GRAVITATIONAL STABILITY IN ISOTHERMAL DIFFUSIONEXPERIMENTS solvents which contain low concentrations of “coordinating solvent.”16r16 The discussion that follows, however, is unaltered if free ions are formed instead of solvent-separated species. If we assume that the addition of phenol leads to the formation of solvent-separated ion pairs in carbon tetrachloride (dielectric constant = 2.2) and methylene chloride (dielectric constant = KO), then our results indicate that the diflerence in free energy between contact and solvent-separated species is the same in both solvents. This effect may be explained in either of two ways.” (a) The short-range properties of ions in solution are independent of the dielectric constant of the medium. (b) Alternatively, the free energies of the two species vary from solvent to solvent, but they do so in parallel. While our experiments do not allow us t o distinguish between these two possibilities, we note that the results are in opposition to continuum theories, 9, lo which would predict that the difference in free energies between contact and solvent-separated species should increase as the dielectric constant of the solvent is lowered. We conclude that the magnitude of the contact interionic interactions in solution must be considerably less than would be expected on the basis of continuum theory, and, as previously mentioned, must be no stronger than specific short-range effects such as hydrogen bonding or ion-dipole interactions. We can imagine certain instances in which these contact interactions would be considerably stronger. Direct hydrogen bonding between cation and anion would
4577
be one case; the formation of a covalent bond between cation and anion would be a second example. l8 Lastly, small cations (e.g., Li+) or multiply charged species might show cation effects on the hydrogen bonding ability of anions. Our results can be readily generalized to include other protic systems. Multiple solvation equilibria should be possible for anions of high charge density in most alcohols and water. Whereas the total amount of ion association will (to first approximation) depend upon the solvent’s dielectric c o n ~ t a n t ,it~ !is~likely ~ that the nature of the associated species mill be strongly influenced by the relative strength of the solvent-anion hydrogen bond. Evans and Gardamlg reached a similar conclusion in their investigation of the conductance of the tetraalkylammonium salts in the straight-chain alcohols. Their results are most consistent with a two-state association model which allows for the existence of both contact and solvent-separated ion pairs. (15) E.D.Hughes, C. K. Ingold, S. Patai, and Y. Packer, J . Chem. Soc., 1206 (1957). (16) See, for example, the work of Smid and coworkers on systems involving ion-dipole interactions: L. L. Chan and J. Smid, J . Amer. Chem. Soc., 90, 4654 (1968),and earlier papers. (17) We thank Professor Spiro for some helpful suggestions regarding this discussion. (18) Some heavy metal halides exhibit this behavior. See G. E. Coates, “Organa-Metallic Compounds,” Wiley, New York, N. Y., 1956,p 151. (19) D.F.Evans and P. Gardam, J . Phys. Chem., 73, 158 (1969).
Gravitational Stability in Isothermal Diffusion Experiments of
Four-Component Liquid Systems’ by Hyoungman Kim Institute for Enzyme Research, University of Wisconsin, Madison, Wisconsin 63706
(Received July 7 , 1Q70)
The conditions for gravitational stability in free diffusion and the diaphragm cell method of studying diffusion in four-component systems are obtained. For each boundary condition, three criteria should be satisfied in order to definitely avoid convective mixing during the diffusion experiments.
Introduction In any diffusion experiment, it is imperative to have gravitatioIla1 stability everywhere in the column of diffusing liquid during the entire duration of the experiment, for otherwise, convective mixing will render meaningless the diffusion coefficients obtained from the
experiment. In the study of diffusion of a two-component system, initial stability at the time of boundary formation will ensure that there is density stability (1) This investigation was supported in part by Public Health Service Research Grant AM-05177 from the National Institute of Arthritis and Metabolic Diseases. The Journal of Physical Chemistry, Vol. 74, No. 26, 1970
4578
HYOUNGMAN KIM
everywhere in the diffusion cell during the entire diffusion process. In the diffusion of multicomponent systems, however, this is generally not the case, especially when the cross-term diffusion coefficients are large. Therefore it is desirable to derive criteria for gravitational stability in diffusion experiments of multicomponent systems. For the case of free diffusion of ternary systems, Wendt2 obtained these criteria and Reinfelds and Gosting3 subsequently simplified them. Here the procedure of Wendt is adopted in order to derive the conditions of gravitational stability for studies of diffusion in four-component systems by free diffusion and diaphragm cell r n e t h ~ d s . ~ ~ ~ The general condition for density stability of any fluid system at time t may be expressed as2
(Wbx),3 0
2
(1)
where x is the position coordinate in the direction of increasing gravitational field. Here d is the density of the fluid which is generally a function of x and t. I n the isothermal diffusion of a multicomponent system the density of the solution is determined solely by the concentrations of the solutes and may be expressed by a linear function of individual solute concentrations if these solute concentrations, Cz, are not far from the mean solute concentrations, C, = [(C,), (CZ)B]/2, of the solutes in the upper and lower solutions A and B placed initially in the diffusion cell. For the fourcomponent systems the expression assumes the form
+
3
d = d(Ci,C2)Cd
+ C H , ( C , - Ct) i=l
(2)
where d(Ci,C2,C3) is the density of a solution in which each solute is at its mean concentration, C,, for the experiment and H , are the density derivatives, (ad/ bC,)c, + I , T , P , where T is the temperature and P is the pressure ( j = 1, 2, 3). Differentiation of eq 2 with respect t o x and introduction of the resulting equation into relation 1 gives the desired density stability condition in terms of the solute concentration gradients 3
c H,(dCi/bX)t 3 0
(3)
1=1
It may be seen that the values of one or two terms, Hi(bC,/bx)t , can be negative without inducing convective mixing as long as the sum of the three terms in relation 3 is either equal to or greater than zero a t all levels in the diffusion cell. Free Digusion. The expression for the solute concentration distribution in four-component free diffusion is4
where
The Journal of Physical Chemistry, Vol. 74, No. 16,1970
R . P. Wendt, J . Phys. Chem., 66, 1740 (1962). G. Reinfelds and L. J. Gosting, ibid.,68, 2464 (1964). H. Kim, ibid., 70, 562 (1966). H. Kim, ibid., 73, 1716 (1969). (0) All the diffusion coefficients here are referred to a volume fixed frame of reference unless specified otherwise. Rigorously these have to be represented as ( D i j ) u ,but, in order t o simplify the equations, the subscript v is omitted. (2) (3) (4) (5)
GRAVITATIONAL STABILITY IN ISOTHERMAL DIFFUSION EXPERIMENTS
lDtjl
Dll = DZI Dai
012
013
DZZ
0 2 3
0 3 2
D33 ,
~
(10)
4579
mixing will occur in the region of the original sharp boundary position and if relation 15 is not satisfied convective mixing will occur near both ends of the diffusion boundary. Even if conditions 14 and 15 are satisfied, this is not sufficient to ensure gravitational stability and the region of intermediate y values must be considered. We consider the regions of y where exp( - u1y2) is negligible compared to exp( - azy2)and exp( - a@). If the value of ( ~ 1 9 . 1 2 ~ 2 9 2 2 H 3 q a Z ) d i in relation 13 is negative and its absolute value is much larger than that of (H1q13 H2922a HsS33)d&, but relation 14 is still satisfied [when (HI% HZ*ZI has a very large positive value 1, then even if exp( - cay2)is much larger than the corresponding exponential for uZ, cases may arise when relation 13 cannot be satisfied. On the other hand, if the condition
+
+ +
+
[Hi*iz
+
+
+ Hzqzz + HaqazIG + + + Ha*sa]d& [H1*13
H2qZ3
2
0 (16)
is satisfied, the last term in relation 13 will always overwhelm the second term since the exponential function of u3 is always greater than that of UZ. If the three conditions 14, 15, and 16 are all satisfied, condition 13 is also satisfied for all values of y. The violation of condition 16, however, will not necessarily bring about convective mixing; this will depend on the relative values of exp [ - a2y2] and exp [ - a3y2]. Therefore relation 16 becomes the sufficient condition. By introducing eq 7 these relations may now be rearranged in terms of HiAC,. From relation 14 3
['/zg1(a) I
c HiACi[Xi(mz(fl) +
i=l
(7) If a o i is negative, *(.\/.iz/) is no longer the probability integral and as y -+ O D , the concentration of solutes becomes infinite. T h e r e fore none of the oi can be negative. If any oi is zero, from eq l l c IDijl becomes infinite, which would create the impossible condition that one or more D i j be infinite. Finally if any two of the three ui are identical gl(o) becomes zero and the solute concentrations become infinite for all values of y. From these considerations it is clear that relation 12 should hold. (8) Relation 15 can also be obtained by dividing relation 13 by . \ / K t exp(- olyz) to give
[Hiqii
+ Hzqzi + Haqsi] t [Hiqia + Hzqzz + Ha*$a] .\//az/aie(ol-02)u2+ + + ~ ~ ~ a a l G & e ( >~ ol - (1') ~ ~ [Hi913
HZq23
From relation 12 it is apparent that when jyl is very large
exp[(m
-
m)y21 >> exp[(ul
-
>> 1
U Z ) ~ ~ ]
and in order to satisfy condition 1' condition 15 has to be met. (9) A similar situation arises in the free diffusion of ternary systems. If relation 20 in ref 2 is not met convective mixing will occur a t both ends of the boundary while violation of relation 21 of ref 2 will induce convective mixing in the vicinity of the original sharp boundary position. Relation 20 together with relation 21 makes the sufficient condition. This differs somewhat from the position taken by Wendt.
The Journal of Phvsical Chemistry, Vol. 74, N o . 26, 1970
4580
HYOUNGMAN KIM
=
(63
+
- U2)Ul~Ul (a1
+
- ‘J3)uZdaz
(a2
-
(21)
al)a3da
and g4(u) = (m2 - U%2)Ul-G ((TI2
-
g3’)aZd;
+ + (a,’ -
al2)g3dG
(22)
From the inequality condition 12, it may be seen that the value of gl(u) is negative definitelo#”and relation 17 may be multiplied by 2gl(a) to obtain
5 H,ACrIXz(H)gz(a) + YI(H)$dU) - g4(a)l
6
0
(23)
Likewise relation 16 is transformed into 3
+
-
i=l
- add;]
+ Yi(H)[(aa- a i ) a z d & +
(a1 - a2)a,d,]
- [ ( ( ~3 ~a 1 2 ) a % d &
(UI
+
Introducing eq 7 into relation 15 and dividing the resulting equation by (UZ - al)/2gl(a), remembering this quantity is positive, we get 3
CHiACi[Xt(H)
%El
HzACz
and from relation 25 one obtains
Yi(H)a3 - a3(gl
+d I >0
H3AC3 0 (31) In free diffusion experiments with ternary systems, the values of AC, are usually made to range from zero to certain positive values so that the concentration difference fractions, CY,, of the solutes on the basis of refractive index3v4will range from zero to unity. Relations 29 through 31 indicate that this range of AC, will guarantee gravitational stability for experiments with fourcomponent systems when the cross-term diffusion coefficients are negligible. For ternary systems studied thus far where the cross-term diffusion coefficients were relatively small, the AC, ranges used also satisfied the condition for the gravitational ~ t a b i l i t y . ~ For , ~ cases where the cross-term diffusion coefficients are very large, l 2 (10) T h e equations 8, 20, 21, and 22 can be transformed into
a3
=
1/41
+ ~ / D z+z 1/Daa
The Journal of Physical Chemietru, Vol. 74, No. $6, 1970
(26a)
- Ud(U1 - u3)(ua - uz) - G*)(Gl - G3)(dZ-
= (UI
(25)
Thus we have three conditions (independent of time and position) which if satisfied ensure gravitational stability in free diffusion of four-component systems. Of these, relations 23 and 25 are necessary conditions and satisfaction of all three inequalities is sufficient to ensure gravitational stability. For a given system at given mean solute concentrations, the value of the diffusion coefficients and the density derivatives, Hi,are fixed so the left-hand sides of relations 23-25 are sums of the three H,AC,, multiplied by appropriate numerical values. Therefore, for a given composition of a given four-component system, the Act are the only adjustable parameters t o be used in satisfying conditions 23-25. When all the cross-term diffusion coefficients in a four-component system are negligible we have the following relations
+ +
(30)
dG+zz3>0
gz(u) =
LTI
H3AC3
>
i=l
CH,AC,{X,(H)
Relation 24 may be likewise transformed into
( G I
+
g4u) = - g z ( u ) [ G z g 4 ( ~ )= gz(u)[uIuz
+
+
UIU~
uzua U
+ ./Z]
./&3
+ 1
6
G 3 )
3
+
0
3
6
3
+
U a G Z ]
From the inequality condition 12, it is immediately clear that
gdu) gz(.) g4u) and
0 0
(11) These transformations were kindly made by Professor L.
J.
Gosting. (12) Although there may be some upper limits as to the size of the cross-term diffusion coefficients relative to the main diffusion coefficients, no theoretical limitation exists so far. T h e relation 0 1 1 0 2 2 DizDsi > 0 for ternary systems is not necessarily a limiting condition because, if one of the cross-term diffusion coefficients is negative, this relationship will hold regardless the size of the cross-term diffusion coefficients. Even if the signs of the cross-term diffusion coefficients are the same, one coefficient can be enormously large if the other one is very small. The same situation also arises in fourcomponent systems.
GRAVITATIONAL STABILITY IN ISOTHERMAL DIFFUSIONEXPERIMENTS this may not be generally the case as shown by one of the following numerical evaluations of the coefficients for H,AC‘, in relations 23-25. We assign illustrative values for the main diffusion coefficients as Dll = lo+;
D22
=
2.5 X
033
=
5 X lo-‘
If we first assume that all the cross-term diffusion coefficients are zero except for (HlIH3)Dla = 2.5 X lo-‘, then relation 23 becomes H1AC1
gree. When the expressions for the concentration distribution are differentiated and introduced into eq 3 one will obtain a condition corresponding to relation 13. This condition will have n terms and each term is a product of exp( - a,y2) and a linear function of n - Ht\kvs. The argument used for the four-component system may be also used here to obtain n conditions for the gravitational stability and if the following inequality can be assumed
mt
+ 0.63HzACz + O.lOHaAC3 3 0
Also relation 24 becomes 0.87HzACz
4581
Q1
> > . . . > Qn-1 > QZ
and relation 25 reduces to
3
0
+ 0.53HzACz + 0.71H3AC3 3 0
From relation 24 HiACl
(32)
(33)
Thus even for this case where (H1/H3)D13 is very large, but with the other cross-term diffusion coefficients equal to zero, one will have gravitational stability as long as HiAC, 3 0. Next, when DI,, D22, and D33 have the values assigned above and (Hl/H2)D12 = 5 X (Hl/Ha)Dia = (Hz/Ha)Dz3 = -2 X lo-‘, (Hz/Hl)Dzl = (H3/H1)& = lo-’, and (Ha/Hz)Daz = lo-’, relation 23 becomes HiACi
>0
then the n conditions have the following general form
+ HaACs 3 0
H3AC3
rJn
where m = 1, 2, . . . , n. Here the necessary conditions are the relation 33 for i = 1 and i = n and the rest of the conditions become the sufficient conditions. Diaphragm Cell Method. With this method the diaphragm itself gives an inherent density stability and the problem of convective flow is far less serious than in free diffusion. However, Stokes1*showed that for accurate diffusion, it is important to place the denser solution in the lower compartment.19 When the steady state is reached in the diaphragm, it is generally assumed that each solute concentration gradient within the diaphragm is constant, and for this case relation 3 assumes the form
+ 8.90HzACz + 6.85H3AC3 3 0
(34)
and from relation 25 -H1AC1 - 1.8H2ACz - 36.SHaAC3
3
0
Here the only way to satisfy all three conditions is to make H3AC3 slightly negative. Thus it is clear that when the cross-term diffusion coefficients are comparable in size to those of main diffusion coefficients, the current practice of preparing the solutions for diffusion experiments may give rise to the density instability. It is therefore recommended that whenever one suspects that one or more of the cross-term diffusion coefficients for the system to be studied are large, the approximate values of the diffusion coefficients (and predetermined H, values) be introduced into relations 2325 in order to determine the safe range of AC,s. The estimate of the diffusion coefficients of nonelectrolytes is very difficult, but for electrolyte systems methods are available for obtaining the approximate values of the diffusion coefficients.13-16 For the general case of a system with n 1 components, the expressions for the solute concentration distributions are similar to the equations for systems with three or four component^.^^'^^^^ For this general case each !Pt5 is a linear function of the n Concentration differences and u% are roots of a polynomial of nth de-
+
The solute concentration differences between the two compartments of a diaphragm cell during the diffusion, AC,(t), may be expressed as4 3
AC,(t)
=
2 z!P&(t/~j)(i = 1, 2, 3) (35) j=l
where
Here IC is the diaphragm cell constant. Equation 35 is now substituted into relation 34 and the resulting relation is divided by e-(“ua)t to obtain (13) L. J. Gosting in “Advances in Protein Chemistry,” Vol. XI, Edsall, et al., Ed., Academic Press, New York, N. Y., 1956; also I. J. O’Donnell and L. J. Gosting in “Structure of Electrolytic Solutions,” W. J. Hamer, Ed., Wiley, New York, N. Y., 1959, p 160. (14) R. P. Wendt, J . Phys. Chem., 69, 1227 (1965). (15) D. G. Miller, ibid., 70, 2639 (1966); 71, 616, 3588 (1967). (16) H. L. Toor, A.I.Ch.E. J . , 10, 448, 460 (1964). (17) W. E. Stewart and R. Prober, Ind. Eng. Chem., Fundam., 3, 224 (1964). (18) R. H. Stokes, J . Amer. Chem. Soc., 72, 763 (1950). (19) Here we are considering only the vertical position of the cell in
which the diaphragm is horizontal.
The Journal of Physical Chemistry, Vol. ‘74, N o . 86,19‘70
HYOUNGMAN KIM
4582 [HiQii
+ HzQzi + Ha%]
X
+
expW(lla3 - l / d l
[HiQiz
+ HzQ2z +
exp[W/m
[HiQi, and from the relation 12 exp[kt(l/a3 - 1/ud
+
H3QazI
X
- 1/41
+
HzQz3
H3Q3al
+
3
0 (37)
H1AC1
13
>1
CHiG 3
i=l
(39)
0
For the other extreme case when t is very large, the first term in relation 37 will overwhelm the other terms (relation 38) and the condition
+
3
0
(40) has t o be satisfied. Upon introduction of eq 7 this condition is transformed into 3
CHtACi[Xi(H)
i=l
H2Q21
+ Y@)ai
Ha931
-
+
H1AC1
(38)
3
+
- 0.63HaACa 3
0
and
exp[W/ua - l / u ~ ) l When t = 0, relation 34 simply becomes
HiQli
tion 41 is a necessary condition. However, within the ordinary experimental range of t this condition may become another sufficient condition.20 The numerical evaluation of the coefficients for HiACi in relations 41 and 43, using the values of Dll = DZZ= 2.5 X D33 = 5 X and (Hi/H3)D13= 5 X lo-', gives
3
+ HzACz - 0.13HaACa > 0
respect,ively. Here all the rest of the cross-term diffusion coefficients are assumed to be negligible. For this case it is not enough to make all the H,AC, positive for the initial condition in order to avoid the convective mixing. It should be noticed that with the same set of diffusion coefficients, HiACt can be all positive in free diffusion. For the ternary systems the condition for the density stability is represented as HiACi(t)
+ llzAc2(t) 3 0
(44) The equation for a solute concentration difference in the diaphragm cell for a three-component system is21-23 (45) AC,(t) = 2 [Ki+e-(k'"+)t+ K i -e - ( k / g - ) t ]
0 (41) where For the intermediate values of t we assume that the last term in relation 37 is negligible compared to the other two terms. If the value of [Hl% H2Q22 Ha!&:] is negative and much larger than that of [HIQII HZQZI HaQ31], whatever the gain made by the exponential function of the first term in relation 37 with the increase in the value of t may not be sufficient to overcome the second term. On the other hand, if we have the condition ai(az
+
ua)]
+
+
[HiQii
+
H2Q21
+ HaQai] + +
+
+ HaQaaz] 3 0
(42) relation 37 is satisfied for all values of 1 as long as both conditions 39 and 41 are met. Relation 42 is now expressed in terms of HtACi by substituting eq 7 and rearranging the resultant expression to obtain [Hi912
i?HiACt{Xi(H)(ui =l Yd(H)g3('J1
-d -
UZ)
H2%2
+ -
In the above equations u+, U-, E , F , G, and H are constants which are related to the diffusion coefficients by the equations (20) When all the cross-term diffusion coefficients are zero, relations 41 and 43 can be converted into
HiACi
- ut2) + m(a12 - U32)lj 6 [U1(ua2
0
HlACl .f H2AC2
0 (43)
Thus relation 39 becomes a necessary condition because the violation of this condition will bring initial convective mixing. The relation 43 is the sufficient condition and the violation of this condition may or may not bring density instability. This will mainly depend on the sign and size of the second term Of relation 42. If one is considering the whole range of t, then condiThe Journal of Physical Chemistry, Val. 74, No. 86,1070
2
(2')
and
3 0
(3')
respectively. These conditions result from the fact that if the faster component, rather than the slower one, diffuses from the upper compartment t o the lower compartment of the cell, a net accumulation in the upper comgartment may result producing the density instability. Again the positive values of HiACa for all components will ensure the gravitational stability. (21) E . R. Gilliland, R. F. Baddour, and D. J. Goldstein, Can. J. Chem. Eng., 3 5 , 10 (1957). (22) F. J. Kelly, Ph.D. Thesis, University of New England, Armidale, New South Wales, Australia, 1961. (23) J. K. Burchard and H. L.Toor, J.Phys. Chem., 66,2015 (1962) *
GRAVITATIONAL STABILITY IN ISOTHERMAL DIFFUSION EXPERIMENTS
and
IDij1’ = DnDiz - DizDzi (49) Introduction of eq 45 into relation 44 and division of the resulting expression by e -(’’L) gives [HJG+
+ H&+I
exp[kL(llu-
- l/u+)l
[HiKi-
+
+ HzKz-]
3
0
> u-, we have the relation exp[kt(l/u- - l/u+)] 2 1
(50)
If we assume u+
(51)
And from the argument used previously for the fourcomponent systems, gravitational stability will be ensured when the following condit’ionsare satisfied.
+ HzKz+ 3 0 HiKi+ + HzK2+ + HiKi- + H2K2HiKi+
(52)
3
0 (53) Equations 46 are now introduced into relation 52 to obtain
HiACi[Dzz - Dii HzACz[Dii
- 2(Hz/Hi)Dzi + U] +
- Dzz
- 2(Hi/Hz)D12
+ U] 3 0
[@2z
- DiJ2 + 4Di2D2i]1’2
(54)
HiAC1 H2ACz 0 (56) For the general case of n 1 component systems, the same argument employed so far gives n conditions for the gravitational stability which have the general form m
+
n.
(57)
m = 1,2,3,
Acknowledgment. The author wishes to thank Professor Louis J. Costing for his assistance and encouragement during the course of this work and for his criticism of the manuscript.
Appendix By using the method similar to the one employed by KirkaldyZ4it will be shown here that eq l l c is positive definite. The phenomenological equation for isothermal diffusion of a four-component system is (JJO
=
(L,l)OXl
+ (Lidox2 + (L,3)0X3 x
(i = 1, 2, 3) (58) where (JJO is the solvent fixed flow of component i, (L,j)ois a solvent-fixed phenomenological coefficient, and X i is the thermodynamic force. The entropy production, u,for this case can be represented asz6
+ (Jz)oXz + (JdoX3
. . . ,n - 1 , n
Discussion The paradox of these conditions for gravitational stability is that this stability can be checked only after values of the diffusion coefficients are experimentally obtained; ideally it would be desirable to find, before performing the experiments, the range of AC, which will
(59)
where T is the absolute temperature. Upon substitution of eq 58 into eq 59 one obtains a quadratic form 3
(55)
and from relation 53
+
not give rise to the convective mixing. One obvious question will be whether it is possible to have a small density instability that leads to false Dij values which stili satisfy the criteria given. There seems to be no simple answer to this. However, the extent of convective mixing will be greatly influenced by different values of AC, and this can be expected to give far greater variations in Dij than result from the usual experimental errors. Therefore if approximate values of Dij are not available in advance, so prior estimation of safe ranges of ACi is not possible, it is advisable to obtain values of the D,, for wide ranges of ACi; if each value of a given Dij agrees within the expected experimental error, then by using the criteria given in the text one may confirm density stability in the experiments.
TU = (Ji)oXi
where
u=
4583
Tu =
3
E C (LiAoXiXj
2=13=1
From the second law of thermodynamics Tu therefore 3
(60)
> 0, and
3
c E(L%,)OX%X,b 0
2=13=1
(61)
This relation has the following sets of necessary and sufficient conditionsz6
b (Lzz)o 3 (L1l)O
(L33)o
0
(624
0
(62b)
>0
(62~)
(24) J. 8 . Kirkaldy, Can. J . Phys., 36, 899 (1958). (25) See, for example, S. R. DeGroot, “Thermodynamics of Irreversible Processes,” Interscience, New York, N . Y., 1952. (26) See, for example, G. Hadley, “Linear Algebra,” AddisonWesley, Reading, Mass., 1961, p 260. The Journal of Physical Chemistry, Vol. 7.4, No. 86, 1970
HYOUNGMAN KIM
4584 According to Prigogine and Defay28 I~cccjl
3
(71)
0
and from relations 64 and 67 it is apparent that IDi*lO 3 0
(72)
The relationship between the solvent-fixed and volumefixed diffusion coefficients are given by27 (Did0 =
(Dlk)"
+
where ijt is the partial specific volume of component i and here the component zero represents the solvent. Introduction of eq 73 into relation 72 brings
la,l"D
+ (Cl~l/~OfiO)+ (C282/cOfiO)
+ (c&/cOfiO)] >0
(74)
where lDillurepresents eq 68 when (DdJ)" are replaced by (Du)v. It should be remembered that the diffusion coefficients discussed in the main text are all although the subscript fi is omitted for the sake of simplicity. If all three partial specific volumes are positive IDdu
3
0
(75)
If, however, one or more partial specific volumes are negative relation 75 holds only when [1
+ (ClSl/COfiO) + (C2fiZ/COfiO) + (Cafia/Coflo)] 3 0 (76)
(27) J . G. Kirkwood, R . L. Baldwin, P. J. Dunlop, L. J. Gosting, and G . Kegeles, J . Chem. Phys., 3 3 , 1505 (1960). (28) I, Prigogine and R. Defay, "Chemical Thermodynamics," Longmans Green and Co., London, 1954, p 224.
The Journal of Physical Chemistry, Vol. 7dI No. $6,1970
