EQUILIBRIUM I N ELECTROLYTE SOLUTIONS
BY FREDERICK GEORGE SOPER
The condition which must be satisfied by all systems in true equilibrium is that the Free Energy change occasioned by a slight displacement of the system from the equilibrium position, is zero. Application of this principle , leads to homogeneous equilibria of the type, A B P Q to the equation FA FB = FP FQ (1)
+ + e + +
+ +
+ +
where the F's denote the partial free energies, as a definition of the equilibrium state. Whilst this equation is probably strictly true of a gaseous system, its application to equilibria in liquid media may occasionally lead to erroneous results, for a slight displacement of the equilibrium may be associated with a slight change in the partial free energy of the liquid medium. The condition of equilibrium is thus more precisely given by
for a small displacement. For equilibria involving neutral molecules this change in the partial free energy of the medium may usually be ignored, but for equilibria involving ionsit may under certain circumstances be considerable, as it comprises 1/3rd of the net partial free electrical energy change of the ions caused by the slight displacement of the ionic eqilibrium. I n order to demonstrate this result which follows directly from the DebyeHuckel theory' of strong electrolytes, the determination of the activity coefficient of an ion in terms of the valence of the b n and of the ionic strength p, will first be considered. I n Debye's evaluation of the total electrical free ecergy of an electrolyte solutionZthe ions are considered as initially uncharged so that if their concentration is c the free energy of the system in the uncharged state will be F, nkTln c/co, where F, is the free energy of the system at the infinitely small concentration c,. The electrical work required to increase the charges of the ions reversibly to their final values is evaluated as A, = Z niz,e $,/3, where z, is the number of unit charges, e, possessed by the n ions of the i'th kind considered and $, is the potential a t one of these ions due to the surrounding ion atmosphere. As was shown by Debye and Huckel (loc. cit.), $, = -za,e K/D, K being a quantity characteristic of the solution and defined by
+
K ~ = -
n"kT
Physik. Z., 24, 185 (1923).
* Debye: Physik. Z., 25, 97
(1924).
8a e2 Np n i z , e -~ '*
- 103DkT
68
FREDERICK GEORGE SOPER
+
The total free energy of the electrolyte solution is thus F, nkT In c/c, nizi2ezK / ~ Dor, for a symmetrical binary electrolyte forming a total number of n ions F = F, nkT In c/c, - nzZe2~/3D (31 The free energy change measured in reversible concentration cells is the free energy of transfer of ions from one solution where their concentration is c1 to another where their concentration is c2, Le., the free energy change is the partial free energy change of the ions. I n order to compare the activity coefficients of ions deduced from these measurements of partial free energy changes, with the theory, Debye calculated the partial free energy of the electrolyte by differentiating the total free energy of the solution with respect to n and 80 obtained the result F = nkT In c/co - nz2e2K/PD (4) Part of the electrical free energy of the electrolyte solution must therefore reside in the medium since the total free electrical energy is - nz2e2~/3D, whilst the partial free electrical energy of the ions is - nz2ez~/2D. This leaves nz2e2/6Das the partial free electrical energy of the medium. Comparison of the partial free energy of the ions with the expression = nkT In f c/co, which defines the activity coefficient f in terms of the solution concentration co of unit activity wefficient, shows that In f = - nz2e2~/2DkT (5) or log f = - 0.505 2 2 4 7 at 25'.
+
+
Conditions of equilibrium in a balanced reaction associated with a change in the partial free electrical energy.-As a simple example of a balanced reaction in which a change in the free electrical energy occurs, an equilibrium of the H' A', may be considered, where a weak acid H A ionises type H A forming ions H' and A'. Let dn mols of H A ionise, forming dn ions each of H ' and A', and let cHA, cH,and cAbe the various equilibrium concentrations. The free energy change of the Eystem, dF, associated with this small disand at equilibrium d F placement, will be dFHA dFH. dFAr dFmdium, = 0 . Thus dn dn - dFoHA - N - RT In C H A dFoH. N - R T In CH. -
e +
+
+
+
+
dnz2e2~/2D f dFoA,
+ dn RT In N
CAI
+
- dnz2eZ~/2D +2
X dn z2ez~/6D= o
and therefore CH. cA,
In-
CHA
=
Const.
+3DkT 22282K
For an equilibrium of the type
+ BZZ + Cz3 + . . a Pz4 + Qz5 + . .
AZ1
69
EQUILIBRIUM IN ELECTROLYTE SOLUTIONS
The activity equilibrium conslant.-The activity of a molecular species A, may be defined by the equation, FA= RTln a A CA,where aA is the activity and CAis a constant. This definition leads to the activity equilibrium constant if ( I ) is valid. As has been indicated (I) is not applicable to ionic equilibria in which a displacement of the equilibrium is attended by a change in the free electrical energy of the system. An alternative proceedure is to define activities by the activity equilibrium constant and so evaluate [‘apparent” H’ A’, this activity coefficients. For the ionic equilibrium HA method would give
+
a +
and since f H . 4 may be regarded as unity, In-
CH.CA,
=
In Const.
-
z
In f l
CHA
where f l is the activity coefficient of a univalent ion. Comparison of equations (6) and (8) gives In f l = z2e2~/3DkT (9) log f = - 0.33 z2 dT at 25’. A similar result is obtained by application of the ionic velocity equation H’ A’. The rate of inter(previous paper) to the equilibrium HA action of the ions is given by
or
+
k c H . c A , e - E,/kT whilst the rate of ionisation is given
v
=
+
1.52
(10)
v’ = k’ C H A e - E,’/kT
(11)
At equilibrium v = v’, and
CH‘ C A ’
or log _ _ = log Const. CHA
Comparison of equation
(I 2)
-
1.52
-
2J303 with (8) , gives
log f l = - 0.33 47 Activity coefficients evaluated from the definition of activity in terms of the partial free energy, F A = CA R T In aA, thus correspond to a limiting law of log f = - 0.j o z2 .\/xwhilst activity coefficients evaluated from the equilibrium constant definition, a A aB . . . ap a~ . . . K, correspond to the limiting law log f = - 0.33 z 2 d \ / y , when the equilibrium is such that neutral molecules are in equilibrium with ions. There are consequently two series of activity values which will in general differ from each other. It may be of interest to consider briefly to which series the various measurements of activity should be assigned.
+
Lewis: Proc. Am. Acsd., 43, 259 (1907); Z. physik. Chem., 61,
129,
(1937).
FREDERICK GEORGE SOPER
70
Activities from solubility.-Since the calculation of activity coefficients from solubility measurements is based on the condition that the ionic activity product is constant, such activity coefficients should show agreement with the formula log f = - 0.33 z2 d r This result is obtained more directly by consideration of the free energy change attending the solution of a small quantity of salt in the saturated solution, remembering that each ion formed contributes z2e2k/6Dto the partial free electrical energy of the medium. Bronsted found for univalent electrolytes’ that the limiting law, log f = - 0.42 ct, satisfied the solubility relationships. Later work,2 however, has been in acbut since when the cordance with the expression log f = - 0.50 z2 47, ionic radii are taken into account the Debye equation becomes - z2B logf = I Aa 47 where A and B are constants, B having the value 0.505 or 0.33,the evaluation of B from experimental data depends on the value assigned to a, the ionic radius. The measurement of the partial vapour pressure of the electrolyte in the solution, as in the case of the halogen acids, may be considered as a solubility method and the activity coefficients obtained by this method correspond to the limiting law, log f = - 0.33 z*vv. I t must therefore follow that the vapour pressure of an electrolyte is not even approximately proportional to the f ~ g a c i t yf‘,, ~as defined by FA = RT In f‘ CA. Activities from freezing points.-In the measurement of activities from the freezing point depression one obtains primarily the activity of the solvents in in the solution. At the freezing point the activity of the solvent in the solution is equal to the activity of the solid solvent and the change in both these activities with temperature is calculable. I n a solution at any desired temperature the activity of the solvent present is therefore known. The partial free energies of the solvent and the solute, F1and FZrespectively, are connected by the equation‘
+
+
dz
+
NI
(g)
-k N z ( g )
+ Cp, then
(s) + (s)
If one defines the activity by FZ= RT In
XI
=
Nz
=
0
and one obtains from this equation the activity of the solute corresponding to the above definition and therefore to the limiting law for the activity coefficient log f = -0.50 z2.\/Fc. summary The equation for the equilibrium of electrolytes in solution has been I. modified to take into account the partial free electrical energy of the medium as deduced from the Debye-Huckel theory of electrolytic solutions. J. Am. Chem. SOC., 44, 938 (1922). Scatchard: J. Am. Chem. SOC.,47, 641 (1925). 3 Lewis: loc. cit. Lewis and Randall: “Thermodynamics”, 207 (1923,)
EQUILIBRIUM I N ELECTROLYTE SOLUTIONS
71
2. It is shown that the two definitions of activity ( I ) in terms of the partial free energy FA = RT In aA CAIand (2) in terms of the equilibrium constant, a A aB . . . / a p a~ . . = K, lead, when applied to electrolytes, to two series of values. 3. Activity coefficients evaluated from values for the activity of the solvent, e.g., from the freezing point depression, correspond to the Debye whilst activity coefficients evaluated from limiting law, log f = - 0.50 zz&, solubility measurements or from the partial v. p. of the electrolyte correspond to the limiting law, log f = - ,033 z*&.
+
Uniuersity College o j Nurtrlh Wales, Bangor June 1 1 , 1927.
