Page 111 - Modern physical chemistry

s. J 3 Concentration Gradients 101

where TJ is the viscosity of the ambient fluid. This is lrnown as Stokes' law.
For one mole of the ith constituent, we have the force

[5.107]
The work done by this force is dissipated. But at constant P and T, the dissipated
energy is Gibbs energy; we have

a p . )
-dpi = - [ -~ dx = .Fi dx. [5.108]
,
ax TP
The force Fi derived from the chemical potential Pi is lrnown as the thermodynamicjorce
acting on the ith constituent.
We will find that the chemical potential is related to the activity a i of the constituent
by the logarithmic relationship:
Pi = PY + RTlnai· [5.109]
Furthermore, the activity is approximately represented by the concentration ci :

Pi ~ PY + RT In Ci . [5.110]
Substituting the derivative of (5.110) into the expression for force Fi from (5.108)
gives us
Fi = _[a Pi ) = _RT[alnC i ) . [5.111]
ax T,P ax T,P
But combining equations (5.107) and (5.105) yields

.Fi = N6mrri[-Di alnCi) . [5.112]
ax T,P

Eliminating Fi from (5.111) and (5.112), then solving for the diffusion coefficient leads
to the result
kT
D.= RT --, [5.113]
~ N67r1}ri 6mrri
which is lrnown as the Einstein-Stokes relationship.

ExampleS.7
Along the axis of a container, the concentration of a solution drops exponentially,
decreasing by one-half in each 10.0 cm section. Calculate the thermodynamic force acting
on the solute at 25° C.
The concentration distribution parallel to the axis of the container is given as

or
lnC=lnco-ax,
whence
alnc
--=-a
ax