Physical Chemistry Third Edition

472 10 Transport Processes

Exercise 10.16

From data in the appendix, calculate the activation energy for the viscosity of glycerol.

Sedimentation

Macromolecules such as proteins can sediment though a solvent in an ultracentrifuge,

which can turn at speeds of several thousand revolutions per second. An object of mass

mthat is at a distancerfrom the axis of rotation is maintained in a circular orbit by a

centripetal force that is given by the formula

Fcmrω^2 (10.4-8)

whereωis theangular speedmeasured in radians per second. The rate of rotation

measured in revolutions per second is equal toω/ 2 π, since one revolution is equal to

2 πradians.

The rotor of an ultracentrifuge has a sample cell with a transparent top and bottom

so that a beam of light can pass through the cell each time it passes the location of

the beam. The value ofr, the distance of the molecules from the axis of rotation,

can be measured by observing the position dependence of the index of refraction,

which depends on composition. After the rotor has been spinning a short time, the

sedimentation speedvseddr/dtwill attain a steady value so that the frictional force

provides the centripetal force given by Eq. (10.4-8).

Since a macromolecule is immersed in a solvent its centripetal force must be cor-

rected for buoyancy, giving a net centripetal force:

Fc(m 2 −m 1 )rω^2 (10.4-9)

wherem 2 is the molecular mass of the macromolecular substance andm 1 is the mass

of solvent displaced by the molecule. Ifρ 1 is the density of the solvent andρ 2 is the

density of the macromolecular substance,

m 1 

ρ 1

ρ 2

m 2 

ρ 1

ρ 2

M 2

NAv

(10.4-10)

whereM 2 is the molar mass of the macromolecular substance andNAvis Avogadro’s

constant. We assume that the frictional force given by Eq. (10.4-1) applies to the

sedimenting molecules and equate the magnitude of the frictional force to the centripetal

force:

fvsed

M 2

NAv

(

1 −

ρ 1

ρ 2

)

rω^2 (10.4-11)

Thesedimentation coefficientSis defined as the ratio ofvsedto the centrifugal

acceleration,rω^2 :

S

vsed

rω^2



M 2

NAv

1 −ρ 1 /ρ 2

f

(10.4-12)

The sedimentation coefficient has the units of seconds and is usually approximately

equal to 10−^13 s. The practical unit of sedimentation coefficients is thesvedberg, defined

so that 1 svedberg 10 −^13 s.

The svedberg unit is named for Theodor
Svedberg, 1884–1971, a Swedish
biophysical chemist who received the
1926 Nobel Prize for his work on
colloids and protein suspensions.