Sept. 5, 1964
ACTIVITYCOEFFICIENTS
DEBYE-HUCKEL
3423
as first found by La Mer and Mason, and
where
qi(U)
- (I/&)
=
-(K/aDkT)(l
(K27112/12D2kT) [(I Ka(1
(1 f Ka)2BeLa=
-('/3)(1
- (1 -
KU)"(KU)
K'U'
+ Ka)(2 +
KU)
+ (1 +
f 2K3a3)e2""E(3Ka)(21d)
We shall be content with the approximate solutions described by eq. 20 and 21, because the system (18) is itself hardly accurate enough to justify further refinement. The corresponding electrostatic corrections to the thermodynamic potentials of the ions are accordingly n n
= ei
[CONTRIBUTION NO.
1754 FROM THE STERLING
CHEMISTRY
+
+
KU)-l
+ Ka)-3 -
f
KU)-4e3xaE(3Ka)]
(24)
represent corrections to The terms proportional to the Poisson-Boltzmann equation, which would actually contribute other terms of comparable to K 3 E ( 3 K U ) when treated more accurately. We note that the term of order K~ has a fairly small coefficient, which is independent of the parameter a. Moreover, its effect differs trivially from that of replacing K~ in the original DebyeHiickel formula by K2(1
+
(KTi2/3DkT))
(25)
This means simply that the opportunities for correlation among the ions are slightly enhanced by the immersion of a charge, provided that the resultant polarization tends to distribute the ionic strength unevenly. In a symmetrical electrolyte there is no first-order effect of this sort, and the correction vanishes. It seems likely that further corrections to \ki,corresponding by reciprocity to *3, etc., will represent r e d u c tions of the effective ionic strength. As was recognized by Bjerrum, pairs of ions close together will not contribute much to the shielding of electrostatic forces
LABORATORY OF Y A L E
UNIVERSITY,
NEW HAVEN,CONNECTICUT]
Theoretical Activity Coefficients of an Ideal Debye-Huckel Electrolyte' BY JAMES F. SKINNER^
AND
RAYMOND M. Fuoss3
RECEIVED MARCH30, 1964 A previous4 approximate integration of the Poisson-Boltzmann equation leads t o a chemical potential which contains two major terms a t low b u t nonzero concentrations. The first is the classical square root term due to long range electrostatic forces, and the second (linear in concentration) is due to short range interactions. I t is shown t h a t the net chemical potential is substantially independent of the distance chosen for the transition from the short to the long range approximation of the electrostatic potential. Also, i t is shown that the deviations from the limiting law which derive from the Poisson-Boltzmann equation are in the opposite direction from those observed in real electrolytic solutions. The calculated activity coefficient for an idealized dilute electrolyte, defined as one whose properties are described by the Poisson-Boltzmann equation, is the product of two terms : the Debye-Huckel activity coefficient for point charge electrolytes times the fraction of ions which contribute to long range interaction (the "free" ions in the Bjerrum sense).
Debye's fundamental contribution to the theory of electrolytes was the introduction of the space charge model, which permitted statement of the mathematical problems in the form of differential equations. For the thermodynamic properties, a solution bf the Poisson-Boltzmann equation was required ; an approximate solution, valid a t low concentrations, was obtained through the familiar series expansion of the Boltzmann factor. The classical limiting law (proportionality between logarithm of activity coefficient and square root of ionic strength) followed iminediately. Most subsequent treatments have been empirical and/or theoretical attempts to increase the con(1) This paper is based on part of a thesis submitted by James F.Skinner to the Graduate School of Yale University in June, 1964, in partial fulfillment of the requirements for t h e degree of Doctor of Philosophy. (2) Sterling Research Fellow, 1963-1964. (3) Grateful acknowledgment is made t o t h e donors of t h e Petroleum Research F u n d , administered by the American Chemical Society, for support of this work. (4) R. M. Fuoss and L. Onsager, Proc. N a t l . Acad. Sci. U . S., 47? 818 (1961).
centration range of the calculated activity functions. Here we shall discuss a solution so dilute that only long range interactions and ptaarwise ionic contacts need be considered; this is an idealized model which will show the magnitude and direction of those deviations from the limiting law which are inherent in the nonlinearity of the Poisson-Boltzmann equation. It is a fictitious system of charged spheres in a contkuum, whose properties are described by the differential equation; of interest is how its properties differ from those of a real electrolytic solution. It is agreed that the limiting behavior a t extremely low concentrations is the same for both. I t will be found that the ideal and reak' systems diverge from the limiting tangent in opposite directions. Another approximate integration of the PoissonBoltzmann equation A$, = ( K ~ E / Dsinh @ ) (t$j/kT)
(1)
was recently4 made in which special attention was
3424
JAMES
F. SKINNER AND RAYMOND M. Fuoss
given to the electrostatic interaction of ions a t short mutual distar~ces.~These configurations are automatically excluded from consideration by the classical approximation of the hyperbolic sine in (1) by its argument, a step which linearizes the equation to the form A$j
(2)
= ~'$j
(Above, $, is the potential at a distance r from a reference ion of species j of charge e in a medium of dielectric constant D at temperature T ; k is Boltzmann's constant and /3 = e2/DkT. This discussion is limited to The symmetrical electrolytes where t f = - e - . ) procedure used was to represent the potential at short distances by the (linear) self-potential e/Dr plus an unknown function + ( r ) +j
=
ej
/Dr
+ +(r)
(3)
Substitution of (3) into (1) gave a second-order nonlinear differential equation which determines 4 ( r )
A4
= (~%/2Di3)exp@/r
+4kT)X
[I - exp(--@/r
+ 24kT)I
A4
(5) which can be solved explicitly to give a potential $s which is valid for small values of r . ("Small" and "large" distances are, respectively, distances of the order of ionic diameters and distances of the order of the radius K - ~ of the ionic atmosphere.) Two arbitrary constants are, of course, involved. One of these is evaluated using the condition that the field strength a t r = a , the contact distance, must reduce to (-c/Da2). Then for large distances, the hyperbolic sine was approximated by the first two terms of its series in (1) to give the differentia1,equation A$L
=
= (~~e/2D@)e'/'
(~'e/D/3) [e$L/kT
which r > /3/2 are defined as free,' in the seiise that no particular ion of the opposite charge is nearby; those pairs for which r < /3/2 present essentially a dipole field to a distant observer. These ions are the ones for which the approximation sinh x = x is emphatically never justified, even in solvents of high dielectric constant, and for which the function 4(r) was introduced. Using this value of R and the Giintelberg charging process, the following expression for the activity coefficient was found
-1nfj
+ ( e $ ~ l k T ) ~ / 6 1(6)
which was solved to give $L, the long range potential. Again, two constants of integration appear; one of these is zero by the condition that the field strength v a n i s h a t r = a. In this way, two expressions, I),and $L, were obtained for the potential, the former describing the field near the ion where screening of the reference ion is negligible, and the latter serving in the region where the ionic atmosphere appreciably screens the charge of the central ion. Two arbitrary constants remain: they are evaluated by requiring that potential and field strength, which must be everywhere continuous, are in particular continuous at a value R of r . The value @/2, the Bjerrum radius, was chosen for R, because pairwise distribution function^^-^ for ions generally have a minimum near r = p/2. Ion pairs for
+-
(5) The model used here to represent the electrolytic solution is charged rigid spheres in a dielectric continuum. The short range potential is m for r < a . Ionic polarization and other short range effects are excluded by hypothesis. (6) N. Bjerrum, Kgl. Danske Vidcnskab. S d s k a b . , M a l , fys. M c d d . , 7 , No. 9 (1926). (7) (a) R. M . Fuoss, Trans. Faraday Soc., S O , 967 (1934); (b) R . M. Fuoss, J . A m . Chem. SOC.,6 7 , 2604 (1935). ( 8 ) H. Reiss, J . Chcm. Phys.. 26, 400 (1956). (9) J. C. Poirier and J. H. de Lap, ibid., SS, 213 (1961).
+ + O(T~)
= ~/(1
T)
(7)
where T is the dimensionless variable
P K / ~=
4.2016 X 106~"'/(DT)"/' (8) F o r d = 4.00 and a t 25' 7 =
c = 4.12&k2/b3
(9)
b = P/a = e2/DkTa
(10)
where
(4)
By approximating the quantity in brackets by unity and neglecting c4/kT compared to @ / r , (4) reduced to the inhomogeneous equation
Vol. 86
= 140.1/026 for
d
=
4.00
(10')
The limiting form of (7) for dilute solutions
agrees exactly of course with the classical result of Debye and is independent of ion size and of the value R of r a t which long and short range potentials and field strengths are matched. Terms of order r 2 c (and higher), however, depend both on a and R. The general nature of the higher terms is to increase the numerical value of In f j , L e . , to decrease the activity coefficient. I n other words, the empirical Bc term in In f which approximately describes the frequently observed miriimum in activity coefficients cannot possibly derive from (1) , high concentration properties will require qualitatively different theoretical sources. The purpose of this paper is to consider the influence of the choice of K on the results and conclusions of the previous analysis.
-
Theoretical Before introducing a variable matching distance R, the approximations made in reducing (4) to ( 5 ) will be reconsidered. For Y small, neglecting the term in exp(-2@/r) compared to unity in (4) is completely justified. The other approximation, exp(ed/kT) = 1, can easily be improved, and since the result introduces the point-charge activity coefficient as a multiplier, the recalculation is worth doing. The first approximationlogives e4/kT
=
2T2[B
+ A@/. + F l ( r ) 1
(12)
where
B
-l/(l
+
(13) is the leading term in the bracketed expression in (12). For small distances (which is the only range where 4 is used), this term dominates the others, and increasingly so with decreasing concentration. We therefore use
e$/kT (10) Ref. 3, eq. 9 aad 24.
-27/(1
T)T
+
T)
(14)
DEBYE-HUCKEL ACTIVITY COEFFICIENTS
Sept. 5, 1964
in (4), giving as the starting equation for + ( r ) A+ = (~~e/2DP)e”’ exp[-2r/(l
where exp[-2rl(l
+ r)1
+ r ) ] (15)
= f+z
4
=
1
+ A P / r + Fl(r) 1
(17)
+
+ Fz(R)l + O ( T )
= +j =
- ~j/Da
+ Fl(a) + FZ(a)]
(T€jKfi2/D)[B’
(19)
Use of (19) to give the electrical part of the free energy, and differentiation of the latter with respect to n, to give the activity coefficient, produces the equation -1nfj
=
(1
+ K R ) - ~ [+T T ’ G ~ ( K R+) T2f*2H(b,P/R)I
(20)
where terms of order r 3 have been dropped, and the function H i s defined by 2H(b,x) = E,(b) - e b ( l
+ b)/b2 -
+
+
E p ( x ) e x ( l x)/x2 (21) The function G 3 ( ~ Rhas ) its origin in the higher
term (in $9 which was retained in the differential equation for the long range potential. Its value, when multiplied by T ~ is, small compared to the other two terms in the brackets in (20) and rapidly decreases as its argument increases. Explicitly
G3(x) = 2 e - 3 x / 3 - 8xe”En(4x)/3
(22)
Numerical values are given in Table I. In order to R )(20) will be replaced simplify the calculation, G ~ ( K in by G 3 ( r ) in constructing Fig. 1 and 2. The error is practically invisible. TABLE I THEFUNCTION Go(%) X
0.02 .04 .06 .08 10 12 14 .16 18 20
Gdx)
X
GI(.%)
0.5176 ,4348 3740 3261 2869 ,2541 2263 2024 1817 1635
0.25 .30 .35 .40
0.12712 .09999 ,07936 06346
.45
.05104
.50 55 .60 .65 .70
,04125 ,03349 ,02728 02230 ,01828
In Table I1 are given the values of H(b,x) which will be used later. The first column gives the values (11) Ref. 3, eq. 9, 10, and 23.
b/2 b/3
5
8
7
6
5765 0535 4506 0018 8 7571 2 1570 2 9169
4.0644 5.5413 8.9385 8.4897 10,2450 3.2672 4.0644
6.5214 7.9984 9.3955 10.9467 12.7020 5.3475 6.2344
10.8786
4
3
0.0000 0 7972 1.4769 2 2741 2.8741 3 6713 4.4253 5.2225 6.1806 6 9778 1.4769 1 3426 2.8741 2.2741
1 3 4 6 7 1 2
5781 0551 4522 0034 7587 5781 3660
2 4 5 7
.....
13.7527
.., .,
17.0592 9.3005 10.3310
i t s asymptotic expansion for b > x >> 1 the same form as the association constant for ion pairs12
Discussion
(18)
For the potential +,*(a) a t the reference ion due to the presence of other ions, combination of (3), (17), and (18) gives +j*(~)
2/3 1/2 2/5
2
r2f2H(b,x) = ( 6 ~ N u ~ e ~ / 3 0 0 0 ) f 2 c (23)
- Gs(KR)
fi21Fi(R)
-
p -
2
( ~ ~ t P f + ~ / 2[B’ D)
+ KR) = - 1 / r
TABLE I1 THEFUNCTION H( a,%) x
where A and Fl(r) are the same as before,” but B’ is left dependent on the choice of R. The long range solution +L is, of course, also the same as before. On matching a t r = R, we find
f+2B’(I
of x,and the headings of the H columns are the values of b. This function, which is defined by (21), has as
(16)
is the square of the activity coefficient. The solution of (15) is
3425
Two kinds of choices are available for R : a value which depends on the solvent v i a the Bjerrum radius P/2 = e2/2DkT and a value independent of dielectric constant. Examples of both will be considered. The chemical potential is most simply presented in terms of the dimensionless variable r , defined by (8). It depends on both concentration and the D T product; two solutions which have equal values of r are in corresponding electrostatic states. Concentration can be expressed also in terms of b and r , as shown by (9). In the figures, a c-scale is given a t the top and a 7-scale a t the bottom of each panel. The Bjerrum parameter b depends on the contact distance a and on the D T product. For convenience, we shall choose the round value d = 4.00 and set T = 298.2’; values of dielectric constant corresponding to a given value of b are then easily obtained from (10’). It is clear that the approximations made in deriving (20) will limit consideration to dilute solutions; we shall use the same criterion as that used in the conductance problem13 KU < 0.2 (24) This choice is based on both theoretical and practicai grounds: (1) if KU > 0.2, the maximum charge in the atmosphere is less than five diameters from the center of the reference ion; a t shorter distances, fluctuation terms will soon become dominant and the approximation by a continuous space charge (which justifies the use of the Poisson equation) ceases to be realistic; (2) for KU > 0.2, terms of order K3u3start to become visible, and the purely numerical approximations (such ) longer are valid. Transas dropping terms in T ~ no lating (24) into other terms, we restrict the discussion to the range of variables where
2r/b
= KU
< 0.2
(25)
i . e . , to the range r < b/10 (26) For convenience in discussion, we rewrite (20) in the form
+ TZ+ T3 = -wj/kT (27) where T I is the leading term r / ( 1 + KR)in activity due --In
fj
=
TI
to long range interionic forces, and T3 is the term which explicitly depends on short range interactions. (As (12) R. M . Fuoss, J . A m . Chcm. Soc., 80, 5059 (1958). (13) R. M . Fuoss and L. Onsager, J. Phys. Chcm., 61, 668, last two paragraphs of Section 1 1 (1957).
3426
JAMES
lo%1.3
52
116
M6
F. SKINNER AND RAYMOND M. Fuoss
1 0 3 ~ 1.5
I 37
61 I
be2
values. For b > 5 (see, for example, b = 7 ) , the positive deviations become rapidly more significant as more and more ions cluster as pairs. (The crossover of curves 1,221 as b increases is due to the term (1 K R ) = (1 n r ) in the denominator and the increasing permissible range of T . ) T o state the result in a different way, in the range of higher dielectric constants (2 < b < 5 ) , the ratio p j / p j * is not a t all sensitive to the value chosen for R. This cpmes about by a compensation between TIand T 3 that has physical significance. We note that a t T = 0: TI is unity and decreases with increasing T , while T3 starts a t zero and increases; compare curves 4, 5 , and 6 with 7, 8, and 9. If we choose a larger value of R,7 3 increases (at fixed r ) ; larger R means counting more ions as short range pairs and fewer as atmosphere ions. To compensate for this, T I decreases, and in such a way that the sum remains almost constant. Consequently, while details of the picture depend on the choice of R, the net chemical potential is substantially independent of R (until, of course, the dielectric constant becomes so small that the exponential nature of H starts t o control). The model t o represent the solvent is a, dielectric continuum, which certainly is not realistic for distances of the order of small multiples of a , however good it may be for distances large compared to a. There has been considerable speculation on the dependence of “effective” dielectric constant on distance from the reference i ~ n . l ~ -In * ~the examples of Fig. 1, the matching distance R depends on the dielectric constant through the Bjerrum radius; as already noted, this choice, for b = 2, automatically eliminates the l(/s from the equations, and only absurd results can be obtained for systems for which a > p / 2 (large quaternary ions in water, for example). Consequently we shall now consider some examples in which the matching distances are absolute. An undoubtedly interesting choice md, where d is the would be to match a t I< = a diameter of a solvent molecule. A simpler choice, which is likewise a pure distance, is t o match a t R = ma, i . e . , multiples of the contact distance. These distances will be of the order of several layers of nearest neighbor solvent molecules, of course. I n Fig. 2 , plots of p j / p j * , T I , and T3 for b = 2 , 3, 5 , and 7 are shown for R = 2a and 3a. Again, the decrease in T1 and the increase in T s with increasing R appears, but now the compensation is nearly perfect; even a t b = 7, curves 1 and 2 practically coincide. Varying the match point between &s and &L between 8 and 24 A. has practically no effect on the calculated thermodynamic potential. The computation was also made for R = 4a, 5n, and 6a; the ratio pJp,* is strikingly insensitive t o the choice of m, for a given value of 6 . Some numerical values of the ratio are shown in Table 111. The result may be stated as follows : the chemical potential of an electrolyte wbich is described by our approximate solution of the nonlinear Poisson-Boltzmann equation contains two large terms, one arising from short range electrostatic interactions and one from averaged long range effects. As the distance a t which short and long range solutions are matched is increased, the first term increases and the second term
+
2 IU% 1.3
0.
3.0 5.3 8.3
t
I
0.5 0.7 Fig. 1.-Potential functions for matching a t multiples of the Bjerrum radius, nfJ/2. Code: 0.0
n
Ti/r
1 3 5
4
T8/7
7 8 9
5 6
0.3
P,/Pj*
1 2 3
already mentioned, TZis small and calls for no further comment.) In order to focus attention on the deviations from point-charge behavior, we shall consider the ratio wj/p,*, where p,* is the chemical potential for the model point-charge electrolyte, with activity coefficient fj* -In fj* = - p j j * / k T
=
T
=
t2u/2DkT (28)
In the figures this ratio is shown as the solid curves; explicitly
When the solid curve falls below unity, the activity coefficient lies above the limiting tangent ( i . e . ,the calculated activity coefficient is nearer unity than the point charge value). The dashed curves represent the long range effects TI/r and the dot-dash curves the short range ones T ~ / T . In Fig. 1, three values of R are represented: p / 2 , 3@/2, and 5p/2, where the code is given by the table in the caption. For d = 4.00, the Bjerrum radius @/2 is numerically equal to 2b X cm. Consider first curves 1, 2 , and 3 for b = 2, 3, and 5 (D = 70, 47, and 28). Curve 1 for b = 2 lies below the 1.00 line, because T3 is zero: the upper and lower limits of the ion pair integral, a and p / 2 , coincide; the solutions are matched a t the surface 7 = a . For this case, GS = t / D a a t 7 = a and IC. = +bL for 7 > a . But for b > 2 , the curves approach unity from above as T goes to zero. There is, however, no startling departure: even for b = 5 at c X In f is -0.234 v s . the limiting law value of -0.200. But note that p,/pj* is greater than unity; in words, the activity coefficient is smaller than the limiting value. We recall that activity coefficients of real solutions a t nonzero concentrations generally are larger than the limiting
Vol. 86
+
+
R M . Fuoss. Ckem. Rev., 17, 2 7 , footnote 3 (1935) (15) H.S.Frank, J . Ckrm. P k y s . , 23, 2023 (195.5). (16) J. Padova i b i d . . 39, 1552 (1963).
(14)
DEBYE-HUCKEL ACTIVITY COEFFICIENTS
Sept. 5, 1964
3427
decreases in such a way that the sum remains practically constant. TABLE I11 THERATIOu , / u , * FOR \'ARUBLE
b
AND
m;
7
= 0.2
bm
2 3 4 5 6
5
3
2
0 8888 9029 92~1 9592 1,0009
1 1 1 1 1
0 9672 9704 9719 9780 9895
1567 1699 1629 1526 1432
7
1.6145 1.6443 1.6302 1.6065 1.5810
Equation 20 can be rearranged into an interesting form which leads t o the heuristic argument that calculation of the concentration of ion pairs by the mass action equation is arithmetically justified, even though the hypothesis of an equilibrium between free and paired ions is no longer needed. Dropping small terms which are not relevant, (20) can be approximated as
We see that the curvatures a t the matching distance
lO3c 1.3 whence
5.2
11.6
20.6
, 1 0 3 ~1.5
6.I
13.7
0.1
02
03
b.2
j, = e - * exp(-rZe-"H)
(31)
For low concentrations, expansion of the second exponential function in (31) gives
j , = e-'(l
- 72e-"H)
(32) 0.0
and using (23) -
fj
C
e '(1
- KAe-2'C)
(33)
KACy2e-2T
(34)
,
1 0 3 ~ ~ 3.0 . ~
29
The mass action equation is
1- y
=
where y is the fraction of free ions. At low concentrations (34) becomes approximately 1- y
KACe-2'
(35)
and combining this result with (33), we finally obtain
f, = ye-'
(36)
T o restate the result, the activity coefficient calculated from (20) is formally the product of e-', the coefficient for point charges, and y, the fraction of ions outsider = R. The approximation given by matching function and first derivative a t R = r is really quite close, as is shown by a consideration of the curvature a t r = R. We show the result for R = p / 2 . Here the second derivatives are +g"
=
+ ~~L,(b)l
(16e/Dp3))1
(37)
0.0
0.1
0.2
Fig. 2.--Potential
0.4
0.5
functions for matching at multiples of the contact distance, ma. Code:
m
Ti/r
Tt/7
2 3
3 4
5 6
+
P,/P,*
1 2
for &(R - 0) and $L(R 0) both equal (16e/Dps) plus terms of order r 2 and higher, and that the coefficients of the T2-terms do not differ greatly. This close match in curvature means that the function goes very smoothly through the point r = R.
