions &/I+and X- are narily no attention is paid to the possibility that in soluspecified by stipulating that, at any temperature and tion the ions of the salt may be hydrated, and any expressure, the activity coefficients of these species shall tensive thermodynamic property X of the solution is related to the corresponding partial molar properties of tend to unity as infinite dilution is approached. The salt and water ( X , and 2,) by the equations standard state thus defined is a hypothetical solution in which the concentrations and activity coefficients” of X = n,Xs nwXw these ions are unity and in which, by virtue of the above mentioned stipulation, there are no interactions be(9) tween solute particles. However, since solute-solvent = T , P , n w; X, = dn, T ,P,nx interactions do not vanish at infinite dilution, any However, explicit recognition of hydration can be made by considering the solution to consist of n,‘ moles of (9) The following discussion owes much t o the arguments of R . A . Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed (revised), hydrated salt (ie.,nl n,‘ moles of hydrated cations and Butterworth and Co. Ltd., London, 1965, pp 238-241. n2n,’ moles of hydrated anions) and n,’ moles of free (10) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd water (water not hydrated to any ion). Then ed (revised), Butterworth and Co. Ltd., London, 1965, p 240.
x,
+
x,+
+
x,
(2)
X = ns’Xe‘
(”>
+ nw’Xw’
(11) Mean concentration and mean activity coefficient should the electrolyte be unsymmetrical.
Volume 78, Number 11 November 1969
R. A. MATHESON
3638 solute-solvent interactions present in real dilute solutions will persist in the standard state. Thus, when a standard state of this type is chosen in each of several solvents by specifying that, in each solvent, the activity coefficient of the solute shall be unity at infinite dilution, the standard chemical potential of the solute cannot be expected to be the same in all solvents. The standard chemical potential of the ions on the molal concentration scale is given by (POM
+
+
POX-)
(m) =
lim W
(pM +
O
+
-
ILX-
RT In mz) (13)
where K is the compressibility of pure water. When, as is common, the standard state for the species MX is chosen by specifying that its activity coefficient shall tend to unity as its concentration tends to zero, the standard chemical potential of this species may depend on the solvent and, in any given solvent, will vary with pressure thus
if the molal concentration scale is used. When the molar concentration scale is employed
and on the molar concentration scale by (POM
+
+
IL"X-) ( c )
= lim (ILM
+
c-+o
+
ILX-
RT In cz) (14)
where m denotes the molality of the ions and c the molarity. Relations between these two quantities and between the molar (y) and molal (y) activity coefficients are given by Robinson and Stokes.'Z It follows from these that, when the density of the solvent is not 1 g rnl-l, the two standard chemical potentials differ as also do the molar and molal activities and molar and molal\ activity coefficients. While the difference between the two activity coefficients is always negligible for dilute solutions, the two activities are related thus a&(,) __
azm
-
my& 1 -= -
do
CY&
and so differ appreciably when the density of the pure solvent (do) is not unity. For a solution of components, 1,2, . . . i
or, since both p and
V are intensive properties
where xi and Vt are, respectively, the mole fraction and partial molar volume of the species i. Since a solution of fixed mole fraction has a fixed molality, differentiation of eq 13 gives
where
70,
=
lim
Vi. Since for
a solution of fixed
m+O
mole fraction
b In c bP differentiation of eq 14 gives
1
av
v ap
When the solution is considered to be made up of hydrated solutes and free water, it is again specified that the solute activity coefficients shall become unity a t infinite dilution so the standard states for the solutes are once more hypothetical solutions in which there may be solute-solvent interactions, but not solute-solute interactions. However, the solvent molecules most affected by the (bare) ions are now regarded as part of the hydration sheath of the (hydrated) ions and therefore as part of the solute. The interaction between the hydrated ions and the free solvent will thus be smaller than the total interaction between bare ions and solvent. Nevertheless, because the hydrated ions are charged particles they will interact t o some extent with the dipoles of nearby free solvent molecules. Consequently, when one is considering solutions of an electrolyte in a series of solvents containing variable proportions of water and an "inert" solvent, one cannot exclude the possibility of some variation of the standard chemical potentials of the hydrated ions with solvent composition if it is specified that, in any solvent mixture, the activity coefficient of the electrolyte shall be unity at infinite dilution. From eq 12, 13, and 14 p ' ~ +(aq)
+
pnx-(sq)
POM
+
+
pox-
+
+
~ILOHZO
where ( p 0 ~ is the standard chemical pox- the stanpotential of the hydrated ions, p%+ dard chemical potential of the ions as ordinarily obtained without considering hydration, and ~ O H ~ isO the chemical potential of pure water at the temperature and pressure in question. The pressure coefficients of (Po ?vl+(as) poX-(aq,) (?n) -and b 0 M b(sQJ pnX-(aq)) ( c ) are obtained by replacing V a l + 7%in eq 15 and 16 by PTohI+(8.q) i.e., by PM Vox- h v o ~ , o . There is no reason to believe that either of these pressure coefficients is generally zero. If pure water at the temperature and pressure in question is chosen as the standard state for water
+ +
(12) Reference 9, p 30 ff.
The Journal of Physical Chemistry
=
+
+
+
+
+
+
THERMODYNAMICS OF
ELECTROLYTE EQUILIBRIA
-poHzO - bP
V'HzO
where V O H ~ O , the molar volume of pure water, is identical with the limiting partial molar volume of water, ~OH,O. Suppose, however, we specify that, for any solution in which the properties of the water do not differ significantly from those of the pure liquid at the same temperature and pressure, UH%O = C H ~ O ,This ( ~ ) that implies a standard chemical potential ~ ' U O H ~ Osuch PH,O = ( ~ O H ~ O ) ( ~ )
+ RT In C H ~ O
When this equation is applied to pure water, it follows that
The right-hand side of this equation is not generally zero, e.g., at 25" for pressures near 1 atm K H ~ O= 4.6 X atm-l while V@H,o/RT= 7.3 X l o d 4atm-'. Pressure Dependence of Equilibrium Constants When calculating conventional equilibrium constants from experimental data it is common to choose standard states for AI+, X- and AIX by requiring that, at any pressure, the activity coefficients of each shall become unity a t infinite di1uti0n.l~ Thus (eq 4,15, and 17)
where K(m)is the conventional equilibrium constant expressed in t,erms of molal concent'ration scale activities and AVO = V@M+ + POx- - VoMx. The corresponding expression for the molar scale is (eq 4, 16, and 18)
I t followsfrom our discussion of standard states that the complete equilibrium constant
KO =
t(as,ax-(aq) alMX(aq)(aHzO)'
is not necessarily independent of pressure and that its pressure coefficient will depend on the standard states upon which the various activities are based. Since for dilute solutions the activity of a hydrated solute and the corresponding ordinary solute activity are the same, the activity quotient ab1 -(aq) ax-(aq)/axx(aq) may be replaced by the conventional equilibrium constant so that (cf. eq 6) P'M+(~)
+
poX-(aq)
- pofirX(aq)
- ~POHZO
=
It must be emphasized that this step implies, in the case
3639 of the hydrated solutes, the choice of a standard state of the type discussed in the previous section of the paper so that each of the terms po M and poMX(sq) is, in general, pressure dependent. If either of the more usual standard states are chosen for water, K/(uH,o)"is pressure dependent. The choice of pure water a t the pressure in question as the standard state for water gives UH*O = 1 at all pressures so that K / ( a ~ , o ) "reduces to K , a quantity which is, in general, pressure dependent. If pure water a t 1 atm is chosen as the standard state, ~ O H ~ is O independent of pressure. This does not make K/(UH,~>" independent of pressure unless t(aq) VOX-(aq) = V O & I x ( a q ) (assuming molal concentration scale activities for the solutes) or V 0 h r Voxteas) = PhIX(aq) RTK (assuming molar concentration scale solute activities). We can see no reason why either of these equations should necessarily hold. However, as mentioned earlier it has been proposed that the quantity K(,,/(CH~O)~ is a complete equilibrium constant which is independent of pressure. For the following reasons we believe that this approach to the formulation of a pressure-independent complete equilibrium constant is questionable. It follows from our discussion that the conventional equilibrium constant is based on standard states such that each of the standard chemical potentials in eq 20 is pressure dependent. Putting a H Z O = CH,O instead of a H Z 0 = l changes the standard state for water but does not alter the standard states for the other substances. Moreover, the relationship UH,O = CHzO implies a standard ~ is pressure dependent. chemical potential ( ~ O H ~ O )which Thus each of the terms on the left-hand side of eq 20 remains prescmre dependent and
+
+
-RT
+
b K(d - In ___ = AVO f (n - ~ ) R T K(21) bP (CHzO)"
Were the various participants in the equilibrium to behave as ideal gases, the molar volume of each would be RT/P so that the right-hand side of this equation would vanish and K/(CHz0)' would be independent of pressure. However, we can see no reason why AVO (n - ~ ) R Tshould K necessarily be zero for a reaction in aqueous solution and therefore conclude that the application of thermodynamics t o the reaction
+
MX(aq)
+ nHzO +M+(aq) + X-(aq)
does not lead to the result that the quantity K/(CH,O)" is necessarily independent of pressure. It follows that the quantity x which makes K/(CH,O)"independent of pressure is not necessarily the hydration number change for the reaction in question. This view is reinforced by the rather large values of x which have been reported for (13) This requirement is explicit when abx+ax-/avx is equated to the infinite dilution limit of the corresponding concentration quotient. It is implicit when it is assumed that, in dilute solution, the activity coefficient of MX is unity and the activity coefficients of M + and X are given by the Debye-Hockel theory.
Volume 78, Number 11 November 1969
R. A. MATHESON
3640
should necessarily be the hydration number for the salt in question. For a number of salts we have calculated q using figures for AVO derived from the appropriate partial molar ionic volumes16and solid-salt densities. l6 We obtained the following: NIgS04 q = 47.8, CaSOd q = 45.6 (solid phase natural anhydrite) or 51.0 (soluble anhydrite), Bas04 q = 45.9, llgCOs q = 48.9, CaC03 y = 51.2 (solid phase aragonite) or 53.5 (calcite), BaC03 p = 55.5, AgCl q = 9.7, AgBr p = 6.4, AgI q = 7.0 (a form of the solid) or 5.0 (P form), LiCl q = 5.0, NaCl q Pressure Dependence of Solubility Products 11.3, KCI y = 11.5,RbClq = 12.1 and CsCl q = 4.7 (all Presuming the dissolved ions to be hydrated, the data refer to 26”). The very large values for the alkaequilibrium between the solid salt AB and the saturated line earth carbonates and sulfates strongly suggest solution of its ions is that for these salts p is not the hydration number. On electrostatic grounds it would be expected thal the AB($ hHzO +A+(aq) B-(aq) lithium ion would be the most strongly hydrated of the Thus alkali ions yet the value of q for LiCl is much less than for NaC1, KC1, and RbC1. Moreover, where the solid PoA+(aq) PoB-(aq) - P 0 A B ( s ) - hPoHzO = can exist in two crystalline forms which have different a A +(aq)aB-(aq) densities, two different values of AVO (and therefore two -RT In U A B ( ~ ) ( ~ H ~ O ) different ~ values of p ) result. However, once in solution ions from the two different forms of the solid are inIf one chooses the respective pure phases a t the pressure distinguishable and must have the same hydration in question as standard states for water and the solid numbers. We are therefore inclined to the view that q AB, then uAB(*) is unity at any pressure and U H ~ O= 1 in is not the hydration number for the salt in question. sufficiently dilute solutions regardless of pressure. Thus “Hydration numbers” have also been derived” from with the usual choice of standard states for the ions plots of log s (s = molar solubility) vs. log CHIO; e.g., a Ka PoA+(eq) PoB-(aq) - P’AB(*) - h l * O ~ z= o - RT value of 28 for Cas04 at 250” was obtained in this way using solubility data for pressurized aqueous solutions. where K s is the solubility product uA+uB-. Each of the Presuming the pressure coefficient of the relevant terms on the left is, in general, pressure dependent. activity coefficients to be negligible, such “hydration Hence where the ionic activities are on the molal connumbers” are identical with the corresponding values of centration scale p and in our view equally deserving of suspicion. the ionization of water (x = 20),14ammonia (x = 28), methylamine (z = 25), and the bisulfate ion (x = 20). We do not believe that these figures are in fact hydration number changes and therefore reject the explanation of the original authors that “these large hydration changes may possibly be related to the “extra” mobility of the hydrogen and hydroxide ions, the structure of water a t room temperature, and their changes when solvent concentrations are varied by pressure alone.”
+
+
+
+
and where they are on the molar concentration scale
Equilibria in Mixed Solvents If it is specified that,, in any solvent, the activity co-
then
efficient of a solute shall become unity a t infinite dilution, the standard chemical potential of that solute can be expected to vary from solvent to solvent. While such variation is well known in the case of ordinary standard chemical potentials (those which result when one makes no specific recognition of hydration), the position is less clear in the case of the standard chemical potential of a solute which is specifically considered to be hydrated. However, we suggested earlier that, given the choice of standard state under discussion, the standard chemical potentials of hydrated ions in a mixed solvent consisting of water and an “inert” component might vary with solvent composition. We now proceed to confirm this suggestion by showing that a satisfactory
There appears to be no reason why the right-hand side of this equation should necessarily vanish or why the ~ quantity q which makes -AVO/RT = (q - 2 ) and ) of pressure which makes K S ~ , , / ( C H4 z oindependent
(14) Reference 5a, Tables I and 11. The “net changes in water of hydration” which appear in these tables are identical with the quantity x defined above. (15) B. B. Owen a n d % T. Brinkley, Chem. Rev., 29,46 (1941). (16) “Handbook of Chemistry and Physics,” 47th ed, The Chemical Rubber Publishing Co., Cleveland, Ohio, 1966. (17) Reference 6a, pp 1541, 1542.
- RT =
+
P0A+
a
- In K 8 ( C= ) AVO
dP
+
P B -
-
- 2RTu
VOAB(s)
v ‘ ~ + FOB-is the limiting partial molar volume of the salt as ordinarily calculated. If a complete solubility product KO, is formulated thus
The Journal of Physical Chmiatrv
THERMODYNAMICS OF ELECTROLYTE EQUILIBRIA
3641
interpretation of the thermodynamic properties of certain electrolytes in such a solvent system is impossible if one assumes the standard chemical potentials of the hydrated ions to be independent of solvent composition. We choose for the solvent system dioxane-water since nmr studies18indicate no solvation of ions by dioxane in 50% (v/v) dioxane-water. The emf of the cell
Assuming the standard chemical potentials of the hydrated solutes to be independent of solvent composition and again choosing pure liquid water as the standard state for HzO
Pt, Hz(g, p
where K ( w ) is the conventional dissociation constant in pure water and a H z O ( ” is the activity of wat)er in the solvent mixture x. We have analyzed the dissociation constants of a number of electrolytes in dioxane-water mixtures and find that the following values of n are required if eq 23 is t o be consistent with the experimental d a t a 9 IVInS04, n = 40; MgSO1, 37; manganese m-benzene disulfonate, 46; LaFe(C?\T)6,50; and acetic acid, 30 to 50 depending on solvent composition. These “hydration number changes” are even more unsatisfactory than the “hydration numbers” previously calculated for hydrochloric acid. Since several of these excessively large values of n and h were obtained from data for solvents of lower dioxane content than the one for which nmr studies indicated ionic solvation by water only, we attribute the unreasonable magnitude of these “hydration” parameters to failure of the assumption that the standard chemical potentials of the hydrated solutes are independent of solvent composition. We do not, however, rule out the possibility of some solvation by dioxane in media of sufficientlyhigh dioxane content. Examination of eq 22 shows that in a mixed solvent system such as dioxane-water there are two factors involved in the variation of conventional equilibrium constants with solvent composition. First, because of the standard states upon which conventional equilibrium constants are based, the standard chemical potentials on the right-hand side of eq 22 vary with solvent composition. Second, because the water activity is not included in the conventional equilibrium constant, the chemical potential of water in the mixed solvent appears on the right, rather than a standard chemical potential of water which is independent of solvent composition. While theories which attempt to relate changes in conventional equilibrium constant with the dielectric constant of the solvent mixture may possibly deal with the first factor in a satisfactory fashion, they ignore the second completely. On the other hand, recent treatments which seek a correlation between the variation in equilibrium constant and the solvent activity disregard the first factor which we have already seen to be significant. Since
= 1atm)/HCl(indioxane-water)IAgCl(s)IAg
has been studied a t several solvent compositions. Having regard to hydration, the cell reaction is ‘/zHz(g)
+ hHzO + AgCl(s)
+
€€+(as)
+ CI-(aq) + Ag
The emf is given by
-FE
=
PH+(aq)
+
PCl-(aq)
+
PAP
-
-
‘/ZPHz(g)
hPHnO
- PAgCl(s)
For each solvent mixture the standard potential E” was calculated on the basis that y --+ 1 as m --*. 0 in that solvent mixture. Thus Eo(‘),the standard potential for any particular solvent mixture of composition x, is related to the standard chemical pot>entialsof the hydrated solutes in this solvent mixture by the equation
-FEo(’) =
(P0H+(aq)
PAg
P°C1-(aq))(’)
-
‘/ZPHz(g)
-
PAgCl(s)
-
hPHzO(’)
where PH~O(‘) is the chemical potential of water in this solvent mixture when m (the molality of hydrochloric acid) --t 0. Choosing pure liquid water as the standard state for HzO
+ RT In
~ H ~ o ( ‘ )= ~ ‘ H ~ O
UH~O(‘)
where ~ O H ~ isO the chemical potential of pure liquid water and UH~O(’) is the water activity in the mixed solvent. Thus, if the standard chemical potentials of the ions are independent of solvent composition
F(E”(”)- Eo(W)) = hRT In U H ~ O ( ’ ) where Eo(W) is the standard potential of the cell when pure water is the solvent. I n order that this equation be consistent with tjherelevant experimental datalgone has to assume the following rather large values of h: 18.4 (z = 20% w/w dioxane-water), 17.9 (x = 45%)) 27.6 (x = 70%)) and41.2 (x = 82%). For the dissociation of MX(aq) (cf. eq 5 )
-RT In K(’)
=
(poY+(aq)
+ (Po?dX(aqj)(Z)
-
-
~PHIO“)
(22)
where K(‘) is the conventional dissociation constant in the solvent mixture x. Both this quantity and the standard chemical potentials of the hydrated solutes are based on the choice of solute standard states discussed a t the beginning of this section. pH2o(’) is again the chemical potential of water in the solvent mixture x.
(18) A. Fratiello and D . C. Douglass, J . Chem. Phys., 39, 2017 (1963).
(19) Standard potentials from Table 8.2 in ref 9. Water activities interpolated graphically from the data of J. R. Goates and R. J. Sullivan, J . Phys. Chem.,6 2 , 188 (1958). (20) Dissociation constants from sources cited in ref 4, p 1539. Water activities as in ref 19. Volume 79, Number 11
November 1969
3642
R. A. MATHESON
(p0MX(aq))(')
- nkoHzO
and since only the last of the terms on the right is independent of solvent composition, we can see no reason to believe that the "complete equilibrium constant" Kc")/ (UH~O('))' is independent of solvent composition. While it may sometimes be possible to find a number p such that K(')/(UH~O(')) is independent of solvent composition, there is no reason to believe that this number is the hydration number change for the reaction in question. Indeed p is identical with the "hydration number changes" which we calculated from eq 23 and which we have seen to be absurdly large. It has been proposedz1 that the conventional equilibrium constant, the hydration number change, and the molar concentration of water in the mixed solvent are related by the equation KO = K/(G",o)" where KO is a complete equilibrium constant assumed independent of C H z O . It follows from eq 22 that
hydrated ions with solvent composition is entirely due to electrostatic interactions between hydrated ions and solvent, the model under discussion leads to the result that
where Akoi(aq) = - por(aq)(w),D(') and D(w) are, respectively, the dielectric constants of the mixed solvent and water, N is Avogadro's number, e is the electronic charge, and r+ am T- are the radii of the hydrated ions, with 2 / r , = l/r+ l/r-. Applying this equation to the hydrated hydrogen and chloride ions, one obtains for the standard potentials of the cell discussed previously the result
+
F(E"(")- Eo("'))= hRT In a H , O ( ' )
-
?(A -A)
When the right-hand side is calculated it is found that, for any reasonable values of h and T , ~ , the result is greatly in excess of the experimental quantity F(E"(') -RT In K ( ' ) / ( C H 2 ~ ( Z = ) ) n(polrqaq) E o ( W ) ) .We suspect that the failure of this equation is kOhIX(aq)(') - ~ ( M H ~ O ( ' ) RT In CH~O($)) at least in part due to our assumption that the standard chemical potentials of hydrated species would be indeAt least in the case of the dioxane-water system it is pendent of solvent composition in the absence of easy to show that (pwZo(') - RT In CH~O(')) depends on electrostatic interactions between these species and the the solvent composition.zz Thus in dioxane-water each solvent. of the terms on the right can be expected to vary with
+
-
solvent composition. As there appears to be no reason why the right-hand side as a whole need be constant we conclude that K/(CH,O)' is not necessarily independent of solvent composition. While it is apparently possible to find a number x such that K/(CH~O)'is independent of solvent composition (the slopes of the linear plots of log K vs. log C H ~ Oreported by Quist and Marshall are such numbers) we can see no reason why this number is necessarily equal to the hydration number change for the reaction in question. Since the standard chemical potentials of hydrated ions appear to vary significantly with solvent composition, a calculation of this variation is desirable. The charged sphere-dielectric continuum model of the electrostatic interactions between ion and solvent is known to be unsatisfactory in the case of the total interaction between bare ions and the solvent. If the main trouble here is the inappropriateness of this model in the case of the water molecules immediately adjacent to the bare ions, it might be expected that, since these water molecules presumably form part of the hydrated ions, the deficiencies of the model would be less serious when one is considering the electrostatic interactions between hydrated ions and free solvent molecules. Assuming that the variation of the standard chemical potentials of the
The Journal of Physical Chemistry
Conclusion Our analysis gives us no reason to dispute the view that solute-solvent interactions are a factor which must be considered in any attempt to account for the variations in conventional equilibrium constants which are brought about by the pressurization of aqueous solutions or by the addition to such solutions of an inert solvent. However, the proper formulation of the effect of such interactions upon the equilibrium constant and other thermodynamic properties is found to be less simple than has sometimes been supposed. I n particular, even if it is assumed that solute-solvent interactions lead to the formation of hydrated solute species each containing a fixed number of water molecules, thermodynamic arguments give one no reason to suppose that for a reaction in which the hydration number change is n either of the quantities K/(uH,o)"or K/(CH~O)' is necessarily a function of temperature only. (21) W. L. ,Marshall and A. S. Quist, Proc. Nat. Acad. Sci. U . S . , 5 8 , 901 (1967). (22) Consideration of the equilibrium between the mixed solvent and its vapor shows that this term is not constant unless PHZO/CHZO is constant (PH~o = partial pressure of water above a mixed solvent containing CHZOmol L - 1 of water). The data of A. L. Vierk (2. Anorg. Allgem. Chem., 261,283 (1950)) show t h a t PHZOICHZO is far from constant in the dioxane-water system.
