Equations for testing Onsager's reciprocal relations for isothermal diffusion depend on the frame of reference chosen for the flows. This subject is considered for certain frames of reference, as is the problem of measuring diffusion coefficients when there is a change of volume on mixing. Frames of reference discussed are those moving with the local center of mass, the local center of volume, the local velocity of the solvent (or of any single component), and that fixed on the diffusion cell. Multicomponent systems, both of strong electrolytes and nonelectrolytes, are considered. An expression is derived which relates the flow of a component in the cell‐fixed frame of reference to the flows in any other frame of reference when there is a change of volume on mixing. This relation is used to show that a flow relative to the cell becomes identical to that in the volume‐fixed frame as the initial differences in concentration within the diffusion cell are made sufficiently small. Throughout this article a special effort has been made to present derivations and final equations in a form well adapted for use in experimental work.

1.
Any reference frame for which the flows are connected by a simple linear relation is a member of this class (see, for example, Secs. 2 and 4 of footnote reference 5). Although the problem of frames of reference for the flows arises in both theoretical and experimental investigations, it frequently has not received adequate attention, especially in experimental work. Different aspects of this problem have been considered by a number of previous workers. Representative references, in addition to footnote references 2–5, include: (a)
L. G.
Longsworth
,
Ann. N.Y. Acad. Sci.
46
,
211
(
1945
);
(b)
G. S.
Hartley
and
J.
Crank
,
Trans. Faraday Soc.
45
,
801
(
1949
);
(c) S. R. de Groot, Thermodynamics of Irreversible Processes (Interscience Publishers, Inc., New York, 1951);
(d)
R. B.
Bird
,
C. F.
Curtiss
, and
J. O.
Hirschfelder
,
Chem. Eng. Prog. Symposium Ser.
, No. 16,
51
,
69
(
1955
);
(e)
O.
Lamm
,
Acta Chem. Scand.
11
,
362
(
1957
).
2.
I.
Prigogine
,
Bull. Classe Sci. Acad. Roy. Belg.
34
,
930
(
1948
).
3.
G. J.
Hooyman
,
H.
Holtan
, Jr.
,
P.
Mazur
, and
S. R.
de Groot
,
Physica
19
,
1095
(
1953
).
4.
G. J.
Hooyman
and
S. R.
de Groot
,
Physica
21
,
73
(
1955
).
5.
G. J.
Hooyman
,
Physica
22
,
751
(
1956
).
6.
L.
Onsager
,
Phys. Rev.
37
,
405
(
1931
).
7.
L.
Onsager
,
Phys. Rev.
38
,
2265
(
1931
).
8.
L.
Onsager
and
R. M.
Fuoss
,
J. Phys. Chem.
36
,
2689
(
1932
).
9.
L.
Onsager
,
Ann. N.Y. Acad. Sci.
46
,
241
(
1945
).
10.
The velocity uSR may be described in terms of the velocity, uSC, of frame S relative to the diffusion cell and the velocity, uRC, of frame R relative to the cell by
uSR = uSC−uRC. (1a)
The velocity of a given frame of reference relative to the cell will, in general, vary both with position and time. For example, a reference frame (or coordinate system) fixed relative to the local center of mass in one part of the cell will in general be moving with respect to a reference frame fixed relative to the local center of mass in another part of the cell. However, the same unit of length is used everywhere in the system to measure distances for every frame of reference considered.
11.
Although we let 0 denote solvent throughout this paper, all the flow equations are still applicable if instead 0 denotes one of the solutes.
12.
The number‐fixed reference frame, N,
[defined byi = 0q(Ji)N/Mi = 0]
is not considered here because it does not seem particularly useful in studies of liquids. This frame is encountered in studies of gases; it should be noted that for ideal gases the volume‐fixed and number‐fixed frames are identical.
13.
This may be shown by substituting into Eqs. (7)–(9) the relation (Ji)C = ci(ui)C, where (ui)C denotes the local velocity of component i relative to the cell. Because (ui)C must be the same for every component when all the concentration gradients are zero, the equations are readily simplified by using Eqs. (5) and (6) and lead to uMC = u0C = uVC = (ui)C.
14.
(a)
R. J.
Bearman
and
J. G.
Kirkwood
,
J. Chem. Phys.
28
,
136
(
1958
);
(b)
R. J.
Bearman
,
J. Chem. Phys.
31
,
751
(
1959
).,
J. Chem. Phys.
15.
Professor Kirkwood expressed considerable interest in this pressure gradient and suggested that in some experiments with large concentration gradients (perhaps in certain cases of diffusion through membranes) it may be of practical importance. He wanted to expand Secs. II and III to include terms in the pressure gradient and in the acceleration term (1/cj)[∂(Jj)C/∂t] in each force. Because the remaining authors are not in a position to complete this extension as he visualized it, we omit these terms in this article and emphasize that the development given here should be applied only to the conventional type of experiment in which these additional terms are exceedingly small. It seems worthwhile, however, to indicate that by using for Xj the relation [see Eq. (5.13) of footnote reference 14(a) or Eq. (12) of footnote reference 14(b)],
Xj = −{(∂μj/∂x)T+(1/cj)[∂(Jj)C/∂t]} (11a)
instead of Eq. (11), and by following the derivation outlined in Eqs. (10)–(22), reciprocal relations between the ij)M in Eqs. (10) still lead to reciprocal relations between the ij)0 for the solvent‐fixed frame. Equations (14)–(17) remain unchanged except that the Xj defined by Eq. (11a) appear in Eq. (15) instead of −∂μj/∂x (the subscript T on ∂μj/∂x in Eq. (11a) indicates that these new derivatives are taken at constant temperature but not at constant pressure). To eliminate X0 from this revised Eq. (15) the generalized form of Eq. (18) (with the pressure term retained),
j = 0qcj(∂μj/∂x)T = ∂P/dx, (11b)
is used to eliminate (∂μ0/∂x)T, and the derivative of Eq. (7) with respect to time
∂(ρuMC)/∂t = j = 0q∂(Jj)C/∂t (11c)
is used to eliminate (1/c0)[∂(J0)C/∂t]. These substitutions lead to a form of Eq. (19) in which −∂μj/∂x is replaced by Xj from Eq. (11a) and there appears the addition term
−[(Ω̄i0)0/c0]{(∂P/∂x)+[∂(ρuMC)/∂t]}
. According to the equation of motion,
(∂P/∂x)+[∂(ρuMC)/∂t] = −∂[ρ(uMC)2]/∂x, (11d)
the term in braces may be replaced by −∂[ρ(uMC)2]/∂x. However, this term may be neglected because a term of order (uMC)2 corresponds to force terms of order higher than the first power; such terms have already been neglected in writing the linear phenomenological relations, Eqs. (10). Therefore it is found that Eqs. (20)–(22) remain unchanged. Equation (11d) may also be used to show that the pressure gradient from inertial effects is very small in the usual experiments for measuring diffusion coefficients.
16.
Because terms for all components are included in Eqs. (12), neither the forces nor the flows are independent. Consequently Eqs. (12) do not uniquely define the ij)M without auxiliary relations, and whether the ij)M satisfy Eq. (13) depends on the choice of auxiliary relations. Suppose that to describe diffusion in a given system one uses for the ij)M a set of permissible numerical values which satisfy the reciprocal relations, Eq. (13). If to each Eq. (12) is then added [see Eq. (18)] the sum
σi = −Kij = 0qcj(∂μj/∂x) = 0
, where each Ki is a constant, a new set of fundamental diffusion coefficients, (Ω̂ij)M = (Ωij)M+Kicj, is obtained; these new coefficients are not subject to reciprocal relations (except perhaps for special values of the Ki). This does not imply that the Onsager reciprocal relations are entirely arbitrary. For a ternary system Eqs. (13) provide three restrictions on the ij)M. One restriction on the set of ij)M for this system is required by Onsager’s considerations of microscopic reversibility; the other two are simply convenient definitions, such as may be achieved by adjusting the Ki. In this paper we need not consider the auxiliary relations required to define completely the ij)M in Eqs. (12); it is sufficient that such relations may be specified in a way so that Eqs. (13) are satisfied (see footnote references 4 and 18).
17.
Reciprocal relations also hold between the new fundamental diffusion coefficients in Eqs. (19) when each (Ji)0 is expressed as moles/cm2/sec, and each μj as the chemical potential per mole.
18.
This is in agreement with Eq. (4.12.7) of Onsager and Fuoss (footnote reference 8), and Eq. (19) of Hooyman (footnote reference 5). Note that for the ij)0 to satisfy Eq. (22) it is not necessary for the ij)M to be defined so that they satisfy Eq. (13); it is sufficient that they may be defined so that Eq. (13) is satisfied. This may be shown by starting with the ij)M defined so that they satisfy Eq. (13) and then adding σi (footnote 16) to each Eq. (12), using values of Ki such that the (Ω̂ij)M do not exhibit reciprocal relations. After repeating the derivation with these (Ω̂ij)M one finds that in the equation corresponding to Eq. (21) the coefficients of Ki and K0 are zero, again giving Eq. (22) as a consequence of ij)M = (Ωji)M.
19.
It will be recalled that for nonelectrolytes the chemical potential per g, μj, of component j can be expressed in terms of the concentration, cj, and molecular weight, Mj, of that component and an activity coefficient, yj, (which may be a function of all solute concentrations)
μj = μjo+(RT/Mj) lnyjcj. (23a)
Here R is the gas constant per mole and T is the absolute temperature. This activity coefficient, yj, is identical with the activity coefficient associated with the molarity scale of concentration; it is customary to let each yj approach unity as all the solute concentrations approach zero. For a given solvent each reference chemical potential, μjo, is a function only of temperature and pressure. By differentiation of Eq. (23a) we see that
∂μj/∂ck = (RT/Mj)/(∂ lnyj/∂ck) (j≠k) (23b)
and
∂μj/∂cj = (RT/Mjcj)[1+(∂ lnyj/∂ lncj)]. (23c)
20.
See Eq. (18) of footnote reference 5.
21.
Our Eq. (30) may be shown to be consistent with Eqs. (30) and (31) of footnote reference 5 by substituting appropriate weight factors into the latter equations.
22.
R. L.
Baldwin
,
P. J.
Dunlop
, and
L. J.
Gosting
,
J. Am. Chem. Soc.
77
,
5235
(
1955
).
23.
P. J.
Dunlop
and
L. J.
Gosting
,
J. Am. Chem. Soc.
77
,
5238
(
1955
).
24.
H.
Fujita
and
L. J.
Gosting
,
J. Am. Chem. Soc.
78
,
1099
(
1956
).
25.
P. J.
Dunlop
,
J. Phys. Chem.
61
,
994
(
1957
).
26.
H. Fujita and L. J. Gosting, J. Phys. Chem. (to be published).
27.
M.
Dole
,
J. Chem. Phys.
25
,
1082
(
1956
).
28.
F. E.
Weir
and
M.
Dole
,
J. Am. Chem. Soc.
80
,
302
(
1958
).
29.
P. J.
Dunlop
and
L. J.
Gosting
,
J. Phys. Chem.
63
,
86
(
1959
).
30.
P. J.
Dunlop
,
J. Phys. Chem.
63
,
612
(
1959
).
31.
D. G.
Miller
,
J. Phys. Chem.
62
,
767
(
1958
).
32.
D. G.
Miller
,
J. Phys. Chem.
63
,
570
(
1959
).
33.
D. G.
Miller
,
Chem. Revs.
60
,
15
(
1960
).
34.
For ternary solutions of electrolytes, a conversion of equations which describe diffusion in terms of ionic species to equations in terms of neutral components has been made by Miller, who considered these transformations for the equation giving the entropy production (see Appendix I of footnote reference 32).
35.
A different description of diffusion of electrolytes (including consideration of electric currents) has been given by
B. R.
Sundheim
,
J. Chem. Phys.
27
,
791
(
1957
).
36.
There is no problem for s = 2 and for s = 3; the neutral solutes are considered to be, respectively, a single salt and two salts with a common ion. For s = 4 two cases may be encountered. If three of the four ionic species have electric charges of the same sign, the neutral solutes are considered to be three salts with a common ion. If two positive and two negative ionic species are present some arbitrariness exists as indicated in the text, where a solution is considered containing ionic species Na+,K+,Cl, and Br. The arbitrariness is generally greater for larger values of s, except when all of the salts have a common ionic species.
37.
Equations expressing forces in terms of flows, instead of flows in terms of forces, are encountered frequently in the literature, and they seem more convenient for the present derivation. Equation (5.13) of footnote reference 14(a) is of this type. Our Eqs. (32) and (33) are identical with equations which have been given by Onsager and Fuoss [see footnote reference 8, Eqs. (4.12.8),(4.12.9), (4.14.1), and (4.15.4), and pp. 2762 and 2763]. By a derivation somewhat similar to that used to obtain Eq. (19) from (12), Eq. (32) may be obtained from the following expression containing flows of all ionic species and of the solvent relative to the mass‐fixed frame,
−∂μ̄i/∂x = j = 0s(Rij)M(Jj)M (i = 0,⋯,s). (32a)
To express the (Jj)M in terms of flows for the solvent‐fixed frame, first Eq. (1) is written for frames M and 0. Then terms in this relation are summed over all components and Eq. (2) applied to obtain u0M. Substitution of this expression back into Eq. (1) for frames M and 0 gives the relation
(Jj)M = k = 0sjk−(cj/ρ)](Jk)0. (32b)
Here δjk is the Kronecker delta (δjk = 1 if j = k and δjk = 0 if j≠k). Substitution of Eq. (32b) into (32a) gives
−∂μ̄i/∂x = k = 0s(Rik)0(Jk)0 (i = 0,⋯,s), (32c)
in which
(Rik)0 = (Rik)M−(1/ρ)j = 0scj(Rij)M. (32d)
Because (J0)0 = 0 [Eq. (3)], Eqs. (32c) for the ionic species are seen to be identical with Eqs. (32). If a restriction proposed by Onsager [see Eq. (7b) of footnote reference 9] may be applied to the (Rij)M in the form
j = 0scj(Rij)M = 0, (32e)
it follows from Eq. (32d) that (Rik)M = (Rik)0. Then the subscripts 0 and M could be omitted from the Rik and reciprocal relations for the (Rik)M would correspond directly to Eq. (33). To avoid any uncertainty the subscripts 0 are retained on the Rik in the text.
38.
S.
Prager
,
J. Chem. Phys.
21
,
1344
(
1953
).
39.
J. Crank, The Mathematics of Diffusion (Oxford University Press, New York, 1956), p. 236 ff.
40.
See, for example, K. Denbigh, The Principles of Chemical Equilibrium (Cambridge University Press, New York, 1955), Eq. (2.114).
41.
R. P.
Wendt
and
L. J.
Gosting
,
J. Phys. Chem.
63
,
1287
(
1959
).
42.
See p. 219 ff. of footnote reference 39.
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