NOTES
3697
Although the principles of an earlier calculation6 leading to AG" of solution of Brz(l) in HzO are correct and the arithmetic of that calculation has been verified, careful consideration of the uncertainties associated with the solubility data and the error from neglect of solubility of HzOin Brp indicates that the earlier AGO = 1.38 kcal/mol is far less reliable than the values cited above. Combination of our AGO = 0.96 kcal/mol with AH" = -0.20 kcal/mol6 for solution of Brz(l) in HzO gives A#' = -3.89 gibbs/mol. Further combination of this AX" with the entropy of Brz(l)8gives so = 32.5 gibbs/mol for the standard partial molal entropy of Brz(aq). The entropy cited earlier6 is in error because of use of an incorrect AG" of solution, while the entropy cited in ref 3 is in error because of an incorrect AH".
Acknowledgments. We are grateful to the National Science Foundation for support of this research, which was suggested by Pitzer and Brewer.* We also thank W. H. Evans for his helpful comments. (8) Pitzer and Brewer, "Thermodynamics," revised by K. S.Pitzer and L. Brewer, McGrsw-Hill Book Co., Inc., New York, N. Y., 1961, problem 20-4.
where a(iu) and ~ ( i u )are, respectively, the polarizability of the adsorbed molecule and the dielectric constant of the continuum solid at an imaginary frequency iu. Equations 3 and 4 have been numerically integrated by Johnson and Klein4 to obtain estimates of 11 and 1 3 . These authors used an approximation for a(iu) which reproduces atom-atom dispersion energies, together with experimental data for the dielectric constant of graphite. The theoretical interaction energy els(z) = -11/z3 gives, for particular atom-surface potential models, values of so, the atom-surface collision parameter,6 and since experimental values exist for the product A , Y ~it, ~ is possible, if the surface area, A , is known, to compare 11 with experiment. Comparison of I3 with experiment is possible via the estimation of an experimental value for 133. In this note these procedures are applied to the inert gas-graphite (P33) system.
Second-Order Interactions If so is to be determined from experimental values of Aso, then A must be evaluated by a method independent of so. Such a method6 gives A = 10.7 m2 g-' (correction given in ref 3) for a monolayer adsorption model or A = 8.32 m2 g-l,' when an allowance for three dimensionality is madea8r9 Experimental values of so determined from Asovalues6 and theoretical values of so are given in Table I. The latter are calculated by equating eq 1 to the long-range limit of the atom-surface 3-p potential
Applications of McLachlan's Theory to Physical Adsorption
by J. D. Johnson' School of Chemical Sciences, University of East Anglia, Norwich, England
giving
Accepted and Transmitted by The Faradau Society (September 3, 1967)
According to McLachlan,2the van der Waals energy between an atom and a continuum surface (distance x from the surface) is given by
while the perturbation to E(r), the bulk-gas interaction energy, can be written for a monolayer as3
where p = r / u , Pm = Z m / g , ~ ( r=) 0, and Em is the distance of the monolayer from the surface. Here I1
)
= I"Jma(iu)( e(iu) - 1 du 2* 0 €(iU) 1
I3 =
+
'> +
I"Jm az(iu)(e(iu) 2n
0
€(iU)
1
du
(3)
(4)
where els* is the atom-surface interaction energy.6 The theoretical values are much smaller (>20% smaller) than experimental values suggest. If the atomsurface potential models considered' are realistic, then the theoretical I1 estimates are too small. (1) This work was supported by the Science Research Council. (2) A. D. McLachlan, Mol. Phys., 7, 381 (1964). (3) J. D. Johnson and M. L. Klein, Trans. Faraday Soc., 63, 1269 (1967). (4) J. D. Johnson and M. L. Klein, unpublished calculations. (5) J. R. Sam,, G. Constabaris, and G. D. Halsey, J. Phys. Chem., 64, 1689 (1960). (6) J. D. Johnson and M.L. Klein, Trans. Faraday SOC.,60, 1964 (1964). (7) D. H. Everett, Discussions Faraday SOC.,40, 177 (1965). (8) A monolayer model gives exactly A / B * = - 2kl21[ k2 - k l B / R T ] . A three-dimensional model gives A / ( B * - a) = -2k12/[kz 3klB/ R T ] (see ref 9 for details). (9) J. A. Barker and D. H. Everett, Trans. Faraday Soc., 58, 1608 (1962).
+
Volume 72,Number 10 October 1968
3698
COMNUNICATIONB TO THE EDITOR
Table I : so Values
(d)for the Inert Gas-P33
Atomsurface
potential model
siB*/R,
103Aso,a
0Ka
om~g-i
System
---Exptl, SO*---A = 1 0 . 7 A = 8 32 m%g-i
Theor
mag-i
SO
Neonb (11/2 = 0.57 X 10-36 erg cm3')
3-9 3-12 3-15 3-18
380b 380b 380*yd 380"d
2.199 2.541
.., ...
2.06 2.37
... .,.
2.64 3.05
... ...
1.61 1.73 1.80 1.85
Argone ( 4 / 2 = 1.95 X 10-36 erg cma ')
3-9 3-12 3-15 3-18
lllOe 2.393 lllOe 2.762 1110d~" . . , 1110d*" ...
2.24 2.58
... ...
2.88 3.32
... ...
1.70 1.82 1.90 1.95
Table 11: Values of 13 for Various zm Distances (Ar-P33)
Krypton (I1/2 = 2.72 X 10-36 erg cm3 ')
3-9 3-12 3-15 3-18
14606 2.406 14608 2.797 .,. 1460dt0 .. 146Odne
2.25 2.61
2.89 3.36
...
...
...
...
1.73 1.85 1.93 1.99
a
1920e 1920e 192Odpe 1920d,e
2.625 3.042
2.45 2.84
...
...
.,.
...
3.16 3.66
...
...
zm,
2.15
1.80 1.92 2.00 2.06
Reference 5 . D a t a are to two significant figures. ReferEstimated. ' Data are to three significant figures.
ence 4.
Third-Order Interactions An experimental estimate of I 3 can be obtained, via a theoretically determined two-dimensional second virial coefficient for the a,dsorbed gas, by analyzing gas-solid adsorption data.3 Such an estimate depends on the
2.73
0.63 5
Pm
106OI~,erg cmo A , m2 g-l
Xenon (11/2 = 3.99 X 10-38 erg cm3 ')
3-9 3-12 3-15 3-1 8
value of xm, the minimum of the atom-surface potential,1° but a reasonable value may be chosen which gives A close to the independent value 10.7 m2 g-' (since a monolayer model is used in the determination of 13; see ref 3). For instance, for Ar-grapFte (P33) l3 (exptl) II 15 X erg cmB(zm = 3.41 A) gives A = 10.5 m2 g-I (see Table 11, which is taken from ref 3 and is extended). The theoretical value for 13 is 5 X 10-60 erg cm6 (ref 4).12 If the adsorption model used to estimate 13(exptl) is realistic, then 13(theor) is again too small for the Ar-graphite system (Ar-graphite third-interaction data are the only published data of any reliability availablel).
-18
0.80 9 -15
A 3.07
3.41
0.90 12 -13
1.00 15 10.5
The application of McLachlan's theory to the inert gas-graphite system gives results which appear to be too low. This may indicate that the continuum model is not a realistic one for graphite surfaces. (10) They may also depend on the bulk-gas potential parameters eo and u. When 0 < p 5 2 p m . l a N 1 6 f ~ u ' p mfor ~ ~ a perturbation A+) = 4 e o q / p . 3 Since q may vary with ea and u by 10% for ArP33,11thenIs'also may be expected to vary by about the same amount. This would not affect the conclusions made here. (11) R. Wolfe and J. R. Sams, J . Chem. Phys., 44, 2181 (1966). (12) This improves a previous estimate of 8 X 10-80 erg cm6.3
C O M M U N I C A T I O N S TO T H E E D I T O R
Comment on "Current Dependence of Water Transport in Cation-Exchange Membranes"
Sir: In a recent paper under the above title, Lakshminar ay an aiah and Subrahmany an1 have presented results of measurements of electroosmotic water transport across cation-exchange membranes in contact with 0.01 N solutions of alkali metal chlorides as a function of current density, in the range 0.32-15.75 mA/cm2. A listing of previous studies of water transport across ion-exchange membranes suggesting no dependency of water transport on current density is also given in this The Journal of Physical Chemistry
report.' However, the above authors appear to have overlooked the fact that similar evidence for a marked dependency of electroosmotic water transport on current density across a polystyrensulfonic acid cationexchange membrane in contact with various solutions of sodium chloride (0.1, 1.0, and 4.0 M ) has been reported previously by Tombalakian, et U Z . , ~ over a much wider range of current density, 1.076-107.6 mA/cm2. (1) N. Lakshminarayanaiah and V. Subrahmanyan, J. Phys. Chenz.,
72, 1253 (1968). (2) A. S. Tombalakian, H. J. Barton, and W. F. Graydon, ibid., 66, 1006 (1962).
