Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.8 MIXTURES INGRAVITATIONAL ANDCENTRIFUGALFIELDS 278

We will assume the densityis uniform throughout the solution.^13 Then integration of Eq.
9.8.11yields

p^00 p^0 D

Zp 00

p^0

dpD!^2

Zr 00

r^0

rdrD

!^2

2

h

r^00

2

r^0

 2 i

(9.8.12)

where the superscripts^0 and^00 denote positions at two different values ofrin the cell. The
pressure is seen to increase with increasing distance from the axis of rotation.
Next we investigate the dependence of the solute concentrationcBonrin the equi-
librium state of the binary solution. Consider a small sample of the solution of massm.
Assume the extent of this sample in the radial direction is small enough for the variation of
the centrifugal force field to be negligible. The reversible work in the local frame needed
to move this small sample an infinitesimal distance drat constantz,T, andp, using an
external forceFcentrthat opposes the centrifugal force, is

∂w^0 DFsurdrD.Fcentr/drDm!^2 rdr (9.8.13)

This work is a contribution to the change dUof the internal energy. The Gibbs energy of
the small sample in the local frame is a function of the independent variablesT,p,nA,nB,
andr, and its total differential is

dGDd.UTSCpV /

DSdTCVdpCAdnACBdnBm!^2 rdr (9.8.14)

We use Eq.9.8.14to write the reciprocity relation

@B
@r



T;p;nA;nB

D!^2 r



@m

@nB



T;p;nA;r

(9.8.15)

Then, usingmDnAMACnBMB, we obtain

@B
@r



T;p;nA;nB

DMB!^2 r (9.8.16)

Thus at constantT,p, and composition, which are the conditions that allow the activity
ac;Bto remain constant,Bfor the sample varies withraccording to dBDMB!^2 rdr.
We integrate from radial positionr^0 to positionr^00 to obtain

B.r^00 /B.r^0 /DMB!^2

Zr 00

r^0

rdr

D^12 MB!^2

h

r^00

2

r^0

 2 i

(9.8.17)

(ac;B.r^00 /Dac;B.r^0 /)

Let us taker^0 as a reference position, such as the end of the centrifuge cell farthest from
the axis of rotation. We define the standard chemical potentialc;Bas the solute chemical

(^13) In the centrifugal field, this assumption is strictly true only if the solution is incompressible and its density is
independent of composition.