Communications to the Editor
1529
described here represents the first demonstrated example of a two-step process and resorts to a reaction involving many water molecules of hydration as given in (3).
Above, n = nl = n2 is concentration, e = el = -e2 is charge, p = e2/DkT,6 is the angle between r21and the field direction
References and Notes
t
(1) See, for example, a review article by R. E. Chao, Ind. Eng. Chem., Prod. Res. Develop., 13, 94 (1974). (2) P. H. Kasai and R. J. Bishop, Jr., J . Phys. Chem., 77, 2308 (1973). (3) J. A. Rabo, C. L. Angell, P. H. Kasai, and V. Schomker, Chem. Eng. frog., Symp. Ser., 83, 31 (1967). (4) 0. H. Olson, J . Phys. Chem., 72, 1400 (1968).
Union Carbide Corporation Tarrytown Technical Center Tarrytown, New York 1059 1
Paul H. Kasal' Roland J. Blshop, Jr.
is insufficient for the complete evaluation of Ao, a coefficient which appears in the integration' of the equation of continuity. The integration is reviewed below; it will be shown that A. = -1 is the limiting value of A. at zero concentration. Consequently, the 1975 derivation in effect contains an unstated approximation: terms of order x2R2 and higher in A. were omitted. These terms in A. would lead to additional terms of order c and higher in the conductance function A(c). The equation of continuity is
divl(fijvjj)+ div2(fjivji)= 0 ( 21 where vij is the velocity of a j ion in an element of volume dV2, given the presence of an i ion in dV1 which is at a distance rZ1from dV2. The distribution functions are defined by
(3)
where
(4) nij = nj e x p ( - e j $ J k T ) gives the local concentration of j ions at dV2and #iis the potential at dV2 due to the i ion and its atmosphere, and congruently for nj,. These functions give the probability of simultaneously finding a j ion in dV2and an i ion in dVl. In the presence of an external field X I1
Jl
+ ftji
(5)
and
Gj
=
$Oj
8nnye2/DkT, n = Nc/lOOO (9) The first term on the right of (7) is the origin of the leading term of order c1I2 in the conductance function, the second term is from the Boltzmann factor approximated by the truncated series x 2
=
r ) C O S 0 + G21 where Qo is defined as the solution of f'21
Sir: The writer is grateful to Professor C. Alden Mead (University of Minnesota) for an analysis which shows that the boundary condition lim ( f f j i / n ' f ) m, c + 0 (1)
f . .= f " .
and
+
Boundary Conditions for Integration of the Equation of Contlnuity
= ninij = 12.11.. I 11 = f j i
(8)
ec = 1 + t2/2 (10) and the last term is a complicated function of the indicated quantities. Equation 7 is a second-order inhomogeneous equation, impossible to solve explicitly; it was therefore treated by the methods of successive approximations. First, f'21 was separated into two parts
Received March 20, 1977
fjj
= e,.$ Oj/kT = e j $ Oi/kT
+ gvj
where the zero-superscripted symbols refer to the spherically symmetric functions in the absence of an external field and the primed symbols represent the perturbations produced by the field. Substitution of explicit values for the velocities and rearrangement converts (2) into eq 2.12 of the 1975 paper: (A - x ' / 2 ) f ' z l= - nZXepexp(- x r ) ( l + x r ) cos 0/r2pkT + x ' t ( 1 + t / 2 ) f f 2 1 / 2+ ~ ( 9 r,x + ' j , f ' 2 1 ) (7)
= Qo( x
(11)
7
a'Qo/ar' + ( 2 / r ) ( a Q o / a r ) - ( 2 / r 2+ x 2 / 2 ) Q o= -n'Cg( In (12) g( x, r ) = exp(-xr)(l + x r)/r2
x ,
r)
(12) (13)
and C = XepIpkT Substitution of (11)in (7) gives (A - x ' / 2 ) ( Q 0cos 0 + G z l )= -n2Cg( x , r ) cos 0 + x 'E(1 + t / 2 ) f r , 1 / 2+ @ ( X , I^,f ' 2 1 , ' j ) I ( 1 5 ) Multiplying (12) by cos '6 and subtracting the result from (15) gives the partial differential equation
+
(a -
2/2)Gzi= [ ( 1 5 ) 1 (16) where the symbol [ (1511 denotes the quantity in brackets in (15). It was shown that Qo is the source of the limiting relaxation term of order c1i2 in the conductance function A(c). Therefore Gzl is the source of terms of order c and higher in A(c). By using Qo cos 6 as the first approximation for f'21 in [(15)] (and the corresponding part Po cos 6 as first approximation to $',) in [(15)], we obtain (A - x ' / 2 ) G Z = 1 ( x 2 t / 2 ) ( 1+ $ / 2 ) Q oC O S 0 + @(~,r,Qo,Po,cos0) (17) where explicit (rather than unknown) functions now appear in the inhomogeneous term on the right. Solution of (17) gives the first approximation to Gzl; when these terms are added to Qo cos 6, we have the second approximation to f'21, as described, Here we are concerned with (121, the partial differential equation which determines Q O ( K , r). The solution is Qo( x , r ) = - (2n2Xep/pkT x 'r')[exp(- x r ) ( l + xr) + A,exp(-qxr)(l + q ~ r ) + Bo e x p ( 4 x r ) ( l - 4 % r)l (18) Since Qo must vanish for r = a,Bo = 0. Since Qo is part off',$, condition (1)requires that lim { [ e x p ( - x r ) ( l + x r ) + A . e x p ( - q x r ) ( l + q x r ) ] / x z r 2 } # c -+ 0 (19) x
00,
The Journal of Physical Chemistty, Vol. 81, No. 15, 1977
Communicationsto the Editor
1530
Expand the exponentials in (19) and let
A o = a.
+
~
1
R%
+ a, x z R z +
(20)
The result is
lim [(l+ ao)/ x + a , R / x + (a2 - a o / 4- 1/2)RZ + (a0q/24t u1/4+ u3 + 1/6)x R 3 + ...I # (211 To keep the quantity in brackets finite at K = 0, a. must equal -1 and al must be 0. However the coefficients a2, a3, etc. remain unknown; condition (1) shows only that Ao= -1 + x + a3 x + . . . (22) For the limit of zero concentration, A. = -1; this value was used in the 1975 derivation, by substitution in the right-hand side of (15). Since Gzl leads to terms of order c and higher in A(c), retention of the terms of order x2 in (20) would generate terms of order c2 and higher in A(c). As corrections to corrections, this effect probably is negligible, but the presence of u2x2in Qo will produce additional terms of order c in A(c). The consequences of the approximation A. = -1 will now be considered. It has been shown3that conductance data at low concentrations can be reproduced by the semiempirical equation A
= A0 -
SC”~
+ EC log c + J ~ +c
(23) where S and E are theoretically predictable coefficients which depend only on DT, 7,and valence type, and 51 and J2 are the intercept and slope of a plot of (24) y (A - A 0 + SC’” - EC log C ) / C J2c3l2
against c1i2. Equation 23 is a three-parameter equation: A = A(c; Ao, J1,J2) where J1 and J2 can be evaluated directly from the data. The 1975 conductance function is also a three-parameter equation: A = A(c; Ao, K, R), where K is the pairing constant and R is the diameter of the Gurney cospheres surrounding unpaired ions. In principle, analyzing a set of conductance data by use of the 1975 equation is equivalent to solving the equations
R ) = J1 f,WY R ) = J z (25) for K and R. Let K’ and R’ be the values obtained for a given set of data using the 1975 equation with A0 = -1. Now if a new equation were derived in which the higher
f l ( K
The Journal of Physical Chemistry, Vol. 81, No. 15, 1977
terms of A(xR) were explicit, the coefficients in f l and f 2 would be changed, because terms of order x2 and higher in A(xR) would lead to additional terms of order c and higher in A(c). Consequently the values K”and R”obtained using the new equation would be different from K‘ and R’, but the limiting conductances will be the same because S and E are unchanged. Furthermore, for several sets of data (for example, a given salt in a series of mixtures of two solvents), one might expect to find the same sequence of values; Le., if K‘, > K‘, > K $ , etc. then K‘I1 > K“, > K’$, etc. Finally, a new boundary condition is proposed to replace (1). The distribution function f i l = n2n21is proportional to the concentration of ions of species 1 at a distance r from a reference ion of species 2. By definition of the 1975 model, unpaired ions are those whose cospheres contain only the reference ion and therefore nZ1is zero for distances a I r I R from unpaired ions of species 2. In terms of f i l , the corresponding boundary condition is
(26)
fZl(R) = 0 which leads to
Qo(R) = 0 (27) as the condition which Qo must satisfy. Substituting r = R in (18) and using (27) gives A o ( t ) =-exp(-t)(l + t)/exp(-qt)(l t q t ) (28) where t = xR. [Note that Ao(0)= -1.1 The corresponding term in the relaxation field is obtained by integration [ref 1, eq 2.451; the result is A X o / X = - O x /6(1 + q ) ( I + t)(I + q t ) (29 1 where /3 = e2/DkT. Work on the evaluation of the higher terms of the relaxation field and of the hydrodynamic terms in A(c) using (26) as the fourth boundary condition is in progress.
References and Notes (1) R. M. Fuoss, J . Phys. Chem., 79, 525 (1975). (2) R. M. Fuoss, J. Phys. Chem., 80, 2091 (1978). (3) R. M. Fuoss, Proc. Natl. Aced. Sci. U . S . A . , 71, 4491 (1974).
Sterling Chemistry Laboratory Yale University New Haven, Connecticut 06520 Received May 9, 1977
Raymond M. Fuoss
