DOC.

12

THERMODYNAMIC DEDUCTION

147

T

for

specific

molecular concentrations

nAB, nA,

and

nB.

We shall call

a

state

of this

kind

a "proper thermodynamic equilibrium."

In

this

state,

radiation of the

frequency

interval dv

decomposes

a

certain

number

of molecules AB

per

unit time. The

same

number

of

molecules AB

must

be formed

anew

from the

products

of

decomposition

per

unit

time,

and in this inverse

process,

the

gas

must

emit

exactly

the

same

radiation

energy

in

the interval dv

as was

absorbed in the

decomposition.

Concerning

the radiation

energy

emitted

in the

recombination,

we

will formulate

two

additional

hypotheses,

the

exactness

of

which

seems

not to

be much in doubt

if

the

radiation

density

is

sufficiently

low.

4.

The number of

recombinations

of molecules

A and

B

per

unit time

is

independent

of

radiation

density.

5.

The

radiation

energy

of

a

given frequency

interval dv emitted

during

the

recombination

of

one gram-molecule

of

A

with

one gram-molecule

of B is

independent

of

radiation

density.

If these

conditions

are

satisfied,

then

they

entail the

following proposition:

If

(1)

P(v),2

nAB,

nA, nB

characterize

a

case

of

"proper thermodynamic equilibrium,"

then there exist

thermodynamic equilibria

that

are

characterized

by

(1a)

p'

=

-.

Vab =

xVab

Va =

VA

Vb =

VB

a

where

a

is

an

arbitrary

constant

independent

of

v,

if the

temperature

of the

gas

mixture in

case

(1a)

is the

same as

in

case (1).

In

fact,

it

follows from

hypothesis

1

and from

hypothesis

4

that the number of

molecules

decomposed

per

unit time

the number of molecules recombined

per

unit

time

are

the

same

in the

two

cases.

In

addition,

it

follows from

2

and

5

that the

distribution of

energy

between radiation and

gas

is invariable if

one

simultaneously

maintains

hypothesis

3.

Thus,

state

(1a)

has

a

stable

existence; hence,

it should be

considered

as a

state

of

thermodynamic equilibrium

which

we can designate as an

"improper thermodynamic equilibrium."

[5]

In order

to

find

the

consequences

of

our

hypotheses,

we

will write down the

equations

that

express

the fact that all the

states

(1a) are

states

of

thermodynamic

equilibrium.

To make this easier

to

do,

we

will

supplement

the

system consisting

of

gas

and radiation with

an

infinitely large

heat reservoir with which the

gas

is

in

permanent

thermal

communication

(by

conduction

of

heat). Suppose

that the

system

is

completely

isolated from the environment. In that

case one

must

have for

every

virtual

displacement:

2

p(v) is

the

density

of

black-body

radiation at

temperature

T,

corresponding

to

frequency

v.