An empirical law quite analogous to Fourier's in the so-called Fickian law of diffusion. Consider a fluid in which there exists a non-uniform space distribution of a dissolved constituent. If S represents the mass of the constituent per unit mass of fluid, then Fick's law for the non-advective flux of this constituent is
where 
is the diffusivity coefficient (with the units
of length times velocity). This relation is presumed to hold
in the absence of temperature and velocity gradients in the
fluid; later a somewhat more general relationship will be
examined in which both temperature and concentration gradients
are simultaneously considered.
From relations (VI-1) and (VI-2) it is required that
The chemical potential 
is, in general, a function of
three thermodynamic parameters, which might be taken as
T, 
, and S. The convention adopted throughout
the text has been that S denotes the total number of grams
of constituent in one gram of fluid. In view of this, the
definition of the chemical potential given by Eq. (V-5)
differs somewhat from the usual definition
, where n is the
number of moles of the constituent per gram of the mixture,
is used in place of S. If m denotes the molecular weight
of the constituent then
and
or
An ideal solution ( ()) is one in which the
potential 
of each constituent is of the form
where R is a positive constant and X is the mole fraction of the constituent concerned and is defined by
the sum extending over all constituents. If the solution
or fluid mixture consists of only two constituents, i.e. the
solute and solvent, then 
will be the sum of two
terms, the mols of solute and the mols of solvent per gram
of solution. The function 
represents the
Gibbs function or free energy function of one mol
of the constituent concerned in its free state, and is
dependent upon T and p only. The potential
for a sufficiently dilute solution is
where 
is a new function of T and and
R' is a new positive constant.
The ideal solution is an oversimplification of real solutions
but it is instructive to consider it in the evaluation of
. The results are probably applicable at least in
the qualitative sense to a real solution and this is our
main interest here. In the absence of temperature and
pressure gradients, it follows from (5.02-8) that
Consequently, for the ideal dilute solution (5.02-2) becomes
where the Fickian law is adopted for 
. The above
relation requires that 
since all other
terms are positive, and is consistent with empirical
findings. Thus, although the relation (5.02-8) may not hold
exactly for a real fluid, the condition that
is positive comes
from the second law in view of the empirical findings of
respect to the sign of .