DISSERTATION ON MOLECULAR DIMENSIONS 177

Einstein's other fundamental

equation

(the

third

equation on p.

21

of Einstein

1905j

[Doc. 15])

follows from

an expression

for the coefficient

of

diffusion D

of

the solute. This

expression

is

obtained from

Stokes's

law for

a sphere

of

radius

P

moving

in

a

liquid,

and

Van

't Hoff's

law for the osmotic

pressure:

R

T

1

D

=

*77^

(2)

6

it

k

N

P

where R is the

gas

constant,

T the absolute

temperature,

and N

Avogadro's

number.

The derivation

of

eq.

(1),

technically

the

most

complicated part

of

Einstein's

thesis,

presupposes

that the motion

of

the fluid

can

be

described

by

the

hydrodynamical

equations

for

stationary

flow

of

an incompressible

homogeneous

liquid,

even

in the

presence

of

solute

molecules;

that the inertia

of

these molecules

can

be

neglected;

that

they

do

not

affect each

other's

motions;

and that

they can

be

treated

as rigid spheres moving

in

the

fluid without

slipping,

under the sole influence

of

hydrodynamical

stresses.[58]

The

hydro-

dynamic techniques

needed

are

derived from

Kirchhoff 1897,

a

book that Einstein first

read

during

his

student

years.[59]

Eq.

(2)

follows from the conditions

for the

dynamical

and

thermodynamical equilibrium

of

the fluid. Its derivation

requires

the identification

of

the force

on a single molecule,

which

appears

in

Stokes's

law, with

the

apparent

force due to

the

osmotic

pressure

(see

Einstein

1905j [Doc. 15],

p.

20).

The

key

to

handling

this

problem

is

the introduction

of

fictitious

countervailing

forces. Einstein had earlier introduced such fictitious forces:

they

are

used in

Einstein

1902a

(Doc.

2)

to counteract

thermodynamical

effects in

proving

the

applicability

to diffusion

phenomena

of

a generalized

form

of

the second law

of

thermo-

dynamics;[60]

they

are

also

used in his

papers on

statistical

physics.[61]

Einstein's

derivation

of

eq. (2)

does

not

involve the theoretical tools

he

developed in

his

work

on

the statistical foundations

of

thermodynamics;

he reserved

a more

elaborate

derivation,

using

these

methods,

for his first

paper

on

Brownian

motion.[62]

Eq.

(2)

was

derived

independently,

in

somewhat

more general

form,

by

Sutherland

in 1905.[63]

To deal

with

the

available

empirical

data,

Sutherland had

to

allow for

a varying

coefficient

of

sliding

friction between the

diffusing

molecule and the solution.

The basic elements

of Einstein's method-the

use

of

diffusion

theory

and the

applica-

tion

of

hydrodynamical techniques

to

phenomena involving

the atomistic

constitution of

[58]

Einstein's

derivation

is

only

valid for

Couette

flow;

a generalization

to Poiseuille flow

is

given

in Simha 1936. For

a

discussion

of

Ein-

stein's

assumptions, see

Pais

1982,

p.

90.

[59]

See Einstein to Mileva

Maric,

29

July

1900 and

1

August

1900

(Vol.

1,

Docs. 68 and

69).

[60]

For

a

discussion

of

this

generalization, see

the editorial note,

"Einstein

on

the Nature of

Molecular Forces,"

p.

8.

[61]

See, in

particular,

Einstein 1902b

(Doc.

3),

§

10.

[62]

This

derivation, given

in

Einstein 1905k

(Doc. 16),

§

3,

is cited

in

a

footnote to Einstein

1905j (Doc. 15), p. 20,

that

was presumably

added after the former

paper

had

appeared.

In

the

same paper,

he also used

eq. (2)

to

study

the

relation between diffusion and fluctuations.

[63]

See

Sutherland

1905,

pp.

781-782.