1330
J. Phys. Chem. 1996, 100, 1330-1335
Correlations and Thermodynamic Coefficients in Dilute Asymmetric Electrolyte Solutions Mark A. Knackstedt and Barry W. Ninham* Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National UniVersity, Canberra ACT 0200, Australia ReceiVed: June 2, 1995; In Final Form: NoVember 7, 1995X
It is implicit in current theories that compare simulation and analytic computation to assume the long-range electrostatic force in electrolyte solutions is an exponential function of separation with a decay length equal to the Debye screening length. Theory, recently verified experimentally, shows that the decay length of the asymptotic form of the double-layer force in mixed and asymmetric electrolytes is markedly different from the classical Debye length. We derive the full (analytic) asymptotic form of the radial distribution function for electrolyte mixtures and extend the result to mixtures of symmetric and asymmetric electrolytes. This allows a simple and accurate determination of the activity and osmotic coefficients of dilute mixed electrolyte solutions.
I. Introduction Extensive literature exists on the primitive model of electrolyte solutions and on double-layer interactions that go, in principle, far beyond Debye-Huckel theory. Yet the simplest theories with extension to include hard core interactions, of the kind suggested in the classic texts like those of Robinson and Stokes1 and Harned and Owen2 and in the work of Pitzer,3 still form the basis of calculations of activity and osmotic coefficients so useful to physical chemists, geochemists, and biophysicists. The modern era began with three seminal papers of Friedman in the first issues of Journal of Solution Chemistry.4-6 Until that time it had hardly been recognized that comparison between theory (in the McMillan-Mayer constant density ensemble) and experiment (in the Lewis-Randall constant pressure ensemble) could only fairly be made by taking into account measured partial molal volumes. Later it was shown that considerable simplification could be made in computation.7 When measured partial molal volumes were taken into account, detailed comparison showed that the primitive model is indeed valid for many salts. The simple theory (Debye-Huckel plus hard core) is as successful computationally as more complicated extensions like HNC with the Carnahan-Starling equation of state. The hydration radii that emerge from fits to the measured activities are additive, as they should be if the primitive model is valid, for salts like most alkali halides but not for salts with Cs+ as cation or for NO3- and SO42- as anion.7 For these, and others, the primitive model fails, in so far as “interpenetration” of ionic shells occurs. Thus, extra “hydration” forces due to water structure must be taken into account self-consistently and analytically.7 That problem remains. Despite the in-principle theoretical advances represented by methods by Kjellander and Marcelja,8 in treating the doublelayer problem at the more sophisticated hypernetted chain (HNC) level for primitive model electrolytes at interfaces, the computational demands are very great. It is almost impossible at a practical level to compute thermodynamic and correlation functions required for many applications, especially mixed and asymmetric electrolytes and those with differing hard core radii. In the analytic theory based on mean spherical approximation (MSA)9 distribution functions can be calculated simply and explicitly. Unfortunately, as we discuss below, within MSA the effective screening length does not correctly describe the X
Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1330$12.00/0
long-range electrostatic screening. Especially in dilute systems the behavior is dominated by long-range electrostatic interactions, and the error in the screening length can affect estimates of thermodynamic coefficients. The current work gives, in the spirit of MSA, a simple and accurate set of results for dilute mixed electrolyte solutions. We derive the full asymptotic form of the radial distribution function for dilute electrolyte mixtures and extend the result to mixtures of symmetric and asymmetric electrolytes. For asymmetric electrolytes the screening length is not given by the usual Debye form because the charge distribution about a given ion at high valence is highly nonlinear. Thermodynamic properties are derived within the primitive model. The difference in the screening length strongly affects estimates of thermodynamic functions. Over 20 years ago Mitchell and one of us showed through a resummation of a diagrammatic expansion that the decay length is an asymptotic expansion that depends on concentration, the leading term of which is the Debye length κ0-1.10 For a primitive model electrolyte Cνz11Aνz22, where C and A denote cation and anion with charge z1q and z2q, respectively, the decay length, κeff-1, expressed on a molar basis has the asymptotic form11,12
[
κeff ) κ0 1 +
]
3 3 2 7 ln 3 1/2 (ν1z1 + ν2z2 ) + O(c ln c) (c ) (ν1z12 + ν2z22)3/2 24x2 (1.1)
where c is the electrolyte concentration in moles/liter and κ-1 0 is the classical Debye length:
κ0-1 )
( ) 4πq2F kBT
-1/2
(ν1z12 + ν2z22)-1/2 )
(
)
2 2 -1/2 0.304 ν1z1 + ν2z2 (1.2) 2 c1/2 expressed in nanometers. The region of validity of (1.1) is κR , 1, where R is the sum of the bare ion radii. The leading term in (1.1) vanishes identically for symmetric electrolytes, but has a large effect for asymmetric systems. For example, for a 2:1 electrolyte the real decay length is reduced by 18% from the Debye length at 10-1 M. The effect is more pronounced for electrolytes of higher valence.
© 1996 American Chemical Society
Dilute Asymmetric Electrolyte Solutions
J. Phys. Chem., Vol. 100, No. 4, 1996 1331
TABLE 1: Comparison of Experimental Asymptotic Decay Lengths for the Double-Layer Forces in Cytochrome c Solutions with the Classical Debye Screening Length k0-1 and with the Asymptotic Prediction keff Given in Eq 1 Calculated for a 12:1 Electrolyte Solution bulk concentration
k-1 0
keff
4 × 10-7 8.7 × 10-7 2.2 × 10-6 3.6 × 10-6 4.1 × 10-6
54.4 36.9 23.2 18.1 17.0
44.7 28.0 15.4 11.0 10.0
-1
experimental decay
TABLE 2: Correction Term to the Classical Debye Screening Length for Simple Asymmetric Electrolytes
45.6 26.0 16.0 11.2 10.5
B (10-2 M)
B (10-1 M)
1:1 2:1 3:1 3:2
1.00 1.00 1.01 1.00
1.00 1.02 1.10 1.04
1.00 1.06 1.31 1.12
1.00 1.18 2.00 1.39
B (10-2 M)
B (3 × 10-2 M)
B (10-1 M)
0.1 M (1:1) + 2:1 0.1 M (1:1) + 3:1 0.1 M (1:1) + 3:2 0.1 M (2:2) + 2:1 0.1 M (2:2) + 3:1 0.1 M (2:2) + 3:2 0.1 M (2:1) + 1:2 0.1 M (2:1) + 1:3 0.1 M (2:1) + 3:1 0.1 M (2:1) + 1:5
1.00 1.00 1.00 1.00 1.00 1.00 1.17 1.15 1.18 1.10
1.01 1.07 1.06 1.01 1.01 1.02 1.12 1.05 1.26 1.09
1.03 1.28 1.16 1.03 1.09 1.08 1.06 1.00 1.42 2.11
1.11 1.79 1.35 1.08 1.46 1.28 1.00 1.30 1.84 5.32
[
i
i i
]
(2.3)
where ci denotes the ionic concentration in moles/liter of species i. The range of validity of the asymptotic form of the pdf is Λ ) κ0βq2/ < 1.11 The result also breaks down when κ0Rij becomes significant compared with unity,11 where Rij ) Ri + Rj and Ri and Rj denote the radii of ions i and j, respectively. At room temperature βq2/ ∼ 7 Å, so that the necessary condition (κ0Rij)2 , 1 is always satisfied for real electrolytes whenever Λ < 1. Clearly at concentrations above 10-1 M the effective screening length formula may no longer be valid. We first discuss the implication of this result to the measurement of the screening length in asymmetric electrolytes. The correction term for two-component electrolytes discussed previously11 is shown in Table 2. The correction term to the Debye screening length Vanishes identically for symmetric electrolytes. For asymmetric electrolytes the screening length is markedly reduced. For a 2:1 electrolyte the real decay length is reduced by 18% from the Debye length at 10-1 M. The effect is very much more significant for electrolytes of higher valence. The correction to the decay length for a number of possible mixtures of electrolytes (symmetric and asymmetric) is given in Table 3. The correction term vanishes again in the case of mixed symmetric electrolytes, but in many cases of asymmetric electrolyte mixtures significant deviations from the classical Debye length are evident.
Aij (2.1) exp-Bx x where x ) κ0r and r ) |r1 - r2|. We consider a solution containing a mixed electrolyte Cνz11, Aνz22, Cνz33Aνz44, ... in an aqueous solvent of dielectric constant with density F, F′, F′′, ..., respectively. Cationic and anoinic species have charges z1q, z3q, z5q, ... and z2q, z4q, z6q, ..., respectively, where q is the unit charge. We have F1 ) ν1F and F2 ) ν2F, F3 ) ν3F′, and F4 ) ν4F′. In the Appendix we show that the decay length has the form
]
B (10-3 M)
κeff (∑i cizi3)2 B) ) (1 + 0.2265) κ0 (∑ c z 2)3/2
gij(x) = 1 -
3 2 Λ ln 3 (∑iFizi ) + O(Λ2 ln Λ) 8 (∑ F z 2)2
electrolyte
to the screening length is given as
The asymptotic behavior of the pair distribution function (pdf) in a general mixed electrolyte system is of the form10
i i i
B (10-3 M)
TABLE 3: Correction Term to the Classical Debye-Screening Length for Mixed Electrolytes
II. Generalization of Pair Distribution Functions for Mixed Electrolytes
[
B (10-5 M)
length12
For electrolytes of large asymmetry the theoretical prediction has been confirmed for aqueous solutions of cytochrome c12 and for insulin.13 The former forms a 12:1 or 8:1 electrolyte depending on pH and provides an excellent test of the primitive model. In Table 1 we show direct force measurements that confirm the remarkable result given by eq 1.12 For simple electrolytes, e.g., Th4+, the Debye length measured by the surface force apparatus (SFA) does not necessarily agree with prediction, a circumstance that reflects interpenetration of hydration shells.14 This is not an issue in this paper, where our goal is to obtain simple analytic expressions for distribution functions and activity coefficients within the primitive model. These expressions will enable tests of the primitive model to be easily carried out.
B) 1+
electrolyte
(2.2)
where Λ ) κ0βq2/. Expressed on a molar basis, the correction
The leading term in eq 2.1 is given by eq 10 in the Appendix. Insertion of (10) and (2.3) into (2.1) gives the asymptotic expression for the radial distribution function. For a simple electrolyte the correlation function is written as
[
gii(r) ) 1.0 - Λzi2
1+
(
)
(
[
3 2
]
exp(-B κ0r) κ0r (νizi + νjzj ) - 2νizi νjzj (zi - zj)
Λ νizi + νjzj ln 3 1.0 + Λ 6 νz2 + νz2 8 i i j j 3
)
νjzj2(zi - zj)2 ln 3 Λ zi2 4 νz2 + νz2 i i
3
j j 3 2
2
(νizi2 + νjzj2)2
2
2
]
ln 3 Λ(zizj) exp(-B κ0r) 4 gij(r) ) 1.0 - Λzizj 3 3 2 3 3 2 2 2 2 κ0r Λ νizi + νjzj ln 3 (νizi + νjzj ) - 2νizi νjzj (zi - zj) 1.0 + Λ 6 νz2 + νz2 8 (νizi2 + νjzj2)2 i i j j Note that the leading terms reduce identically to Λzi2 and Λzizj, respectively, for symmetric electrolytes.
(
)
1+
(2.4)
(2.5)
1332 J. Phys. Chem., Vol. 100, No. 4, 1996
Knackstedt and Ninham
Figure 1. Comparison of the radial distribution functions for simple 2:1 electrolyte given by the primitive model and by the asymptotic form given by eq 2.1 at c ) 10-2 M. g12(r): (s) primitive model, (‚‚‚) asymptotic form. g11(r): (- - -) primitive model, (‚‚‚) asymptotic form. g22(r): (-‚-) primitive model, (-‚‚‚-) asymptotic form.
Figure 2. Comparison of the radial distribution functions for simple 3:1 electrolyte given by the restricted primitive model and by the asymptotic form given by eq 2.1 at c ) 10-3 M. g12(r): (s) primitive model, (‚‚‚) asymptotic form. g11(r): (- - -) primitive model, (‚‚‚) asymptotic form. g22(r): (-‚-) primitive model, (-‚‚‚-) asymptotic form.
The radial distribution function plays the central role in statistical mechanical theories of ionic solutions since all thermodynamic quantities can be expressed in terms of it. Thermodynamic properties for electrolytes are often derived and simulated in terms of the restricted primitive model (the simple Debye-Huckel theory decorated to include hard sphere repulsion) of an ionic solution. In this model the radial distribution function has the form
(2.6)
Figure 3. Comparison of the radial distribution functions g11(r) and g22(r) for simple 1:1 electrolyte with a small concentration of added asymmetric (X:1). The dotted curve gives the restricted primitive model prediction for both cation g11 and anion g22; the solid line, the asymptotic form for g11(r); and the dashed line, the asymptotic form for g22(r): (a) c1:1 ) 0.1 M, c2:1 ) 0.01 M; (b) c1:1 ) 0.1 M, c3:1 ) 0.001 M; (c) c1:1 ) 0.l M, c8:1 ) 5.0 × 10-5 M.
A comparison of eqs 2.4-5 and 2.6 are shown in Figures 1 and 2 for simple asymmetric electrolytes at very dilute concentrations. The effect of the enhanced screening in (2.45) is evident. In the discussion leading to (2.4-5) we have assumed point charges; one can choose instead to consider a system of charged hard spheres of diameter Rij. The hard sphere contribution to the radial distribution function is small, however, since we are considering very dilute solutions (the term exp(κ0Rij)/(1 + κ0Rij) is very small for very dilute solutions; for κ0-1 large).
In Figure 3 we compare the form of eq 2.1 with that of eq 2.6 for mixtures of symmetric 1:1 and an asymmetric electrolyte (X:1). In all these cases we consider a small quantity of the added asymmetric electrolyte. The distribution function g11 for the co-ion of the ion of high valence is barely different from the prediction of the restricted primitive model. A large difference in the counterion distribution g22 is noted due to the highly nonlinear charge distribution about the multivalent ion. An effect on mixtures of two symmetric electrolytes is also noted. In Figure 4 we show the prediction of the radial distribution functions (g11, g22) for the system of 1:1 with added dilute 2:2.
exp(κ0Rij) exp - [κ0r] gij(r) ) 1.0 - Λzizj 1 + κ0Rij r
Dilute Asymmetric Electrolyte Solutions
J. Phys. Chem., Vol. 100, No. 4, 1996 1333
Figure 4. Comparison of the radial distribution function g11(r) for simple 1:1 electrolyte (c1:1 ) 0.1 M) with a small (c2:2 ) 10-2 M) concentration of added 2:2. The dashed curve gives the restricted primitive model prediction; the solid line, the asymptotic form.
Figure 5. Osmotic coefficient calculated from the virial equation for a simple 2:1 electrolyte. The dashed curve gives the prediction of restricted primitive model, while the dotted curve gives the prediction from integration of (3.6) with the asymptotic form of gij(r).
III. Derivation of Thermodynamic Properties The thermodynamics of the electrolyte can be readily calculated using the radial distribution functions derived in the previous section. Assuming that the potential energy of the N-body system is pairwise additive, all thermodynamic functions can be written in terms of g(r).15 The excess thermodynamic energies and osmotic pressures are given by the energy and virial equations:
E*
)
NkT
E
3 1 ∞ cicj∫0 uijgij(r)4πr2 dr - ) ∑ NkT 2 2ckT i,j
(3.1)
∂uij gij(r)4πr2 dr ∂r
(3.2)
and
p* ) p - ckT ) -
1 6
∑ cicj∫0 r ∞
i,j
where zi and ci are the valence and bulk concentration of ionic species i, c ) ∑ ci, N ) ∑ Nj, and uij is the interionic potential. Taking the interionic potential to be
uij(r) ) ∞, r < R
(3.3)
zizje2 , r>R r
(3.4)
∑ cicjzizj∫R gij(r) dr ckT i,j
(3.5)
uij(r) ) (3.1) and (3.2) become
E* NkT
)
2πe2
∞
and
p* )
e2
∑cicjzizj∫R gij(r)4πr dr + 6 i,j ∞
2π kTR3 3
cicjgij(R+) ∑ i,j
(3.6)
The osmotic coefficient and activity coefficients are easily derived from either equation.7,16 The two routes to the thermodynamics, while formally equivalent, generate different results. Their relative merits can be assessed only a posteriori, a matter beyond our brief report.17 In this section we illustrate the implication of the form of g(rij) derived in the previous section to calculate the osmotic coefficients via the virial equation. We compare the predictions
Figure 6. Effect of a small concentration of 2:1 electrolyte on the osmotic coefficient of 0.1 M symmetric electrolyte. The dashed curve gives the prediction of the restricted primitive model, while the dotted curve gives the prediction from integration of (3.6) with the asymptotic form of gij(r).
derived from (2.1) with model calculations based on the restricted primitive model. The comparison is shown in Figure 5. The primitive model result (the simple Debye-Huckel theory decorated to include hard sphere repulsion) provides an excellent description of thermodynamics of symmetric electrolytes,7,16 often as good as any of the modern theories (MSA, “exact” Monte-Carlo, etc.). For simple asymmetric systems the corrected asymptotic form of the correlation function leads to a marked difference in the osmotic coefficient still at very low concentrations within the primitive model. For a mixed asymmetric system (1:1 electrolyte with a dilute concentration of Z:1) the thermodynamics is strongly affected by the added asymmetric salt, as shown in Figure 6. To apply the results here to real electrolyte solutions requires applying the LewisRandall to the McMillan-Mayer conversion and taking into account measured partial molal volumes.7 More importantly, extensive experimental data for dilute asymmetrics and mixed electrolytes are required. IV. Conclusion Long-range electrostatic interactions dominate the behavior of electrolytes at low concentrations, both for bulk solutions and in double-layer interactions. The asymptotic form of distribution functions therefore provides the leading contribution
1334 J. Phys. Chem., Vol. 100, No. 4, 1996
Knackstedt and Ninham where q is the unit charge. We have F1 ) ν1F, F2 ) ν2F, F3 ) ν3F′, F4 ) ν4F′, ... and ∑ Fizi ) 0, the last being a consequence of charge neutrality.
to the thermodynamic functions. What we have shown is that, within the primitive model, the distribution functions can be calculated explicitly and analytically. The effect of the nonclassical form of the Debye length is quite marked for asymmetric electrolytes and thus can affect integration of measured thermodynamic functions and surface forces. It does appear necessary to use the correct asymptotic forms, especially in double-layer problems that involve mixed electrolytes, before we can arrive at any conclusion about “hydration” effects in surface forces, currently the subject of great debate. Finally, in some problems that involve charged proteins, with electrolytes, some curious results sometimes emerge that appear to defy thermodynamics. Thus with added electrolyte, say with cation as counterion, the protein appears to “unbind” on addition of electrolyte, in defiance of the law of mass action. When the self-energy of a protein is calculated, as a function of model electrolyte, it is necessary to use the correct Debye length for the entire system, protein plus electrolyte, to achieve sensible results. It is clear then that there is indeed no inconsistency. We remark again that the results we give are valid only for the primitive model. For many real asymmetric electrolytes it is necessary to go to the higher order “civilized” model1,7 to obtain sensible results.
The electrostatic interaction in electrolytes at large distances is governed by the asymptotic behavior of the pair distribution function gij(r). The pair distribution function has the asymptotic form gij(x) = 1 - Aij/x exp-Bx, where x ) κ0r and r ) |r1 - r2|. The indirect correlation functions hij(r) ) gij(r) - 1 are related to the direct correlation functions cij(r) through the OrnsteinZernike relations
hij ) cij + ∑ Fk∫cik(r - s) hkj(s) ds
(1)
k
In Fourier space hij are solutions of the system of linear equations (n2 equations for n species).
hij ) cij + ∑ Fkcik hkj
(2)
k
which can be expressed in terms of the matricial equation
(1 - xFiFj cik)(1 - xFkFj hjk) ) 1
Appendix Here, the asymptotic form for the pair distribution function in a mixture of electrolytes is derived. The derivation is a direct generalization of Mitchell and Ninham.11 Consider a solution containing a mixture of m ) n/2 electrolytes, Cνz11, Aνz22, Cνz33Aνz44, ..., in an aqueous solvent of dielectric constant with density F, F′, F′′, ... Cationic and anoinic species are denoted by C, C′, C′′, ... and A, A′, A′′, ... with charges z1q, z3q, z5q, ... and z2q, z4q, z6q, ..., respectively,
(3)
The indirect correlation functions hij have the solution
hij )
cij + ∑k Fk [cik ckj - cij ckk]
(4)
|1 - xFiFj cij||
where the denominator is the determinant |1 -
xFiFj cij|.
Through resummation of a diagrammatic expansion Mitchell and Ninham11 showed that when κ0Rij , 1, the Fourier transform of the direct correlation functions (cij) has the form
cij ) -
-1 4πΛ 2πΛ2 2 2tan (Ω/2) z z + z z i j i j Ω κ03Ω2 κ03
(5)
where Ω ) k/κ0 is the normalized wave vector, and Λ ) κ0q2/kBT. As in ref 11 the constant B in the eq 1 is determined by the zero of the determinant |1 - xFiFj cij|. To lowest order we require the zero of -1 Λ tan (Ω/2) 1 1+ 22 Ω Ω
(∑iFizi4)(∑iFizi2) +
1 n-1 n FiFjzi2zj2(zi - zj)2 ∑i)1 ∑j)i+1 Ω2 + ... ) 0 (∑iFizi2)2
(6)
To first approximation the root of the above equation is Ω0 ) i(i2 ) -1) and tan -1(Ω0)/2) ) i/2 ln 3. Hence to lowest order
[
Ω0 ) i 1 +
]
4 2 n-1 n 2 2 2 Λ ln 3 (∑iFizi )(∑iFizi ) - ∑i)1 ∑j)i+1 FiFjzi zj (zi - zj) + O(Λ2 ln Λ) 8 (∑ F z 2)2 i i i
(7)
This equation is valid for κ0Rij , 1. Note that the correction term Vanishes identically for a mixture of symmetric electrolytes. The pair distribution function therefore has the asymptotic form gij(x) ∼ e-Bx/x, where B ) κeff/κ0 is given by 4 2 n-1 n 2 2 2 Λ ln 3 (∑iFizi )(∑iFizi ) - ∑i)1 ∑j)i+1 FiFjzi zj (zi - zj) + O(Λ2 ln Λ) B)1+ 8 (∑ F 2)2
(8)
i izi
This result reduces to (2.2) in the main text. Aij is given by 2πi times the residue of the integrand of eq 4 at the pole Ω ) iB.
Aij )
1
[ [
(2π)2i
lim ΩfiB
Ω - iB
]
|1 - xFiFj cij|
[cij + ∑ Fk[cik ckj - cij ckk]]2πi k
(9)
Dilute Asymmetric Electrolyte Solutions Evaluated at B f 1, this leads to
J. Phys. Chem., Vol. 100, No. 4, 1996 1335
[
1+
Aij ) Λzizj
[
]
∑k*ik*j Fkzk2(zk(zi + zj - zk) - zizj)] ln 3 Λ zizj + 4 ∑Fz2
[ ]
[
i i i 2
Λ ∑iFizi ln 3 2(∑iFizi )(∑iFizi ) - 3(∑ 1+ Λ 6 ∑Fz2 8 (∑iFizi2)2 i i i 3 2
References and Notes (1) Robinson, R.; Stokes, R. Electrolyte Solutions; Butterworths: London, 1954. (2) Harned, H.; Owen, B. The Physical Chemistry of Electrolyte Solutions; Rheinhold: New York, 1943. (3) Pitzer, K. J. Phys. Chem. 1973, 77, 268. (4) Friedman, H. J. Solution Chem. 1972, 1, 387. (5) Friedman, H. J. Solution Chem. 1972, 1, 413. (6) Friedman, H. J. Solution Chem. 1972, 1, 419. (7) Pailthorpe, B.; Mitchell, D.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1984, 80, 115. (8) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1986, 127, 402. (9) Blum, L.; Hoye, J. J. Phys. Chem. 1977, 81, 1311.
4
2 2 i,jFiFjzi zj (zi
]
- zj)2)
(10)
(10) Mitchell, D.; Ninham, B. Phys. ReV. 1968, 174, 280. (11) Mitchell, D.; Ninham, B. Chem. Phys. Lett. 1978, 53, 397. (12) Kekicheff, P.; Ninham, B. Europhys. Lett. 1990, 12, 471. (13) Nylander, T.; Kekicheff, P.; Ninham, B. J. Colloid Interface Sci. 1994, 164, 136. (14) Pashley, R. Personal communication. (15) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976. (16) Olivares, W.; McQuarrie, D. Biophys. J. 1975, 15, 143. (17) While the contact term in (3.6) is not treated in an entirely consistent manner, at the concentrations we consider the contribution of the term is negligible.
JP951505S
