D O C . 2 8 3 Q U A N T U M T H E O R Y O F I D E A L G A S 2 7 7

. (1a)

The number Δs of cells that belong to a particular elementary region ΔE of the en-

ergy is consequently

. (2)

Given an arbitrarily small , one can always choose V sufficiently large so that

Δs be a very large number.

§2. Probability of State and Entropy

We now define the macroscopic state of the gas.

Let there now be present in the volume Vn molecules of mass m. Δn of them have

energy values between E and E + ΔE. These are distributed among the Δs cells.

Among the Δs cells, one finds contained in

p0Δs, no molecule,

p1Δs, 1 molecule,

p2Δs, 2 molecules,

etc.

The probabilities belonging to the

sth

cell are then obviously[3] functions of the

number of cells s and the integral index r, and they should therefore be more com-

pletely designated in the following as . Obviously, for all s,

. (3)

For given and given Δn, the number of possible distributions of the Δn mole-

cules over the energy region under consideration is equal to

,

which according to Stirling’s theorem and equation (3) can be substituted by

,

for which one can also set the product

Φ V

4

3

--π( - 2mE)3 /2 ⋅ =

[p. 262]

Δs 2π----

V

h3

-(2m)3 /2E1 /2ΔE =

ΔE

E

-------

pr

pr

s

s

r

¦pr

1 =

pr s

Δs!

pr sΔs)! (

r 0 =

r=∞

∏

----------------------------

1

sΔsprs

r

∏pr

--------------------- -