Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.8 MIXTURES INGRAVITATIONAL ANDCENTRIFUGALFIELDS 279

potential under standard state conditions on a concentration basis at this position. The solute
chemical potential and activity at this position are related by

B.r^0 /Dc;BCRTlnac;B.r^0 / (9.8.18)

From Eqs.9.8.17and9.8.18, we obtain the following general relation betweenBandac;B
at an arbitrary radial positionr^00 :

B.r^00 /Dc;BCRTlnac;B.r^00 /^12 MB!^2

h

r^00

2

r^0

 2 i

(9.8.19)

We found earlier that when the solution is in an equilibrium state,Bis independent
ofr—that is,B.r^00 /is equal toB.r^0 /for any value ofr^00. When we equate expressions
given by Eq.9.8.19forB.r^00 /andB.r^0 /and rearrange, we obtain the following relation
between the activities at the two radial positions:

ln

ac;B.r^00 /

ac;B.r^0 /

D

MB!^2

2RT

h

r^00

2

r^0

 2 i

(9.8.20)

(solution in centrifuge

cell at equilibrium)

The solute activity is related to the concentrationcBbyac;B Dc;B (^) c;BcB=c. We
assume the solution is sufficiently dilute for the activity coefficient (^) c;Bto be approximated
by 1. The pressure factor is given byc;Bexp



VB^1 .pp/=RT



(Table9.6). These
relations give us another expression for the logarithm of the ratio of activities:

ln

ac;B.r^00 /

ac;B.r^0 /

D

VB^1 .p^00 p^0 /

RT

Cln

cB.r^00 /

cB.r^0 /

(9.8.21)

We substitute forp^00 p^0 from Eq.9.8.12. It is also useful to make the substitutionVB^1 D
MBv^1 B, wherevB^1 is the partial specific volume of the solute at infinite dilution (page 234 ).
When we equate the two expressions for lnåac;B.r^00 /=ac;B.r^0 /ç, we obtain finally

ln

cB.r^00 /

cB.r^0 /

D

MB

1 v^1 B

!^2

2RT

h

r^00

2

r^0

 2 i

(9.8.22)

(solution in centrifuge

cell at equilibrium)

This equation shows that if the solution densityis less than the effective solute density
1=vB^1 , so thatvB^1 is less than 1, the solute concentration increases with increasing distance
from the axis of rotation in the equilibrium state. If, however,is greater than1=vB^1 , the
concentration decreases with increasingr. The factor

1 v^1 B

is like a buoyancy factor
for the effect of the centrifugal field on the solute.
Equation9.8.22is needed forsedimentation equilibrium, a method of determining the
molar mass of a macromolecule. A dilute solution of the macromolecule is placed in the cell
of an analytical ultracentrifuge, and the angular velocity is selected to produce a measurable
solute concentration gradient at equilibrium. The solute concentration is measured optically
as a function ofr. The equation predicts that a plot of ln.cB=c/versusr^2 will be linear,
with a slope equal toMB

1 v^1 B

!^2 =2RT. The partial specific volumevB^1 is found
from measurements of solution density as a function of solute mass fraction (page 234 ). By
this means, the molar massMBof the macromolecule is evaluated.