COMMUNICATIONS TO THE EDITOR
4453
where Z = distance of the plane from an arbitrary reference plane, H , = plate height for particle diffusion control, a = K A Y / K B Yx, = mole fraction of B, and C1 = constant of integration.
1 1
By requiring
Zdx = 0, Hagiwara calculated C1 = 1.
With the slightly different, but more useful experimentally, boundary condition that Z = 0 at x = 0.5, we obtain C1 = In 2 and
H, =
- 1)(Z) In [(l - x)~/x] (In 2)(a - 1) (ff
+
(2)
Z is negative when x > 0.5. Similarly, Hagiwara’s plate height for film diffusion control can be written as
Hf =
(ff - 1)(Z) In [(l - x)/xa] - (In 2)(a - 1)
(3)
The equation derived by Powell and Spedding from countercurrent extraction theory is Ri (21 - 2,) In a In- = h R2
(4)
where R1 = ratio of A to B at a plane with coordinate 21, R2 = ratio of A to B at a plane with coordinate Z2, and h = theoretical plate-height from countercurrent extraction theory. Replacing R with (1 - x)/x and using the same boundary condition as before h=
Table I --a
= 1.10Hrlh
0
Hp/h
0.01 0.10 0.30 0.49 0.51 0.70 0.90 0.99
1.034 1.022 1.009 1.000 0.999 0.990 0.978 0.967
0.967 0.978 0.990 0.999 1.000 1.009 1.022 1.034
---a
Hp/h
= 1.50-Hi/h
1.148 0.865 1.088 0.903 1.029 .0,948 0.988 0.985 0.985 0.988 0.948 1.029 0.903 1.088 0.865 1.148
--a
= 2.0-
Hp/h
Hf/h
1.256 1.138 1.033 0.965 0,959 0,900 ,0.833 0.779
0.779 0.833 0.900 0.959 0.965 1.033 1.138 1.256
values of H,/h and Hr/h as a function of x have been calculated for several values of a as shown below in Table I. The h in Powell and Spedding’s work is usually determined from the slope of a log R vs. Z plot. The values of h can be either higher (data weighted toward values of 2 > 0.5) or lower (x < 0.5) than H , values obtained by curve fitting. In any case, plate heights calculated from countercurrent extraction and rate theories must be in close agreement because the mathematical forms are similar, not necessarily because “the elution system had arrived at a state of dynamic equilibr um .’’ SAVANNAH RIVERLABORATORY E. I. DU PONT DE NEMOURS AND Co. AIKEN,SOUTHCAROLINA29801
WILLIAM H. HALE,JR.
RECEIVED APRIL20, 1970
Z In a
(5)
In [(I - 5)hI
Dividing eq 2 and 3 by eq 5 gives
Semiideal Behavior of Solutions of
In [(I - x)/xI
h In [(I -
%)/XI
h
(6)
Solvated Nonelectrolytes
(7)
Sir: Garrod and Herrington‘ have claimed that the water activities and activity coefficients of sucrose solutions are inconsistent with eq 40 of their paper so that sucrose solutions cannot be described as “semiideal dilute.” Since we have demonstrated2 that an essentially similar semiideal model unrestricted to dilute solutions does adequately describe the behavior of sucrose solutions, it is necessary to discuss their argument. They find for the case where a solute s interacts with water to form various hydrates, denoting the actual mole fractions of each species at equilibrium by qw, qs, qsw, etc. (Nw, N O ,N1, etc., in the notation of our previous paper2),that q s qw < 1, which is indeed obvious from the definition of mole fractions, as long as at least one other species such as sw is present at all. In terms of their “semi-ideal dilute” solution model they
Because (a - 1) is the first term in the power series expansion of In a, H,/h and H f / h approach 1 as a approaches 1 If points 1 and 2 are any two points in the A-B overlap region, more general equations for H J h and H f / h can be written as follows
HP -
a-1
h
In a
+
a-1
(9)
In a With the boundary condition, x2
=
0.5 at Z2
=
0,
(1) J. E. Garrod and T. M. Herrington, J . Phys. Chem., 74, 363 (1970). (2) R. H. Stokes and R. A. Robinson, ibid., 70,2126 (1966).
The Journal of Physical Chemistry, Vol. 74, No. 86,1970
COMMUNICATIONS TO THE EDITOR
4454 then correctly put in their eq 37 qw = a,, where a, is the conventional water activity, and incorrectly, q s = yx,,.XS,where X , js the stoichiometric molefraction of solute, and yX,,is “the activity coefficient of the solute on the Henry’s law mole fraction scale”. It is apparent from their eq 39 that their yx,ais to be identified with the conventional mole fraction scale activity ~oefficient.~ Irrespective of what activity scales are used, the total Gibbs energy of a fixed quantity of a given solution is invariant. For simplicity let us consider the case of a solute s in equilibrium with a single hydrate sw
chemical potentials in the standard states for B and s. We evaluate this constant in the usual way by finding the limit of (qa/m)as all solute concentrations tend to 1. In this limit, qs (0.001 MI).^, zero and y and hence -+
A
= lim (qs/m) = lim (O.OOIM~)(ms/m)=
(0.001Ml)/(l
a,
A
G
= (0.001M1
+ ms - m)pw +
which, since paw= pa
G
m,p,
+ (m - ms)paw
+ pw at equilibrium, reduces to
a,
a, =
PB
(1)
= Pls
+ RT In + RT In qs
fs
(2)
where y is the conventional stoichiometric molal activity coefficient, and superscript zeros denote the appropriate standard states. It should be noted that (1) does not implyp~O= pB0 in the standard states; the standard state for B is a hypothetical solution of total solute molality m unity and molal scale activity coefficient unity, containing both the anhydrous solute and the solvate, while the standard state for s is a hypothetical solution of molefraction X , = 1, containing the anhydrous solute s only. Likewise, that for sw is a hypothetical solution with X,, = 1, containing no anhydrous solute. At infinite dilution, a, = 1, the solute consists of s and sw in the ratio lim (m - m,)/m, = K , lim m,/m = 1/(1 K ) , and all activity”c0efficients are unity, K = qsw/(qsaw)being the mole fraction scale equilibrium constant for the hydration process. On Garrod and Herrington’s “semiideal dilute” model, fs = 1 at all concentrations so that (2) becomes
+
7, =
Amr
+ O.OOIMl(l - h)m
(5)
+ + ts
+
tsw= a,
+ ss(l + Kaw)
= 1
(6)
These equations, together with eq 3 and our value of A , give Y =
1 + K 1 ~ a , 1
+
1
+ O.OOIM~(I - h)m 1 + K 1 Ka,
+
1 - a, O.OOIMlm
(7)
By similar methods it can be shown that if there are several stages of hydration
+
+
+
where h = u / X , Z: = q s ( l Ka, K2aW2 ---), 20 = qe (1 K K2 - - - ) , u = qe (Ku, 2 K2aw2 - - -). Corresponding with eq 4, we have for the case of multiple hydration
+
+ +
+
+
0.001Mlm~< 20
+
(9)
For aqueous solutions of sucrose,2 with 11 sites available for hydration, K = 0.994 and Z0 = 11.61.
(3)
where A is a constant involving the differences of The Journal of Physical Chemistry, Vol. 74,No. 16, 1970
0.OO 1Mlm
1
where h = a/Z and, if there is only one stage of hydration, 2 = qs (1 Ka,) u = q,Kaw, and a,
+ RT In m + RT In y = p,O
- 0.001Mlhm 1 + 0.001Ml(l - h)m 1
1-a,=
I n terms of activities, this becomes
BO
+ 0.001Mlmy < 1
or
+ mpB
so that
(4)
The correct inequality (4) is easily satisfied with an appropriate value of K , whereas their eq 40 is not. Our earlier treatment2 of semiideal solutions was based on the equations
+ mp,
G = (O.OOIMi)pw
+ 0.001M1*-1 my
