Corn r n unicat ions to the! Editor
1241
(2) IR. C. Marshall and D. W. James, J. lnorg. Nucl. Chem., 32, 2543 (1970). (3) El. N. Figgis, Trans. faraday SOC., 56, 1553 (1960); E. Konig and A. S.Chakrevarty, Theor. Chim. Acta, 9, 151 (1967); M. Gerloch, J. Chem. SOC. A, 2021 (1968); B. N. Figgis, M. Gerloch, and R. Mason, Proc:. Roy. SOC.,Ser. A, 279, 210 (1964). (4) D. J. Robinison and C. H. L. Kennard, Cryst. Struct. Commun., 1, 185 (1972)' K G . Shields and C. H. L. Kennard, ibid., 1, 189 (1972). (5) C . E. Schtiffnr, Structure Bonding, 5 , 68 (1968). (6) ID. W. Srnith,J. Chem. SOC.A, 1708, 2529 (1969): 176 (1970). (7) E. U. Condon and G. H. Shortley, "Theory of Atomic Spectra," Cambridge University Press, New York, N. Y., 1951. (8)
I::,
A . Coulscin, "Valence," 2nd ed, Oxford University Press, London,
1961. (9) M , Wolfsberg and L. Helmholtz, J. Chem. Phys., 20, 837 (1952). (10) 6.K . Jorgmsen, "Orbitals in Atoms and Molecules," Academic IPress, London, 196Z1. i l l ) %J. H. Van Vieck, "Theory of Electric and Magnetic Susceptibilities," Oxford University Press, London, 1932. II, 7, 155 (1964). (12) h ~ J.. O.P O ~ ~ ComputerJ., (13) A. Bose, bl. 5. Ghosh, R. N. Bagchi, and A. K. Pal, Ind. J. Phys., 38,367 (1984).
(14) R. Chidambaram, A. Sequeira, and S . K. Sikha. J. Chem. Phys., 41, 3616 (1964). (15) F. 0. Ellison and H. Shull. J. Chem. Phys., 23, 2348 (1955). (16) D. W. James and R. C. Marshall, to be submitted for publication. (17) A. K. Pal, R . N . Bagchi, P. R. Saha, and R. K. Shaha, lnd. J. Phys., 41, 856 (1967). (18) For the situation when ( z 2 ) ( x 2 '- y2) is the first transition, the -f
(19) (20) (21) (22) (23)
difference between the u and x perturbation energies is almost inconsequential with regards to the magnetic properties because the maximum difference of 900 c m - ' would not significantly alter the calculated magnetic susceptibilities and g values. D. W. Smith, J. Chem. SOC.A, 1496 (1970). E. Clementi, iBM Technical Report, No. R.J.-256 (1963). J. W. Richardson, W. C. Niewpoort, R. R. Powell, and W. F. Edgell, J. Chem. Phys., 36 1057 (1962); J. W. Richardson, R. R. Powell, and W. C.Niewpoort, ibid., 38, 796 (1963). M. A. Hitchman, J. Chem. SOC.A, 4 (1970). D. E. Billing, B. J. Hathaway, and P. Nichoils, J. Chem. SOC. A ,
316 (1969). (24) C. K. Jorgensen, "Inorganic Complexes," Academic Press, London. 1963. (25) D. E. Billing and B. J. Hathaway, J. Chem. Sac. A , 1516 (1968) (26) M. Gerioch and J. R. Miller, Progr. /norg. Cham., 10, 1 (1968).
COMMUNICATIONS TO THE EDITOR
Notes on the Intermolecular Energy of Fluids at the Critical Temperature and Its Dependence on the Tempera tu re Publication costs &misted by the American Petroleum Institute
Sir: Tt is well known that the second virial coefficient derived from the hard-sphere equation of state with a van der Waals attraction term is
4b(T) - a(T)/(RT) (1) where b(T) = 1/6~N,3ue3, NO is the Avogadro number, and =:
is the effective collision diameter that varies with the temperature. n(T) and u are related to the critical temperature P and the corrected liquid molar volume V* a t T / F = 0.6 as f o l l ~ w s ~ - ~
u,
1 ~ R N & = 0.434V*
a' = 4.65RTCV*
(2)
where u is thtt value of cr, a t T = 0 and @ is the value of a(T) a t T = '7%. halogous relations are obtained by using the critical volume V instead Qf V*. Thus, the values of p/V* of different substances at the critical temperature, Pc/V*, should be identical. This is, however, not true. Recently, LdaX and Spencer3 obtained the values of u, and the pair potenbials u(r) for n-alkanes (C, to Cg) by the Monte-Carlo simulation of chain molecules. The values of (ue)3 are surprisingly small and they vary approximately lrnearlg with (V*)2/3, not with V*. The line passes through the origin and this result implies that b ( V M c (V*)2/3and
-
pc/(p)2/3
=
C]
-
C,(V*)'/3
(3)
where the constants evaluated by the method of least squares for 12 substances are c1 = 3.41 and c2 = 4.121. The values of pc were taken from the recently published tables.4 The results shown in Figure 1 confirm the validity of eq 3. This result also implies that the product F V * in eq 2 is in fact F(V*)1/3(V*)2/3,where F(V*)1/3 i s proportional to the minimum value of u(r). It has been shown elsewhere5 for several substances that Tc(V*)3/3/T0c(V O * ) ~ / ~ = i i / ~ o o ,where a / k are the minimum values of u(r) obtained for the Kihara potential and subscript 00 denotes a reference substance (argon, tLoo/k = - 166 K). Equation 3 is valid a t T = Tc only, According to Rowlinson's theory of noncentral forces,6 in general a = &(l-iv / ( k T ) ) and v / k can be estimated from Pitzer's acentric factor7w q / k = 0.833wT"
(4)
Accordingly
a(T) = 4.65RT0'(1
+ &)V"
(5)
where ToC= P ( l + S / ( ~ P )and ) - eq ~ 5 simplifies to eq 2 for T = P . Equations 4 and 5 were recently applied to the evaluation of the excess functions of mixtures2 with satisfactory results. Below, the values of 1)/ k obtained from observed energies of vaporization of liquids U* at T / P = 0.6 are compared with v / k calculated from eq 4. u*is equal to the residual energy (u-- Uid) where u,d is that of a perfect gas. The part due to attraction is obtained from the relations - Uh = U" - ( U , - U1d) (6) where (U, - u d ) is the residual energy of hard spheres
u
The Journal ot Physical Chemisfry, Vol. 78, No. 12, 1974
Communications to the Editor
1242
These results show that the hard-sphere equation of state with a van der Waals attraction term is suitable a t high densities of a fluid. It is unclear why 4(4) "degenerates" to the form of eq 3 a t very low densities.
6 n.Alkanes @ Neopentdne
References and Notes
3
4
(1) A. Kreglewski, R. C. Wilhoit, and 8.J. Zwolinski, J . Chem. Eng. Data, 18, 432 (1973); J. Phys. Chem., 7 7 , 2212 (1973). (2) A. Kregiewski and R. C.Wilhoit, prepared for publication. (3) M. La1 and D. Spencer, J. Chem. SOC., Faraday Trans. 2, 1502 (1973), (4) 8. J. Zwolinski, et a/., "Selected Values of Properties of Hydrocarbons and Related Compounds," API Research Project 44, TRC Data Project, Texas A & M University, College Station, Tex. (5) A. Kreglewski, "Mixtures of Fluids," AP144-TRC Publications, Thermodynamics Research Center, Texas A & M University, Coilege Station, Tex. 1973. (6) J. S. Rowlinson, "Liquids and Liquid Mixtures," Butterworlhs, London, 1959. (7) The values of acentric factors are given by R, C. Reid and T. K. Sherwood, "The Properties of Gases and Liquids," McGraw-Hill, New York, N. Y . , 1966. (8) G. A. Pope, P. S. Chappelear, and R. Kobayashi, J. Chem. Phys., 59, 423 (1973).
5
Thermodynamics Research Center Texas A & M University College Station, Texas 77843
Figure 1.
TABLE I: Some of the Intermolecular Energy ~
_
I
I
_
_
_
_
_
_
Aleksander Kreglewskl
Received February 21, 1974
_
d k
(w4)
(Uh-uld)/R (es 7 Y
q/k
(eq 8 )
_ _ l l _ . l l _ . _
Argon Krypton Xenon Methane Ethane Propane n-Butane n-Pentane n-Hexane Neopentane Tetrafluoromet ham Rerfluoro-n-hexane Methanol Ethanol a
-0.3 -0.3 4-0.5
2.1
6.9 3.8 4.2 7.1 26 50 71
27 47 71 99 123 70 36
102
183
281
237 273
263 388
45 63 81 55 79 81 81 77 73 70 58 46 152 129
136 73 38
For the liquid a t T/!P = 0.6.
(for references see ref 1)
(e', -
U J I R ' S ) = --2T(d(/T),[(l
+
- f)-2
(1 -
-(U
Equivalent Conductances of Univalent Counterions and Coions in Polyelectrolyte Solutions
'PI
(7)
-
U,)/(RT) = 4.65V*T,"(1 4- 2 v / ( K T ) ) / ( T V ) (8) where I; = b(T)/'Vand (a[/aT),were evaluated as bef0re.l V was set equal to the observed liquid molar volume at T / F 0.6 that for small molecules is slightly larger than 3
V*. The values of q / k obtained from eq 8 are given in Table I. ( U h - a / ; d ) is, positive and increases the absolute value of ( U - Uh)so that the values of s / k of the rare gases are positive. They are certainly better than the values of a / k otrtained from vapor pressures and P ( i e . , from w ) where the hard-sphere correction cannot be evaluated. It is known thiat the Kihara potential fails to reproduce well the values of p for fluorocarbons and other gases with specific interactions. 'The above results suggest the following relation lor a / k suitable in application of the Kihara potential
+
ii/h = -(P3635'T'(V*)1/3(1 C 7~/(hT)X1 7~/(kT'))-' (9) fitting ii/k = -166 I< for argon8 ((V*)1'3 = 3.028 cm molI
The Journal of Physical Chemistry, Vol. 78, No. 72, 1974
Sir: Nagasawa, Noda, Takahashi, and Shimamotol have recently published detailed data on the equivalent conductances of Na+ and C1- in solutions of poly(acry1ic acid) neutralized to various degrees by NaOH. Their results, which contain several interesting features, including a transition with decreasing NaCl concentration from positive to negative equivalent sodium conductances, have motivated us to extend a previous c a l ~ u l a t i o n ~of- ~countenon transport in a solution of polyelectrolyte salt to a solution which contains an additional component of simple salt. We will restrict attention to univalent counterions and coions and reserve for another context the extension to multivalent species. Extension of the theory to the case of added salt i s straightforward, since this problem has already been solved for the tracer diffusion coefficients of counterion and coion.2 We recall the starting point of the diffusion theory j , = ('kT/{,o)[VnFl
+
nFiZ,V($F
- n F ' F , / h T ) ] (1)
In this equation j , is the local flux of counter- or coions of species i with valence z c ,f Z 0 is the friction constant of such ions at infinite dilution in the pure solvent, F , is an externally imposed uniform force on each ion i, n ~ is ' the local concentration of i as perturbed by this external force, and $F = e&/kT, where e is the protonic charge and &F is the locally inhomogeneous electrostatic potential set up by the polyions and perturbed by the influence of the external force on the ionic atmospheres. In the diffusion theory the introduction of F,is merely a device to allow later ap-
