Simple Nature - Light and Matter

into an ice cube sitting in some warm water!

By the way, note that although we’ve redefined temperature,

these examples show that things are coming out consistent with the

old definition, since we saw that the old definition of temperature

could be described in terms of the average energy per atom, and

here we’re finding that equilibration results in each subset of the

atoms having an equal share of the energy.

Entropy of a monoatomic ideal gas
Let’s calculate the entropy of a monoatomic ideal gas ofnatoms.
This is an important example because it allows us to show that
our present microscopic treatment of thermodynamics is consistent
with our previous macroscopic approach, in which temperature was
defined in terms of an ideal gas thermometer.
The number of possible locations for each atom isV/∆x^3 , where
∆xis the size of the space cells in phase space. The number of pos-
sible combinations of locations for the atoms is therefore (V/∆x^3 )n.
The possible momenta cover the surface of a 3n-dimensional
sphere, whose radius is

√

2 mE, and whose surface area is therefore

proportional toE(3n−1)/^2. In terms of phase-space cells, this area

corresponds toE(3n−1)/^2 /∆p^3 npossible combinations of momenta,

multiplied by some constant of proportionality which depends on

m, the atomic mass, andn, the number of atoms. To avoid having

to calculate this constant of proportionality, we limit ourselves to

calculating the part of the entropy that does not depend onn, so

the resulting formula will not be useful for comparing entropies of

ideal gas samples with different numbers of atoms.

The final result for the number of available states is

M=

(

V

∆x^3

)n

E(3n−1)/^2

∆p^3 n−^1

, [function ofn]

so the entropy is

S=nklnV+

3

2

nklnE+ (function of ∆x, ∆p, andn),

where the distinction betweennandn−1 has been ignored. Using
PV=nkTandE= (3/2)nkT, we can also rewrite this as

S=

5

2

nklnT−nklnP+..., [entropy of a monoatomic ideal gas]

where “...” indicates terms that may depend on ∆x, ∆p,m, and
n, but that have no effect on comparisons of gas samples with the
same number of atoms.
self-check C
Why does it make sense that the temperature term has a positive sign
in the above example, while the pressure term is negative? Why does

Section 5.4 Entropy as a microscopic quantity 331