Physical Chemistry Third Edition

10.4 Transport Processes in Liquids 469

particles (particles roughly 3 to 1000 nm in diameter) nearly obey Stokes’ law, and their

effective radii are nearly equal to their actual radii.

Around 1905 Einstein devised a theory ofBrownian motion, the irregular motion

of colloidal particles suspended in a liquid. Einstein assumed that a colloidal particle

is bombarded randomly by the molecules of the solvent and was able to show for a

spherical colloidal particle that the mean-square displacement of the particle in thez

direction in a timetis given by

〈

z^2

〉



kBT

3 πηr

t (10.4-3)

wherekBis Boltzmann’s constant,Tis the absolute temperature,ris the radius of

the particle, andηis the viscosity of the solvent. Comparison of this equation with

Eq. (10.2-19) shows that the diffusion coefficient of a macromolecular or colloidal

substance is given by

D 2 

kBT

f



kBT

6 πηreff

(macromolecular or colloidal substance) (10.4-4)

Using a dark-field microscope, Perrin was able in 1908 to measure repeatedly the

displacements of colloidal particles and verified Eq. (10.4-3) experimentally. For many

skeptics this was considered to be the definitive verification of the existence of atoms

and molecules, since Einstein’s derivation of Eq. (10.4-3) depended on the assumption

that the colloidal particle was bombarded randomly by solvent molecules. Perrin was

able to obtain an approximate value of Boltzmann’s constant from Eq. (10.4-3), and

thus calculated a value of Avogadro’s constant using the known value of the ideal gas

constant.

Brownian motion is named for Robert
Brown, 1773–1858, the most prominent
British botanist of his time. He observed
a jittery motion of pollen grains and
showed that this phenomenon was not
due to biological motility by observing it
in mineral particles, including some
taken from the Sphinx. Brown is also
credited with discovering the cell
nucleus around 1833.

Jean Baptiste Perrin, 1870–1942, was a
French physicist.

EXAMPLE10.16

The diffusion coefficient of hemoglobin in water at 20◦C is equal to 6. 9 × 10 −^11 m^2 s−^1.

Assuming the hemoglobin molecule to be spherical, calculate its effective radius. The vis-

cosity coefficient of water at this temperature is equal to 1. 002 × 10 −^3 kg m−^1 s−^1.

Solution

r

kBT

6 πηD



(

1. 3807 × 10 −^23 JK−^1

)

(293 K)

6 π

(

1. 002 × 10 −^3 kg m−^1 s−^1

)(

6. 9 × 10 −^11 m^2 s−^1

)

 3. 1 × 10 −^9 m3100 pm31 Å

Exercise 10.15

a.Estimate the molar volume of hemoglobin from the molecular size in the previous example.

b.Human hemoglobin has a density of 1.335 g mL−^1 (a little larger than typical protein densities,

which run around 1.25 g mL−^1 ). It has a molar mass of 68 kg mol−^1. Calculate its molar

volume and compare with your answer from part a.