ACTIVITY COEFFICIENTS AND MASS-ACTIOK LAW I N ELECTROLYTES‘ B Y L. ONSAGER
In two recent papers,2 F. G. Soper claims that in considering chemical interaction in dilute solutions of electrolytes, the activity coefficients should be taken according to the formula: e2 In f i = - zi2 K 3DkT He thus rejects the original formula given by Debye and H i i ~ k e l which ,~ is: In
fi
=
-
zi2 K
e2 2DkT’
-.
z =valence; e = charge of an electron; D =dielectric constant; k = Boltzmann’s constant; T = abs. temp.; K = mean inverse radius of the ionic atmosphere,
given by the equation:
when ni means the number of ions of the kind i present in I cm3 of the solution, and the sum is taken over all kinds of ions present in the solution. The value found by Soper for log f is just $ of that found by Debye and Huckel. Of course, both the expressions ( I ) and (2) cannot be right a t the same time. I am going to show, that Soper’s formula ( I ) is wrong, and depends upon an erroneous deduction from some of the results given by Debye and Huckel. In his first paper, Soper argues as follows: “The total free electric energy of a solution of n ions, found by charging them reversibly” is n z e where $ is the potential a t one of these ions due to the surrounding “ion atmosphere.” Part of this electrical free energy resides in the medium, but change in the partial free electrical energy of the ions is attended simultaneously by a change in the electrical free energy of the medium. The potential energy possessed by a pair of ions with respect to their combination will be the difference of the total work done in charging the ions A and B and that done in charging the complex formed by collision of h and B. If the ions have charges ZA and Z B , the complex will have the charge Z A + Z B and the potential energy of h and B will be given by:
+
Be
( z A $A
+
ZB $B
-
(zA
+
zB) $A,BJ
Contribution from the Johns Hopkins University, Baltimore, Md. J. Phys. Chem., 31, 1790 (1927); 32, 67 (1928). Physik. Z., 24, 185 (1923). Debye: Physik. Z., 25, 97 (1924).
+,
(3)
L. ONSAGER
1462
The potential given by :
+i,
a t an ion i due to its surrounding ionic atmosphere is
where n, is the number of ions of the i’th kind per cc. in the solution, e the charge on an univalent ion, N the Avagadro number and k the gas constant per molecule.” In citing the above, a couple of obvious misprints in the formulae (3) and (4) have been corrected. These formulae were not numbered in Soper’s paper. Soper says “the total potential energy possessed by an ion in virtue of its charge is z e He means the potential energy with respect to the surrounding ions; however, he goes too far in treating the energy as a particular property of the ion. The influence of the electrical forces between the ions, a t equilibrium, is given by the change 6 W in the total free electrical energy W involved by a small displacement of the equilibrium. Let this displacement involve changes 6N,, 6N1,. . . 6N, in the total numbers No,N1 . . . N. of the molecules and ions of each kind 0, I , . . , s presefit (the index o indicates the solvent). The change:
+ +.”
6W
=
a w 6Ni z dN, ~
,=o
is obviously completely given when the partial free energy: =
(E)p, T, No, N 1 . . .
Xi-1, K ’ i + l . .
. hTs
is given for each kind i of molecules (ions) entering the reaction: because, for the others, 6x1= 0. Now, the change in the free electrical energy IV involved by introducing an ion1 is given (practically) by the work necessary to charge it in the solution. However, the sum of the partial free electric energies of the ions is not equal to the work required to charge them all either simultaneously or successively, that is, to the total free electric energy. And, accordingly, the partial free energies cannot be obtained simply by distributing the total free electric energy between the ions even in the most reasonable manner, because the sum cannot possibly fit, a t a j y rate, not as long as the square root law holds. The partial free energies wi are obtained by derivation of the total free energy, as shown above. The differenceZ between the work necessary for charging the ion in the electrolyte and that nccessaryfor charging it in the neutral solvent amounts to : 1 In reality, we cannot determine the partial free energy of a single ion, but only the sum of the partial free energies of such a combination of ions for which the sum of the charges is zero, or differences between the partial free energies for ions of the same charge or combinations of the same total charge. Fi = $ eizi+i as long as +i is proportional to z,, which is true for small concentrations.
x=zi
I n general we must put: Ti =
J X=0
of the charge.
$ (he) d(ke) considering the potential as a function
ACTIVITY COEFFICIESTS AND MASS-ACTION L.4W
I463
+
and not 5 e z, $). Thus the partial free electrical energy of the ion is e z, +, (+ a constant which does not vary with the concentration). However, the total free electrical energy of the electrolyte is only 8
IT =
5
Z N>.zie+,= l = I
e2K
-
Z
3 Di=i
(6)
r\j,z?
when N1. . Xi . . N, mean the total numbers of the different kinds of ions present. This is easily seen by considering the process of charging the ions one by one. The work of charging the first ions will not a t all differ from that required to charge them in pure solvent. When so many ions are charged, that the ionic strength has increased to a fraction X of its final value, the work of charging an ion of the i’th kind will be:
as the mean radius I / K ’ of the ionic atmosphere a t that moment will amount times its finalvalue I/KnThichis not reached until all ionsare charged. to I Let us charge the ions in such an order, that we have approximately equal fractions of the ions of each kind charged a t the same time. Thus, a t a certain stage of the process, numbers AN1, . , . XNi , , . AN, of each kind are charged. The work needed for charging further N1dX . . . Xi dX . . . N, dX ions of each kind will be : dW=
E
i = i
S,dX
( - S K ~ )’ =
e2K -2 DiX = , Nz z2i
dX
the work needed for charging these ions separately in pure solvent being subtracted. The total work is found by summing up all steps. This gives us: X=1
The total free electrical energy W can be considered as a function of the pressure p , the temperature T, the numbers 9 1 , 9 2 . . . N, of the different kinds of ions, and the number No of the molecules of solvent. In these variables,
must be equal to
aw ;
~
dN,
because the change in
aw
W involved by
6S,= . I , and this dS, dN change in W must equal the work W,required to carry out this change of the system. Now, this is easily controlled by deriving the expression for W given by (6). I n carrying out this derivation, we must remember that K is a function of the numbers IC,: introducing one more ion of the i’th kind is
~
~
L. ONSAGER
1464
a
~2
-4
Nizi2 =
=*
I)kT,=,
I: N, z? i = x ~
r
DkT
V
.
’
here, V means the total volume of the system. Owing to the fact that we have to deal with a dilute solution of small compressibility, we have until now neglected the change in the volume involved by introducing and charging the ions. It may be shown that this approximation involves but a small error, and it has nothing to do with those principal features of the theory which we are discussing here. Therefore, we will assume that the volume is given by the solvent molecules alone : V = No vo ; vo meaning volume per molecule of the solvent. Thus we have: 8
( i = ~ , z.
j = x
.
.s)
Now the derivation of W gives:
L: Nj zjz
2
,=I
Now, according to (6) and (7) the free electrical energy is an extensive property of the system. Every such function obeys a relation of the form:
Comparing this with (8) and ( 6 ) , it is easily seen that Yo =
cannot be dN.3 zero. In this respect, therefore, one may say that part of the free electric energy “resides in the medium.” W, is easily obtained:
According to ( 7 ) : _a K= - -
dN0
K 2
KO
ACTIVITY COEFFICIENTS AND MASS-ACTION LAW
1465
Thus:
Now it is easy to see that (9) is fulfilled. According to (8) and (6) :
Thus, making use of
(IO)
: s
Now; Soper says: “Part of this electrical energy resides in the medium, but change in the partial free electrical energy is attended simultaneously by a change in the electrical free energy of the medium.” The meaning seems to be that a change in the number of ions of the i’th kind alters the partial free energy of the medium. This is true, but the partial free energies of all kinds of molecules, ions as well as solvent, are changed. When we derive :
it follows from the definition: =
- aw wi dNi
that:
a%
Z NjdNi
= O
j=,
so that all the changes in the partial free electrical energies just compensate each other. This may be enough to show that Soper’s results depend upon his misunderstanding of the term “partial free energy,” and that the theory of Debye and Huckel is satisfactory in all points attacked by him, as far as the deductions are concerned. Soper tests his own formulae on experiments performed by various authors on reaction velocity. He finds them to fit better than the formulae of Debye and Hiickel, applied to reaction velocity by Bronsted. However, these experiments are made a t so high concentrations that large deviations
1466
L. ONSAGER
from the limiting formulae given by Debye and Huckel must be expected. At such high concentrations, a square root formula may fit for interpolation over a certain range, just as a tangent fits a curve. But it should be emphasized, that the only sound application of the square root formula is to use it with the theoretical coefficient as an asymptotic formula for small concentrations. Univalent binary electrolytes in water may fit the limiting formulae fairly well up to an ionic strength of 0.01;for others, the limit must be taken still lower. Summary: The considerations of F. G. Soper, which lead to other limiting formulae for the activity coefficients of electrolytes than those given by Debye and Huckel, were analyzed and shown to be erroneous. The fact that Soper’s formulae agree better with the experiments considered by him is attributed to his applying the limiting square root formulae a t too high concentrations.
