J. Phys. Chem. 1990, 94, 8412-8471
8472
vancement of Pure Research (NWO). We thank Mr. R. W. W. Hooft for his assistance in writing the program for the numerical solution of the Poisson-Boltzmann Equation. Finally we thank Miss M. Uit de Bulten and Miss M. de Groot for typing the manuscript. Appendix. Derivation of the Adsorption Equation for Ions in an Electrical Double Layer (Eq 15) Consider a system with a constant volume V, consisting of n,
moles of species i, with valence number zi. Inside the system is a charged surface, with an area A, in contact with the solution via an electrical double layer. The charged surface has a constant curvature, which need not be zero. Similar to the approximations that are made in the PB equation, we will assume that only electrical forces are present in the double layer. Of course, when we only consider the electrical forces, the electrical double layer is unstable and would not form spontaneously. In real physical systems a driving force, such as the chemical adsorption of a potential-determining ion, is present. However, we omit the specification of a driving force because it is not essential for this derivation and would cause unnecessary complications. The situation inside the system is conveniently described by defining excess quantities that are due to the presence of the electrical double layer. Following the Gibbs treatment of the surface tension, we define
(A3) where the superscript r refers to the reference system, for which the concentrations of the ions are constant throughout the system and equal to the concentrations far from the charged surface in the actual system. The superscript s refers to the excess quantities due to the presence of the electrical double layer. The differential of the excess free energy the double layer d F , at constant surface charge density CT, is given by S=S+S'
d F = -Ssd T +
Epi dn: + fel
dA
(A4)
I
On the right-hand side of eq A4 the first term gives the temperature dependence of F , the contribution x i p i dn: originates from the excess of the ions in the interfacial region, and the contribution fer dA is due to the excess electrical energy in the double layer. The excess Helmholtz free energy is a homogenous function of first order in A and n:. Therefore we can integrate over these extensive variables at constant T, yielding
Fs = f e , A + E p i n : i
(A5)
By subtracting from eq A4 the equation obtained from general differentiation of eq A5, we obtain the Gibbs-Duhem equation Ssd T + Cni dp: + A df,, = 0 ('46) I
A rearrangement of eq A6 finally yields, for constant T and u dpi df,l = -XI', i
(A7)
Comparison of a Primitive Model Perturbation Theory with Experimental Data of Stmple Electrolytes Kwong-Yu Chan Department of Chemistry, University of Hong Kong, Pokfulam Road, Hong Kong (Received: January 23, 1990)
Perturbation theory of the restricted primitive model is applied to some simple electrolytes and compared with experimental data of activity coefficients. The theory fits the data well with only one parameter, the diameter of the ion. However, the fitted diameter is inconsistent with Pauling's diameter.
Introduction
Debye and Huckel's theory of simple electrolytes] has been well tested against many experimental data. Their theory has been accepted to be exact in the infinite dilute solution limit. For higher concentrations, typically >0.002 M, the DebyeHuckel limiting law is found to be incorrect. Over the years, numerous theories have been introduced to improve the Debye-Huckel theory. These include modifications of the Debye-Huckel equation, empirical approaches, and statistical mechanics based theories. So far, none of the theories has been completely satisfactory. The representative empirical theories are Pitzer's virial developmentZ and NRTL theory of Chen et al.' These are being used inindustry. However, the parameters in empirical correlationsare not for many electrolytes, and the extrapolation to other conditions is difficult. Statistical mechanics theories centered around two Hamiltonian models: the primitive model in which the solvent is a continuum, ( 1 ) Debye, P.;Huckel, E. Phys. Z 1923, 24, 185. (2) Pitzer, K. S. J . Phys. Chem. 1973, 77, 268. (3) Chen. C. C.; Britt, H. I.; Boston, J. F.; Evans, L. A . AIChE J . 1979, 25, 820.
0022-3654/90/2094-8412$02.50/0
and the ion-dipole model in which the solvent is modeled as dipolar hard spheres. Variations exist in the techniques of solving the corresponding partition function and distribution functions in the modeled system. Several reviews have appeared on this s ~ b j e c t . ~ Perturbation theories were developed by Stell and LebowitzS for the primitive model and by Henderson et ale6for the ion-dipole model. Waisman and Lebowitz' and Blum* have solved the mean spherical approximation (MSA) for the primitive model and Blumg and Adelman and Duetsch'O have solved the MSA for the iondipole model. The hypernatted chain (HNC) integral equation of the primitive model was solved by Rasaiah and Friedman," (4) Hakkjoid, B.; Stell, G. In The Liquid Stare ofhfatrer: Nuids, Simple and Complex; Montroll, E,W., Lebowitz, J. L,,North-Holland,: 1982; p 175. Friedman,,, H, R ~phys. ~ ,,-hem. 1981, 32, 179. (5) Stell, G.; Lebowitz, J. L.J. Chem. Phys. 1968, 48, 3706. (6) Henderson, D.; Blum, L.; Tani, A. In ACS Symposium Series No. 3W,
Chao. K.C.; Robinson, R. L., American Chemical Society: Washington, - DC..
1986. (7) Waisman, E.; Lebowitz, J. L. J . Chem. Phys. 1972, 56, 3086, 3093. (8) Blum, L. Theor. Chem. 1980, 5. 1. (9) Blum, L. Chem. Phys. Letr. 1974,26,200. Blum, L. J . Chem. Phys. 1974, 61, 2129. (IO) Adelman, S. A.; Deutch, M. J. J . Chem. Phys. 1974, 60,3935.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990 8473
Perturbation Theory Applied to Simple Electrolytes and the ion-dipole HNC equations have been solved by Levesque et al.12 Computer simulation work was also performed to test the accuracies of the theories. Card and ValleauI3 have reported the MC simulations for the primitive model, and recently Chan et aI.l4 and Caillol et a1.I5 have reported results for the ion-dipole model. These different theories and computer simulation results tested the consistency of the mathematics in solving the thermodynamics of a given Hamiltonian. Comparison with experimental data is necessary to assess the correctness of the Hamiltonian. The HNC theory has been compared with experimental data." The MSA has been applied to several electrolyte systems by Planche and Renon.I6 The perturbation theory is in general easier to apply since interactive procedures are not required and integration and differentiation can be easily performed to obtain other thermodynamic variables. Furthermore, perturbation theory is easier to visualize physically, and the error can be traced to the order of the parameter being truncated. The test of perturbation theory, although relatively simple, has not. been reported previously. This paper will report such a preliminary test.
is the square of the inverse Debye length, pi is the number density of the ion i, p is the total number density of ions, AHS is the Helmholtz free energy of hard spheres, and hYz and h:23 are the total pair and triplet correlation functions of hard spheres at a given density. Using the analytical expressions of the structural properties from the Percus-Yevick equation, Larsen et aI.I9 gave the following expressions for the various integrals in ( 5 ) :
+ 0.4581p* - 0.268p*' + 0.1543p*' 0 . 0 7 3 3 ~ *+~ 0 . 0 1 6 8 ~ * ~ ) u2q = ShY2r = uz(-0.5 + 0.5760p* - 0.591 l ~ +* ~ 0 . 5 4 2 8 ~ *-~0 . 4 1 6 3 ~ *+~ 0.1630p*') u3w = 1 h P 2 r Z= a3(-0.333 31 + 0.7418p* - 1 . 2 0 4 7 ~ *+~ UP
= h92 = a(-1
+
1.613 9 ~ -* 1~. 5 4 8 7 ~ * ~ 0.6626p*')
+
= JSSh;l23 = ~ ~ ( 1 . 52.3431~* 2.8107*'-
U ~ S
+
3 . 1 2 9 2 ~ * ~ 2 . 6 8 9 9 ~ *-~1 . 0 0 8 1 ~ * ~(10) )
Perturbation Theory Perturbation theories have been reviewed by Henderson" and Gubbins.'* In general, the Helmholtz free energy is expanded into a series of succesive higher order terms of a certain parameter starting from the equation A - A. = -kT In ( Q / Q o )
where p* = pa3 is the reduced density. The first term of (5) is exactly equal to the Debye-Huckel limiting law. The series converges very slowly, and a Padi approximation can be formed: A =--
Nk T
(1)
where A and A. are the free energies of the system and reference system, Q and Qoare the configuration partition functions of the system and the reference system, k is the Boltzmann constant, and T is the absolute temperature. The configuration partition function depends on the Hamiltonian of the system:
12rP* K3a3
(
)
+ n1Ka 1 + dlKa + d 2 ~ a ~ 1
(11)
where n , , dl, and dzare constantsZorelated to the various integrals in (7)-( IO). Stell and Wuzo gave the following equations: n , = ( 1 . 5 ~- 0.5s
+ 2.25pq + 0.4218p3)/(0.5625p2 + 1.5q) (12)
+ 1.12Spq)/(0.5625p2 + 1.5q) (13) d2 = ( - 1 . 1 2 5 ~+ ~ 0 . 3 7 5 ~+~2.25q2)/(0.5625pz + 1.5q) dl = ( 1 . 5 ~- 0.5s
where the integration is over the whole coordinate phase space, and the Hamiltonian for the primitive model of electrolytes is (3)
(4)
where uij is the pair interaction energy between i and j , zi and zj are the charge valencies of i and j , rij is the distance between i and j , e is the charge of an electron, D is the dielectric constant of the medium, and ai = aj = ai, is the case of the restricted model. Perturbation theory of the restricted primitive model (RPM) has been solved by Stell and L e b o ~ i t z .Their ~ result is
(14)
-
-
In expanding the Padi, the series ( 5 ) will be reproduced to sixth order in KU. The correct limiting behavior of K U 0 and KU m are also satisfied. For the special case of d l = 1, dz, n , = 0, the Debye-Huckel equation with diameter dependence is produced. Equation 11 has been tested favorably against MC datal9 and was found to give results indistinguishable from MSA. Slight deviations from H N C results of Rasaiah was observed by the same authors. Experimental data of electroyte solutions are available in osmotic coefficients and activity coefficients. To perform a direct test of (1 1) with experimental data, chemical potential is needed. The chemical potential of species i is Pi = a ( A p / N ) / a p i
(15)
The original series ( 5 ) can be differentiated term by term, and then a new Pad6 can be formed. Alternatively, (1 1) can be differentiated and then checked for proper limiting behavior. Since a / a p i = ( a / a p ) ( a p / a p i ) = a/+, where p is the total density of all ions, the latter method yields
where K~
= 4re2/(kTD)
all ions
z?pi
(6)
I ) Rasaiah, J. C.; Friedman, H. L. J . Chem. Phys. 1968, 48, 2742. 2) Levesque, D.; Weis, J . J.; Patey, G. J . Chem. Phys. 1980, 72, 1887. 3) Card, D. N.; Valleau, J. P. J . Chem. Phys. 1970, 12, 6232. 4) Chan, K. Y.; Gubbins. K. E.; Henderson, D.; Blum, L. Mol. Phys. 1, 66, 299. 5) Caillol, J . M.; Levesque, D.; Weis, J . J. Mol. Phys. 1990, 69, 199. 6) Planche, H.; Renon, H . J . Phys. Chem. 1981, 85, 3924. 7) Henderson, D. Ann. Reo. Phys. Chem. 1974, 25, 461. 8) Gubbins, K. E. Fluid Phase Equilib. 1983, 13, 35.
-
where the superscript A of I.L denotes the excess over the reference hard-sphere fluid. The proper limits are obtained as KU 0 and KU m . Again the leading order term is the Debye-Huckel
-
(19) Larsen, B.; Rasaiah, J. C.; Stell, G. Mol. Phys. 1977, 33, 987 (20) Stell, G.; Wu, K. C. J . Chem. Phys. 1975, 63, 491.
8474 The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
1
I
Chan
t
00 t
3708
I
t r
+
05c -
1
c +
t
I
2758
+
G G-
f
2,
KCI
1
a
c5c
I
+*++
-c
I 1
2 554 S6P
2 754
-050
%6P
-
3 704 Y 63
0 00
I
I
t
- 1.00
L
- ' 52 , P
-
50 2Cd 25C 300
'23
mo a r o 3
Figure 2. Activity coefficients of bromide salts.
limiting behavior and for the special case of nt = d2 = 0 and d , = 1, the first term gives the full DebyeHuckel equation. To apply (16), the chemical potential of the hard-sphere reference system is needed. Several approximations can be used. The Carnahan-Starling equation2' yields
+ (3 - q ) / ( 1 - q ) 3
sqrtic)
rro aro'
sqrtk)
Figure 1 Activity coefficients of chloride salts.
p l H S / k T= In (pA3) - 3
171
-Lb
030 0 5 0 '00 ' 5 0 200 2 5 0 306
(17)
i
100 -
+-
050 -
312A
where A is the thermal de Broglie wavelength and q is the packing factor (a/6)p* and 1,= PIA +
3 wA SL6P
(18) Notice the density and q in (17) are those of total ions, although the chemical potential is of an individual ion. The molar activity coefficient 6, can be obtained by finding the limiting function of ( 1 8) as p 0 and then subtract it from pl: lim p, = pI* = kT In (pA3) (19) P-0
c
LII
A
KI
I
1
PIHS
-
~
2 406
9w
312A
YEP
-0 50
I
t
'
--1 5 000 0 I0 5 0 100 o 150 2 00 0 2 5 0 3 0L 0
sqrtk)
molaro5
Figure 3. Activity coefficients of iodide salts.
For the restricted primitive model, the ions have the same size. Therefore, the mean ionic coefficient is equal to the individual activity coefficient:
E*
= (E+E-)0.5 = F+ =
E-
(22)
Comparison with Experimental Data
The RPM assumes ions of equal diameter and charges. While simple symmetric electrolytes have cations and anions of the same charge, most of the cations and anions have different sizes. Assuming the mean ionic activity can be represented by a mean ion of a given size, the comparison of RPM with data is meaningful. Of the common electrolytes, activity increases in the order H+ > Li+ > Na+ > K+ > Cs+ in reverse order of their sizes. On the other hand, activity increases in the order I- > B i > CI-, in the same order of their sizes. Some combinations of these electrolytes are chosen for testing the perturbation theory. The data from Robinson and Stokes22and Parsons23were used. Conversion (21) Eyring, H.; Henderson, D.; Stover, B. J.; Eyring, E . M. Staristical Mechanics and Dynamics; Wiley: New York, 1982; p 268. (22) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Academic Press: New York, 1955. (23) Parsons, R. Handbook of Electrochemical Constants; Butterworths: London, 1959.
from the molality scale to molarity scale is done according to the equations from Robinson and Stokes:
E* = (mdo/c)7* c = md/(l
+ 0.001mWg)
(23)
(24)
where E+ is the activity coefficient in the concentration scale of moles/liter, yt is the activity coefficient in the scale of moles/ kilogram of solution, m is molality, c is molarity, d is the density of the solution, do is the density of the pure solvent, and W, is the molecular weight of the solute. The actual densities of the solution at various concentrations are from L a ~ ~ g e . The * ~ converted data are listed in Table 1. The Stell and Lebowitz sixth-order expansion with Pad6 resummation (SL6P) expressed in (21),(16),and related equations is compared with the experimental data in Figures 1-3 for the halides and in Figure 4 for the Li salts. The wide range of data can be very well fitted by SMP with only one parameter, viz., the diameter of the ion. In the dilute limit, there is exact agreement between the theory and data since it is the region of (24) Lange, N. A. Lunge's Handbook of Chemistry; McGraw-Hill: New York, 1987.
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990 8415
Perturbation Theory Applied to Simple Electrolytes
TABLE I: Experimental Activity Coefficient Data in Two Scales” m Y* d c E+ m Y+ d c €+ m Y+ d c E+ (a) LiCl (cation diam = 1.20 A. anion diam = 3.62A. mol wt = 42.40) 0.00I 0.963 0.999 0.001 0.964 0.300 0.744 1.006 0.298 0.749 1.200 0.796 1.026 1.172 0.815 0.002 0.948 0.999 0.002 0.949 0.400 0.740 1.008 0.396 0.747 1.400 0.823 1.031 1.362 0.846 0.005 0.921 0.999 0.005 0.922 0.500 0.739 1.010 0.495 0.747 1.600 0.853 1.035 1.551 0.880 0.010 0.895 0.999 0.010 0.896 0.600 0.743 1.013 0.593 0.752 1.800 0.885 1.039 1.738 0.917 0.020 0.865 0.999 0.020 0.867 0.700 0.748 1.015 0.690 0.759 2.000 0.921 1.043 1.923 0.958 0.050 0.819 1.000 0.050 0.821 0.800 0.755 1.017 0.787 0.767 2.500 1.026 1.054 2.381 1.077 0.100 0.790 1.oo 1 0.100 0.793 0.900 0.764 1.019 0.884 0.778 3.000 1.156 1.063 2.830 1.225 0.200 0.757 1.003 0.199 0.761 1.000 0.774 1.022 0.980 0.790 3.500 1.317 1.073 3.270 1.410
4.000 4.500 5.000 5.500 6.000 7.000 8.000 9.000
(b) LiBr (cation diam = 1.20 A, anion diam = 3.90 A, mol wt = 86.86) 0.300 0.756 1.017 0.297 0.763 1.200 0.837 1.070 1.163 0.864 0.400 0.752 1.023 0.396 0.761 1.400 0.874 1.081 1.350 0.907 0.500 0.753 1.029 0.493 0.763 1.600 0.917 1.093 1.535 0.956 0.600 0.758 1.035 0.590 0.770 1.800 0.964 1.104 1.718 1.010 0.700 0.767 1.041 0.687 0.782 2.000 1.015 1.115 1.899 1.069 0.800 0.777 1.047 0.783 0.794 2.500 1.161 1.142 2.345 1.238 0.900 0.789 1.053 0.879 0.808 3.000 1.341 1.168 2.779 1.447 1.000 0.803 1.058 0.974 0.825 3.500 1.584 1.193 3.203 1.731
(c) Lil (cation diam = 1.20 A, anion diam 0.600 0.838 1.056 0.586 0.857 1.000 0.700 0.852 1.065 0.682 0.875 1.200 0.800 0.870 1.074 0.776 0.897 1.400 0.900 0.888 1.083 0.870 0.918 1.600
E+
d
c
1.510 1.741 2.020 2.340 2.720 3.710 5.100 6.960
1.082 1.091 1.100 1.109 1.117 1.133 1.148 1.163
3.701 4.124 4.538 4.944 5.342 6.115 6.859 7.576
1.632 1.900 2.226 2.603 3.055 4.247 5.949 8.268
4.000 4.500 5.000 5.500 6.000 7.000 8.000 9.000
1.897 2.280 2.740 3.270 3.920 5.760 8.610 12.920
1.218 1.242 1.266 1.289 1.312 1.355 1.397 1.437
3.616 4.020 4.414 4.798 5.173 5.899 6.593 7.258
2.098 2.552 3.104 3.749 4.546 6.835 10.448 16.022
0.944 0.998 1.060 1.127
1.800 2.000 2.500 3.000
1.127 1.198 1.418 1.715
1.163 1.180 1.221 1.261
1.687 1.861 2.288 2.700
1.203 1.287 1.550 1.905
2.000 2.500 3.000 3.500 4.000 4.500
0.573 0.569 0.569 0.572 0.577 0.583
m
Y+
0.00I 0.002 0.005 0.010 0.020 0.050 0.100 0.200
0.966 0.954 0.932 0.909 0.882 0.842 0.796 0.766
0.999 0.999 0.999 1.000 1.000 1.002 1.005 1.011
0.001 0.002 0.005 0.010 0.020 0.050 0.100 0.199
0.967 0.955 0.933 0.910 0.883 0.844 0.799 0.771
0.100 0.200 0.300 0.400 0.500
0.815 0.802 0.804 0.813 0.824
1.009 1.018 1.028 1.037 1.047
0.100 0.198 0.296 0.394 0.490
0.819 0.809 0.814 0.826 0.840
0.001 0.002 0.005 0.0IO 0.020 0.050 0.100
0.965 0.952 0.927 0.902 0.869 0.816 0.770
0.999 0.999 0.999 1.000 1.000 1.001 1.004
0.001 0.966
0.002 0.005 0.010 0.020 0.050 0.100
0.953 0.928 0.903 0.870 0.818 0.773
(d) KCI (cation diam = 2.66 A, anion diam = 3.62 A, mol wt = 74.56) 0.200 0.718 1.008 0.199 0.723 0.900 0.610 1.039 0.876 0.627 0.300 0.688 1.013 0.297 0.695 1.000 0.604 1.043 0.971 0.622 0.400 0.666 1.017 0.395 0.674 1.200 0.593 1.051 1.158 0.615 0.500 0.649 1.021 0.492 0.659 1.400 0.586 1.060 1.343 0.611 0.600 0.637 1.026 0.589 0.649 1.600 0.580 1.068 1.526 0.608 0.700 0.626 1.030 0.685 0.639 1.800 0.576 1.076 1.707 0.607 0.800 0.618 1.034 0.781 0.633
1.083 1.102 1.121 1.138 1.155 1.171
1.886 2.323 2.747 3.159 3.557 3.944
0.608 0.612 0.621 0.634 0.649 0.665
0.001 0.002 0.005 0.010 0.020 0.050 0.100
0.965 0.952 0.927 0.903 0.872 0.822 0.772
0.999 0.999 0.999 1.000 1.001 1.003 1.007
0.001 0.002 0.005 0.010 0.020 0.050 0.100
0.966 0.953 0.928 0.904 0.874 0.824 0.776
(e) KBr (cation diam = 2.66 A, anion diam = 3.90 A, mol wt = 119.01) 0.200 0.722 1.015 0.198 0.728 0.900 0.622 1.071 0.871 0.643 2.500 0.593 1.188 0.300 0.693 1.024 0.297 0.701 1.000 0.617 1.079 0.964 0.640 3.000 0.595 1.221 0.400 0.673 1.032 0.394 0.683 1.200 0.608 1.094 1.149 0.635 3.500 0.600 1.253 0.500 0.657 1.040 0.491 0.669 1.400 0.602 1.109 1.331 0.633 4.000 0.608 1.284 0.600 0.646 1.048 0.587 0.661 1.600 0.598 1.124 1.511 0.633 4.500 0.616 1.313 0.700 0.636 1.056 0.682 0.653 1.800 0.595 1.139 1.688 0.634 5.000 0.626 1.342 0.800 0.629 1.064 0.777 0.648 2.000 0.593 1.153 1.863 0.637
2.288 2.699 3.096 3.479 3.849 4.206
0.648 0.661 0.678 0.699 0.720 0.744
= 4.32 A, mol wt = 133.85)
0.910 0.955 1.007 1.063
1.092 1.110 1.128 1.146
0.964 1.148 1.330 1.510
(f) KI (cation diam = 2.66 A, anion diam = 4.32A, mol wt = 166.02) 0.001 0.952 0.999 0.001 0.953
0.005 0.0IO 0.020 0.050 0.100 0.200
0.928 0.903 0.872 0.820 0.778 0.733
0.999 1.000 1.001 1.005 1.011 1.022
0.005 0.010 0.020 0.050 0.099 0.198
0.929 0.904 0.874 0.823 0.783 0.741
0.005 0.010 0.020 0.050 0.100 0.200 0.300 0.400
0.920 0.900 0.860 0.809 0.756 0.694 0.656 0.628
0.999 0.005 1.000 0.010 1.001 0.020 1.005 0.050 1.01 1 0.099 1.024 0.198 1.036 0.296 1.049 0.393
0.922 0.902 0.862 0.812 0.760 0.701 0.665 0.639
0.100 0.200 0.300 0.400 0.500 0.600
0.754 0.694 0.654 0.626 0.603 0.586
1.015 1.031 1.048 1.063 1.079 1.095
0.099 0.198 0.295 0.392 0.488 0.583
0.759 0.701 0.664 0.639 0.618 0.604
0.100 0.200 0.300 0.400 0.500
0.754 0.692 0.651 0.621 0.599
1.019 1.038 1.058 1.077 1.096
0.099 0.197 0.294 0.390 0.485
0.759 0.701 0.663 0.637 0.618
0.300 0.400 0.500 0.600 0.700 0.800
0.707 0.689 0.676 0.667 0.660 0.654
1.034 1.046 1.057 1.068 1.080 1.091
0.296 0.392 0.488 0.583 0.677 0.770
0.718 0.703 0.693 0.687 0.682 0.679
0.900 1.000 1.200 1.400 1.600 1.800
0.649 0.645 0.640 0.637 0.636 0.636
1.102 1.112 1.134 1.155 1.176 1.196
0.863 0.954 1.135 1.312 1.486 1.657
0.677 0.676 0.677 0.680 0.685 0.691
2.000 2.500 3.000 3.500 4.000 4.500
0.637 0.644 0.652 0.662 0.673 0.683
1.216 1.264 1.309 1.353 1.395 1.434
1.825 2.232 2.622 2.995 3.352 3.695
0.698 0.721 0.746 0.774 0.803 0.832
4.500 5.000 5.500 6.000 7.000 8.000 9.000
0.474 0.475 0.477 0.480 0.486 0.496 0.503
1.470 1.511 1.551 1.589 1.662 1.729 1.792
3.763 4.103 4.430 4.744 5.339 5.893 6.411
0.567 0.579 0.592 0.607 0.637 0.673 0.706
(h) CsBr (cation diam = 3.38 A, anion diam = 3.90 A, mol wt = 212.83) 0.700 0.571 1.110 0.676 0.591 1.400 0.510 1.214 1.310 0.545 0.800 0.558 1.126 0.769 0.580 1.600 0.500 1.243 1.483 0.539 0.900 0.547 1.141 0.862 0.571 1.800 0.493 1.271 1.654 0.537 1.000 0.538 1.156 0.953 0.565 2.000 0.486 1.298 1.821 0.534 1.200 0.523 1.185 1.133 0.554 2.500 0.474 1.364 2.225 0.532
3.000 3.500 4.000 4.500 5.000
0.465 0.460 0.457 0.455 0.453
1.427 1.487 1.544 1.599 1.651
2.612 2.982 3.336 3.675 3.999
0.534 0.540 0.548 0.557 0.566
(i) CsI (cation diam = 3.38 A, anion diam = 4.32 A, mol 0.600 0.581 1.115 0.579 0.602 1.000 0.533 1.188 0.700 0.567 1.133 0.671 0.591 1.200 0.516 1.223 0.800 0.554 1.152 0.763 0.581 1.400 0.501 1.258 0.900 0.543 1.170 0.853 0.573 1.600 0.489 1.291
1.800 2.000 2.500 3.000
0.479 0.470 0.450 0.434
1.324 1.356 1.434 1.507
1.624 1.785 2.173 2.541
0.531 0.527 0.518 0.512
(8) CsCl (cation diam = 3.38A, anion diam = 3.62A, mol wt = 168.37) 0.500 0.606 1.061 0.489 0.619 1.600 0.509 1.188 1.498 0.544 0.600 0.589 1.073 0.585 0.604 1.800 0.501 1.210 1.672 0.539 0.700 0.575 1.085 0.679 0.592 2.000 0.495 1.232 1.843 0.537 0.800 0.563 1.097 0.773 0.582 2.500 0.485 1.284 2.259 0.537 0.900 0.553 1.109 0.867 0.574 3.000 0.479 1.333 2.658 0.541 1.000 0.544 1.120 0.959 0.567 3.500 0.475 1.381 3.041 0.547 1.200 0.529 1.144 1.142 0.556 4.000 0.474 1.426 3.409 0.556 1.400 0.518 1.166 1.321 0.549
wt = 259.82) 0.943 0.565 1.119 0.553 1.291 0.543 1.459 0.536
‘ m in moles/kilogram of solvent, corresponding activity coefficient is y+; c in moles/liter of solution, corresponding activity coefficient is E+; d in kilogram/liter of solution; solvent diameter is 3.0 A.
Debye-Huckel limiting behavior. The simple equation (21) can be computed directly with an assumed diameter of the hard sphere
and other known constants. This is very suitable for the development of engineering correlations. Other theories, e.g., MSA,
8476 The Journal of Physical Chemistry, Vol. 94, No. 22, I990 +
' -1
A
42 5 A
t
100 r
.oo
r-
Chan
1.50 t 1
t
t
1
0.53
t
LiCl
A
~iBr
0
Lll
0.50 i
t
LiCI
A
KCI
0
CSCl
aJ 3 706 Y6P
000
r
-
i
SLW
390h SLBP
1
25A SL6D 4
-0.50 L
-050
. -M + H S
t
1
2.55A
-1.00 1
- 1.50
-150,,,,,,,,,.1, , , ' # 2 ~ 1 00 5 0 '00 ' 5 0 2 0 3 2 5 0 300 ,
.
,
,
,
sqrt(ct
,
-
'
,
I
'
0.000.50 1.00 1.502 00 2.50 300
sqrt(c) molaro5 Figure 6. Comparison of S M P and Debye-Huckel plus hard sphere.
molarc5
Figure 4 Activity coefficients of lithium salts.
__
_ _ ~ _ _ ~ _ _ - TABLE 11: Assignment of Best-Fitted Diameters for Individual Ions i
a = 0.5((r+ + u-)
os
u-,
c+, A C*
2 cs
Li+
K+ cs+
r>
%B-=
-2'
~
-z- 5 r
l i _ u
' C O 150 2 C C 2 5 C ~ CI'C
033 35C
s3
+
c
T
:3'
Figure 5. Comparison of contributions to activity coefficient.
require an iteration scheme to solve a set of high-order algebraic equations2s and is less convenient to apply. The excess (over ideal solution) chemical potential of an ion can be interpreted as composed of an attractive part and a repulsive part. In eq 21, the first part p,P/kT is due to the attractive charge-charge interaction expressed by (1 6), whereas the second part is due to hard-sphere repulsion. This second part is always positive as shown in (21) since I) < 1. The negative attractive part has the Debye limiting behavior at low concentration, whereas at high concentration it approaches a limit: A
Ki lim -(SL6P) m-+m k T
(
= --"I) 2DkTa 3 d2
and at high concentration, the Debye-Huckel theory with pendence approaches the limit
(25) u de-
piA e2 lim -(DH) = -kT 2DkTu The pF(SL6P) and piA(DH) are therefore very similar as they have the same limit at infinite dilution and approached similar
( 2 5 ) Sanchez-Castro, C.; Blum, L. J . Phys. Chem. 1989, 93, 7479.
3.90 1.90 1.1
CI3.62 8, 3.76 2.76 2.36
A 14.32 A 4.1 1 3.1 1 2.71
Br3.90 A 3.90 2.90 2.50
values at high concentration. For the several cases of ion diameters considered, the difference between (25) and (26) is very small, as shown in Figure 5. Also shown is the hard-sphere contribution, which is the dominating term in [* as u becomes large. It seems that SL6P is not a significant improvement over the Debye-Hiickel equation except that the hard-sphere fluid property is formally introduced to give the repulsive behavior. If the hard-sphere property is added to the Debye-Hiickel equation, it is a simpler alternative to S M P for correlation of activity data. Figure 6 shows that there is little difference for the two methods in fitting the data of chlorides. While these simple theories are successful in fitting the data, there is a severe defficiency. The best fitted diameters are not comparable to the Pauling diameter of the ions. For the halide salts, experimental In [+ decreases with the size of cations, in the opposite trend predicted by all the hard-sphere, S M P , and Debye-Hiickel contributions. The limiting behavior of Debye-Hiickel theory is exact in the region 0.002 M. One would expect the transition from the limiting behavior to real ions to be continuous. While the theories approach the Debye-Hiickel limiting law as u 0, the data of halides show the opposite direction. It is possible to assign a modified diameter to a particular ion so that the activity of an electrolyte pair can be calculated on the basis of the averaged diameter of the cation and anion. An example of such an assignment is shown in Table 11. The anion diameters were the same as their Pauling's diameters, whereas the cations ave diameters very different from Pauling's diameters. This assignment of ionic diameters is similar to the result of the best fitted diameters of a nonprimitive MSA model applied to nine electrolytes.'6 The authors have anion diameters equal to Pauling's diameters, and the best fitted cation diameters increase in the order sodium, lithium, and hydrogen. They attributed the deviation of cation diameters to hydration effect. Further comparison of theories and data must be made to investigate the abnormal values of best fitted cation diameters. A civilized model may give ad-
-
J . Phys. Chem. 1990, 94, 8471-8482
ditional insights to the solvation or other effects.
Conclusions Explicit expressions of the chemical potential and activity coefficient of the Stell-Lebowitz perturbation theory are reported. The one-parameter equation can fit experimental data well. The improvement over the Debye-Hiickel equation comes mainly from the hard-sphere properties. For some cations, the ionic-diameter
8477
dependence of experimental data is opposite from the theoretical prediction. Correlations developed upon this simple result are possible, whereas improvement in the RPM is necessary. Work is being carried out t o apply the recent perturbation iheory of Henderson et a1.6for mixtures and compare it with experimentaldata. Acknowledgment. I thank Dr. Douglas Henderson for reading the work and pointing out some major errors.
Dependence of the Standard Thermodynamic Properties of Isomer Groups of Benzenoid Polycyclic Aromatic Hydrocarbons on Carbon Number Robert A. Alberty* and Kuo-Chih Chou Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: March 28, 1990)
The benzenoid polycyclic aromatic hydrocarbons can be organized as an infinite number of homologous series with an increment of C4H2between the successive isomer groups in each series, rather than CH, for the usual homologous series. The standard thermodynamic properties in the range 298.15-3000 K have been calculated for the isomers in 20 isomer groups in the first six series. Each of the standard thermodynamic properties has been found to be linear in carbon number n. The parameters A(A,Go) and B(A,Go) for representing the dependence of the standard Gibbs energy of formation on carbon number, A&", = A(A,Go) + B(A,G")n, are of special interest because the equilibrium distribution within a homologous series at constant T, f,PHIdepends on A(A,Go) and the equilibrium distribution at constant T, P,PCIH, PH2depends on B(A@). It is especially interesting to find that the A parameters for A C , , Ago,,, AfH",, and A&",, are the same for all six series. This has made it possible to determine parameters for representing the temperature dependences of all of the thermodynamic properties of all six series by linear regression of the data on the six series simultaneously. Thus any property for any isomer groups in the first six series at any temperature in the range 298.15-3000 K can be calculated by using the 4 2 parameters involved and the I O parameters for representing the temperature dependences of the properties of graphite and molecular hydrogen.
Introduction The infinite number of isomer groups of benzenoid polycyclic aromatic hydrocarbons can be organized into an infinite number of homologous series in which the increments between successive isomer groups are C4H2, rather than the CH2 of ordinary homologous series.'v2 In making equilibrium calculations on benzenoid polycyclic aromatic hydrocarbons, it is advantageous to use isomer groups to reduce the number of species and estimate thermodynamic properties at higher carbon numbers, where data are l a ~ k i n g . ~The , ~ molecular formulas for isomer groups in the first six series are given in Table I with the ranges of carbon number n. Tables of standard thermodynamic properties of all of the isomers in the first several isomer groups in these six series in the range 298.1 5-3000 K have been calculated by using the Benson group additivity method5 with parameters of Stein and Fahr.68 The numbers of isomer groups studied are given in Table 1. The published tables of the standard properties of isomer groups show that all of the properties are linear in carbon number. This is illustrated by Figures 1-3, which show the isomer group properties A$",, AfHon,and A@", at IO00 K as a function of carbon number n. It is striking to find that the plots for these properties have a common intercept for the six series. The lines in the figures were obtained by linear regression by the plotting program. The plots for A C P nare similar but have relatively more (1) Dias, J. R. Handbook of Polycyclic Hydrocarbonr; Elsevier: New York. 1987: Part A. (2) Alberty, R. A.; Reif, A. J . Phys. Chem. Ref. Data 1988, 17, 241. (3) Alberty, R. A. J . Phys. Chem. 1983,87, 4999. (4) Alberty, R. A. J . Phys. Cliem. 1989, 93, 3299. ( 5 ) Benson, S.W. Thermochemical Kinetics; Wiley: New York, 1976. (6) Stein, S.E.; Fahr, A. J . Phys. Chem. 1985,89, 3714. (7) Alberty, R. A.; Chung, M. B.; Reif, A. J . Phys. Chem. Ref. Dora 1989, 18, 77. (8) Alberty, R. A.; Chung, M . B.:Reif, A. J . Phys. Chem. Ref. Data 1990, 19, 349.
0022-3654 f 90 f 2094-8411$02.50f 0
TABLE I: Formulas for Isomer Groups in Six Homologous Series of Benzenoid Polvcvclic Aromatic Hvdrocarbons no. of isomer
series benzene pyrene naphthopyrene coronene naphthccoronene ovalene
formulas CnHn/2t3 CnHn/2+2 CnHn/2tl C,H,/2 CnHn12-l CnHn/2-2
values of n 6, 10, ...
groups studied
16, 20, ... 22, 26, ... 24, 28,
...
30, 34, ... 32, 36, ...
6 4 3 3 2 2
noise because of the small magnitude of AfC",. When there is a common intercept for certain properties for the six series, there will be different intercepts for other properties, as will be shown here. In order to discuss the linear dependences on carbon number within a given homologous series, eqs 1-9 are used. The bars AfCop, = A(AfC",) A g o , = A(ApSO)
+ nB(A&Op)
+ nB(A,,S") AfHon= A(AfHo) + n B ( A f H o ) A@", = A(A,Go) + nB(A@") ~ o f n= A(Cof) + n E ( C p ) So, = A(So) + nB(So) H", - Ron.298= A(Ho - R"298)+ nB(Ho - H o 2 9 8 ) - H" = A(Ro - H",,)+ nB(R" - H",,) Go, - Ron,,,= A(Go - R",,) + nB(G" - R",,) PO,
n,scr
(1) (2) (3) (4)
(5) (6) (7) (8) (9)
indicate molar properties, but no bar is used on the formation properties because the subscript f indicates that 1 mol of the isomer group is formed. The parameters A and B are functions of temperature only for a given homologous series. The labeling of 0 1990 American Chemical Society
