Chemistry Reference
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a three-dimensional (3D) phenomenon. However, many of the experimental methods
used to analyze diffusion restrict it to a one-dimensional process, making it much
easier to study its mathematical treatments in one dimension (which then may be gen-
eralized to a 3D space).
The resolution of Equation (2) for a unidimensional process is much easier if we
consider
D
as a constant. This approximation is applicable only when there are small
differences of concentration, which is the case of our open-ended conductimetric tech-
nique, and of the Taylor technique [20, 21]. In these conditions, it is legitimate to
consider that our measurements of differential diffusion coef¿ cients obtained by the
techniques are parameters with a well de¿ ned thermodynamic meaning [20, 21].
In our research group, we also have measured mutual diffusion for multicompo-
nent systems, that is, for ternary and, more recently, quaternary systems.
Diffusion in a ternary solution is described by the diffusion equations (Equations
3 and 4).
∂
c
∂
c
−=
() ( )
J
D
+
( )
D
1
2
(3)
v
∂
x
v
∂
x
1
11
12
∂
c
∂
c
−=
() ( )
J
D
+
( )
D
(4)
1
2
2
21
v
∂
x
22
v
∂
x
∂
∂
c
∂
∂
c
J
J
where
are the molar fluxes and the gradients in the concentra-
tions of solute 1 and 2, respectively. The index
v
represents the volume fixed frame
,
,
and
2
x
1
x
1
2
of the reference used in these measurements. Main diffusion coefficients give the flux
of each solute produced by its own concentration gradient. Cross diffusion coefficients
D
12
and D
24
give the coupled flux of each solute driven by the concentration gradient
in the other solute. A positive
D
ik
cross coefficient (i k) indicates co-current coupled
transport of solute i from regions of higher to lower concentrations of solute k. How-
ever, a negative
D
ik
coefficient indicates counter-current coupled transport of solute i
from regions of lower to higher concentration of solute k.
Recently, diffusion in a quaternary solution has been described by the diffusion
equations
1
(Equations 5-7) [19].
1
An aqueous quaternary system, which for brevity we will indicate with ijk, not indicating the solvent 0,
has three corresponding aqueous ternary systems (ij, ik, and jk), and three corresponding aqueous binary
systems (i, j, and k). The main term quaternary diffusion coefficients can then be compared with two ternary
values,
ij
D
ii
and
ik
D
ii
, and with one binary value; similarly for the other two main terms
ijk
D
jj
and
ijk
D
kk
. The
quaternary cross diffusion coefficient
ijk
D
ij can be compared only with one ternary diffusion coefficient
ij
D
ij
;
this is also true for all the other cross terms. The comparison between the diffusion coefficients of system ijk
with those of the systems ij, ik, and jk, permits to obtain information on the effect of adding each solute to
the other two. The comparison between the diffusion coefficients of the quaternary system with those of the
systems ijk, and with those of the systems i, j, and k, permits to obtain information on the effect of adding
each couple of solutes to the other one.
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