Page 110 - Modern physical chemistry

100 Entropy and the Second Law

whence
dG _ " II. dni . [5.99]
dn -~rt dn
i
where
[5.100]

the total number of moles in the system.
If the concentrations are kept constant throughout the process, then dn/dn and dG/dn
are constant and
dni = ni dG G [5.101 ]
dn n dn n
Substituting into (5.99) and canceling n yields

G = LJ.li ni' [5.102]
i
Thus, Pi equals the contribution per mole of the ith constituent to the Gibbs free energy,
As a consequence, it is called the partial molar Gibbs free energy of constituent i in
the solution.

5. 13 Concentration Gradients
When a constituent diffuses through a viscous medium under the influence of a con-
centration gradient, dissipation of energy occurs. When the temperature and pressure
are kept constant, this energy is Gibbs energy.
Consider a solution in which the concentration c i of the ith constituent is a function
of coordinate x alone. Suppose that this concentration gradient causes the i molecules
to travel through a given cross section at an average speed Xi' If the molecules contin-
ued to travel at this speed, they would travel distance Xi in unit time. So the molecules
passing through unit cross section at position x in unit time equal

[5.103]
in number.
Now, increasing the negative concentration gradient should increase J i ; decreasing it
should decrease J i . We set
aC'
J. =_D·_t [5.104]
t t ax'
where D j is called the diffusion coefficient for the ith constituent. Combining equations
(5.103) and (5.104) gives
Xi = - Di aCi = -Di alnci . [5.105]
Ci ax ax

We expect Di to vary with the other constituents present and with temperature T and
pressure P. When these influences are kept fixed, we expect it to vary only slowly with
c i • The empirical Fick's law states that Di is independent of c i .
In a first approximation, one may consider each i molecule to behave as a hard sphere
moving through a viscous continuous fluid. But the forcefneeded to maintain a sphere
of radius rat speedx is
f =6rrrrrX, [5.106]