L. A. WOOLF
1166
Multicomponent Diffusion in a System Consisting of a Strong Electrolyte Solute at Low Concentrations in an Ionizing Solvent by L. A. Woolf Difusion Research Unit, Research School of Physical Sciences, Australian National University, Canberra, A.C.T., Australia (Received August 3,1971) Publication coats assisted by L. A . Woolf
Diffusion in a solution composed of a strong electrolyte and a solvent which also ionizes must be described by the flow equations for a three-component system when the ions of the solute and solvent are at similar concentrations. A theory is developed to predict values of the diffusion coefficients for the flows of neutral components. The theory is adequate to describe diffusion in the system (particularly when the concentrations of the solute ions are either much greater than or much less than those of the solvent ions) and indicates large values for the cross-term diffusion coefficients when all ionic concentrations are comparable.
It is usual t o describe diffusion in a two-component system consisting of a strong electrolyte solute, 1, dissolved in an ionizing solvent, 0, by Fick’s first law (JJR
=
- ( D I ) R ( W ~ Z ) (i = 0,1)
where the flows J, are measured with respect to a reference frame, R, D, denotes the diffusion coefficient, and (i3ci/hx) is the local gradient of concentration ci. Following a suggestion by Woolf, Miller, and Gosting, experiments by Mills2showed Fick’s law was inadequate t o describe the flows of the neutral components when the stoichiometric concentration of 1 was close to that of the ions which come from partial ionization of the solvent. Since then theoretical methods have been developed to predict diffusion coefficients in systems of three (or more) components when two ureak or strong electrolyte components have an ion in This paper shows how a theory based on that of ref 3 can be used to calculate values of diffusion coefficients in a three-component system consisting of a partly dissociated solvent and two fully dissociated electrolytes not having any ions in common but each having an ion in common with the solvent. I n eo doing the qualitative prediction of ref 1 is confirmed and given a quantitative basis.
Avl,DV,, = C J % 9 a=
+ + V~~B’S
VIA‘‘ VZ&%
(3)
v 1 7 D Z 7
(4)
On either description of the composition of the system it must contain five constituents-the neutral molecules 0 and the four ions, 4 , 5 , 6 , and 7. However, the number of independent components for diffusion is two? Because the theoretical development requires interrelationships between ionic and neutral component chemical potentials, it is necessary to describe the system in terms of the three neutral solutes 0, 1, and 2 which ionize according t o (l),(3), and (4). Reaction 1 is assumed to be sufficiently rapid to ensure local chemical equilibrium? All flows and related coefficients are referred to the solvent-fixed frame of reference.’ The chemical potentials p i and their gradients X i (= -grad pi) of the solutes 0, 1, and 2 are defined in terms of ionic chemical potentials by 7
7
xi = with
V06
=
V07
=
j=4vijxj
VIS
=
VIE
=
(i = 0, 1, 2)
(6)
= 0.
Conserva-
v24
=
v27
Theory The system to be considered contains a partially ionized substance, 0, with a degree of dissociation, 01 (composed of v,, ions of A of valence 24 and v,, ions of B of valence 25) and a completely ionized substance, 3, C,,, Dv3,. These dissociate according to eq 1 and 2
+ vo5BZfi CY8BDv3, = vasCZo +
Avo,B~,, = vo4AZ4
~37D%
L. A. Woolf, D . G. Miller, and L. J. Gosting, J . Amer. Chem.
(2) R.Mills, J. Phys. Chem., 66, 2716 (1962). (3) R. P. Wendt, ibid., 69, 1227 (1965). (4) D . G.Miller, ibid., 71, 616 (1967). (5) There are three independent relations between the five oonoen7
cizj = 0,
+
7
c#j
=
1, and Kw =
C ~ ~ ~ ~ C ~ ~ ~
(1)
trations:
(2)
where Kw is the equilibrium constant for (1) and the ci’s are concentrations in mol cm-3. Also see ref 7. (6) Compare with W. H. Stockmayer, J . Chem. Phys., 33, 1291 (1960). (7) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, ibid., 33, 1505 (1960).
The system can also be described as consisting of substance 0 and the substances 1 and 2, which ionize fully as, respectively The Journal of Physical Chemistry, VoE. 76, No. 8,1979
(1)
Soc., 84, 317 (1962).
i=4
C O ~ O
i=4
1167
MULTICOMPONENT DIFFUSION OF STRONG ELECTROLYTE AND SOLVENT tion of charge and mass for (l), (3), and (4) provide, respectively, the restrictions
and the Lr5 corresponding phenomenological coefficients. These Coefficients are related by
7
vfjzj
(i = 0, 1, 2)
0
(7)
3'4 2
=
c5
C i-0
where
( j = 4, 5, 6, 7)
cv0tYfjCf
k 5
where a01
= am = 1 and
a00 = a!
(9)
The dependence of the macroscopic velocities ut of the neutral components 0, 1, 2 on those of the constituent ions may be defined by
=
( k t j = 1, 2)
&k,/bcl
(18)
In the following section the Et, of (15) are related to the Dt5of (16). Relations between Dt5and l$5. The condition of zero net electric current flow requires here that 7
CzJ9 = 0 j=4
(19)
and enables the term f E in (15) to be replaced so that The flows J a of the neutral components or ions are related to these velocities
J$ =
(i
CtUt
1, . . ., 7)
7
7
j = 4 k=4 1=4
(11)
C C
na=4 n = 4
Thus (8)-(11) provide
JmnZmZn
(i = 4, 5, 6, 7) (20) The xi in (20) are replaced using (7) in the form
(The terms in J o have been omitted from (12) since Jo = 0 is used here to define the solvent-fixed frame of reference for the flows and velocities.) Entropy Production. For this system of four ions the entropy production due to isothermal diffusion is 7
Tu =
j=4
2%= a t 7 z 7
with
a47
v2~)a57,
= -v~/v14, a57 = - ( V O ~ / Y O a~67) ~= U ,- (v25/ a77 = 1. Using (21) and defining
and
7
j-4
(13)
$5
JtX,
Tr = i=l
The linear laws relating the flows and forces may therefore be written in terms of either ionic or neutral component phenomenological coefficients. In the former instance 7
C l t j ( X j + ZjfE)
j=4
C C lmnam7an1 (i, I
j=l
=
4, 5, 6, 7) (22)
enables (20) to be rearranged in the form 7
Ji =
C Q t i X l 2=4
(i = 4 , 5, 6, 7)
(23)
The ionic chemical potentials X 1 in (23) are related through (6) by
where the coefficients at, have the values = c~~~= = a70 = a 7 1 = ~7~ = 0, a41 = 1/V14, a60 = 1/vo5, a51 = - (vo4/v14)a41, a60 = -vd(v05v2d, a61 = - (v04/ V I ~ ) ~a6B 2O = ,l/m. Using (24) and the identity8 a62
(i = 4,5, 6, 7) (15)
where the lij are ionic phenomenological coefficient^.^ Diffusion in multicomponent electrolyte systems is almost invariably described in terms of flows of neutral solutes. Experimental and theoretical flow equations for the present system may be written as 2
7
m=4 n=4
2
Jt = - E D
(21)
7
+f E C
JJ5
where u is the local entropy production per unit volume, T the absolute temperature, f = 107F( F is the Faraday) and E is the local electric field. By using (12), (7), and (6) in (13) the ionic flows are replaced and the coefficient of f E eliminated
Ji =
(i = 4, 5, 6, 7)
7
C ~ L= 0~ Q (i =~4,5,~6, 7) 1-4
(25)
in (23) givesg
2
gradc, = C L i 5 X j
(i
= 1, 2)
(16)
j=l
where the Di5define neutral solute diffusion coefficients
(8) Reference 3, footnote 12. 2 (9) The Gibbs-Duhem equation in the form X O = - ( l / c o ) C cjXj j=1 has been used to eliminate the term in XOfrom (24).
The Journal of Physical Chemistry, Val. 76,No. 8,19'7.8
L. A. WOOLF
1168 2
Jc =
7
2c
j=1 z=4
(i = 4, 5, 6, 7)
ar,'QttXj
(26)
where Ul,'
- (c,/co)alo
= azj
The flows of neutral solutes are obtained through (12) (puttingp = i 3)
+
Because results are available for the flows of bromide ion in the H20-MgBr2 system, it is convenient to write an explicit expression for J7 using 4,5,6, and 7 to denote the hydrogen, hydroxyl, magnesium, and bromide ions, respectively. The expression is shortened by using the equation for the limiting value of the tracer diffusion coeficient Dt of an ion i, D f o = [RT/(F2 X 107)](Xt0/ Izt\), where R is the gas constant, and by noting that a t very low concentrations it is reasonable to assume -xt
These latter flows are related to the neutral solute diffusion coefficients using (16) and (18)
=
(&*/ax)
=
(aPl/w(%/w; pfo
=0
lit =
(i # j )
(i, j
Xt0ct/(l~JF2 X lo7)
= 4,
5, 6, 7)
(i = 4, 5, 6, 7)
Here X f o is the limiting equivalent conductance in cm2 ohm-' equiv-l and the Faraday is in coulomb equiv-l. Use of the first of (29) in (22) gives Q ~= z
6t1Etg
-
Ztclzzaf7a
dS
(i, I
= 4,
5, 6, 7) (30)
where 7
8= and 6 t z
=
ak'I2&k
k-4
1 (i = I) or 0 (i # E ) .
Discussion I n relating the preceding equations with experiment
it is convenient to examine three concentration regions. Mutual Digusion. When c6, c7 >>. c4, c5 all of Qf4 and !&S (i = 4-7) in (23) may be shown via (30) to be much less than any of at6 or Qf7 (i = 4-7) so that J j = 0 ( j = 4, 5 ) and Jt = Q&Y7 (i = 6, 7) (31) Because of the requirements of electrical neutrality J a and J7 are then not independent and may be shown by a similar route to that used by Miller* t o give a limiting law expression for the mutual diffusion coefficient. This result is in qualitative agreement with experiments of Rlills who found that with decreasing solute concentration mutual diffusion persisted in two aqueous solumol cm-a. tions to solute concentrations of about Tracer Diffusion. When C 6 , c7
