Robert J. Hunter Univers~tyof Sydney Sydney, Australia
Calculation of Activity Coefficient from Debye-HijtkeI Theory
I n most undergraduate physical chemistry courses the Debye-Huckel theory of strong electrolytes is introduced and an attempt is made to calculate the activity coefficient as a function of ionic concentrac tion on the basis of this theory. The development starts from the linearised form of the Poisson-Boltzmanu equation ( 1 , f) and arrives at an expression for the potential around an ion (regarded as a point charge) :
where q = z+e is the charge on the central ion (with sign included), K is the permittivity of the solvent, and K is the usual DebyeHiickel parameter. This discussion will be restricted to symmetrical electrolytes to avoid unnecessary algebraic complications, in which case
where n is the concentration of either ion type in ions per cc and B is a constant independent of concentration and charge. To calculate the effect of the electrostatic interaction between the ions on the activity of the electrolyte, it is necessary to split the chemical potential into an electrical part and an ideal part, it being assumed that all of the departure from ideality can he attributed to this interaction. It can then be shown that (1, 3): G+"'per ion = kT in r+ (3) where 6 + * 5 s the electrical contribution to the partial molar free enthalpy of the positive ion due to its interaction with the negative ions and y+ is the activity coefficient. The final step in the process is to determine the way in depends on the electrostatic potential. The which 6+*'
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purpose of this paper is to show that the usual solutions to this problem, though leading to the correct result, are logically somewhat unsatisfactory and obscure some of the important features of the interaction. An alternative procedure which obviates these difficulties, a t the expense of a slight increase in algebraic manipulation is presented. Although this procedure has been described previously in the literature (4, 5), and it was essentially that adopted in the original calculations of Debye (6), there does not appear to be a readily accessible description which preserves the essential features with a minimum of mathematical complexity. The present paper is an attempt to remedy this situation using an approach which is similar to that described by Verwey and Overheek (7) in connection with the analogous problem of calculating the free enthalpy change involved in establishing the electrical double layer a t the interface b e tween a colloidal particle and an electrolyte solution. We shall, however, begin by considering the simple charging process first introduced by Muller (8) and Giintelberg (9)and used as a basis for the analysis in most undergraduate texts ( 1 4 ) . The Muller-Guntelberg Charging Process
The relation between and ++ is evaluated by writing the potential in equation (1) in the form:
the higher terms in the expansion being neglected. The first term is, of course, the potential due to the presence of the central ion. The second term is the potential due to the oppositely charged ionic atmosphere which behaves as though it were a sphere of charge - g and radius 1/~. We now consider the work done when an initially
uncharged particle is introduced into the solution and is gradually charged to its final charge of q by the transport of small increments of charge qdX from infinity up to the particle in the presence of the field due to the oppositely charged ionic atmospheve. The increments of charge are being moved in the direction of the field so that the system can he made to do work during this process given by:
and the potential in the neighhourhood of type C particles due to the presence of the (negative) type A particles is:
with a similar expression for J.- (X). The total work done by the system over a11 the ions in each step of the process will be given by the product of the number of ion pairs and the electrical work done in each step: -m[*-(X)- *+(A)lqdX
where
The total work done by the system during the entire charging process is The parameter X characterizes the extent to which the charging process has been carried out and so varies from zero to unity as the charge is built up. This work can then be equated to the change in free enthalpy of the system (per ion) due to t,he electrical interactions, so that
Therefore: (total) = 2qSBna'a/3K
and so Though this result is correct, the above procedure does not carry much conviction. I t is true that the introduction of just one extra charged particle cannot appreciably affect the value of K, so that it may he treated as a constant in integrahg equation ( 5 ) . The fact that only one charged particle is produced by the process, however, means that if it is repeated often enough the transport of charge will have to be done against an ever increasing potential due to the neglect of the requirement of electroneutrality. One could overcome this objection by producing ions of opposite sign in successive charging proccsses. Since the charge enters into eq. (6) as a square t,erm, the creation of each such charged particle would contribute addit,ively to the change in partial free enthalpy. Even then, however, the procedure leads to a breakdown of the concept of the ionic atmosphere since, during the build up of the charge, it is not possible to maintain a balancing charge a t an average distance of 1 / from ~ the central ion. Since we go to considerable trouble in electrolyte theory to point out that we are only interested in work terms subject to the requirement of electroneutrality (10) it is somewhat difficult to justify the above procedure. I n the following charging process, electroneutrality is always preserved and the separate contributions to the work done on or by t,he system can be readily visualized. The "Simultaneour'' Charging Process
We consider a solution containing n uncharged particles per cc of each of two different types ( A and C) and we will establish a symmetrical electrolyte containing n positive and n negative ions per cc of charge q and -q respectively, by transferring increments of negative charge -qdX from each of the particles of type C, to a corresponding particle of type A so that the former will become cations and the latter anions. At any intermediate stage in the process when the charge on the ions of type C is qh, the value of the parameter K is , from (2) :
a@' am
@"(per ion pair) = - =
G.1 (per ion)
-@Bn'/2 li
-qlx
= -2K
When the charge is developed in this way there are three important work terms to be considered:
1. The work done on the system when moving an increment of negative charge away from the positive particle and towards the negative particle. This is the energy involved in establishing the charges themselves in the absence of a n y interaction. This work term would he undiminished even if the particle concentration approached infinite dilution. Since this latter situation is taken as the standard for ideal behavior, this work term cannot he involved in the departure from ideality. (The same term would appear in the first charging method (8, 9) and is neglected for the same reason.) 2. The work term calculated above. 3. The work done by the ions of one sign as they move into the field of the oppositely charged ions as the charging proceeds. As X increases, KA increases and so 1 / which ~ ~ measures the mean separation between the ion and its oppositely charged counterion, decreases. As, for example, a, negative ion moves closer to its positively charged partner its potential energy is lowered due to the increased electrical field in which it finds itself: 6U
=
(8)
-q($~' - +I,)
represent two successive (average) where J.,'and potentials experienced by the ion. At the same time, however, the entropy of the system is reduced by a compensating amount (7a) : 6S = -k in n h l
(9)
where n, and n2 are the local ion concentrations at the points where the electrical potential due to the oppoVolume 43, Number 10, October 1966
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sitely charged ion is $,' and $z' respectively. Since, by Boltzmann's theorem ns = nl exp ( d h ' - #l')lkT)
(10)
it follows that aF = aU = TaS = 0. The change in free enthalpy aG will, for this condensed system, also be zero in the process. This energy change is therefore not available as external work and so does not contribute to a reduction in the free enthalpy of the system. To put it another way, if the charging is carried out at constant temperature this term manifests itself as a flow of heat out of the system (7). Literature Cited (1) MOORE,W. J., "Physied Chemistry," 4th ed., Longmsn~, London, 1963, p. 3.52, el sep.
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(2) CASTELLAN, G. W., "Physiezl Chemistry," 1st ed., AddisonWesley Publishing Co., London, 1964, p. 332. (3) EGGERS,D. F., JR., el al., "Physical Chemistry," John Wiley & Sons, London, 1964, p. 3 i l . MACINNESS, D. A., "Principles of Electroehemistry," Dover, N. Y., 1961, p. 137. (4) WALL,F. T., "Chemical Thermodynamim," 2nd ed., W. H. Freeman & Co., London, 1958, p. Mi. (5) Fowmn, R., AND GUGGENHEIM, E. A,. ''Statistical Thermodvnilmics." 1st ed.. Cambridee Cniversit.7. Press. New -.---, -.-.,r
~
(6) UERYE,P., Physik. Z., 25, 9 i (1924).
(7) VERWEY,E. J. W., A N D OVERBEEK, J. TH. G., "Theory of Stability of Lyophobic Colloids." American Elsevier Publishing Co., Inc., Nex York, 1948, p. 56. (7s)Ibid, p. 5 4 (8) M ~ L L E R H., , Physik. Z., 28, 324 (1927). (9) G~~NTELBERG, E., Z . Physik. Chem., 123, 199 (1926). (101 ROBINSON.R. A,. .aND STOKES.R. H.. ''Electrol~teSohtions," Bntteraorths, Inc., London 1955, p. 25.
