34
THERMODYNAMICS OF RADIATION
By (25-2) the change of entropy of the gas is
S 2 - S x = 8S = [ d ( T) dQ (27*4)
= | 9? J d ( T ) dT + J d (T) pdv
= §fni l2 d(T)dT + 91P T9 ( T) - ...(27-5).
J T x JVi. v
Now $ 2 — S x can depend only on the initial and final temperatures and
volumes of the gas. This follows from the theory of gases according to
which two specimens of the same gas at the same temperature and density
are alike in all their properties. Or it can be deduced more generally from
the reversibility. For if S 2 — S x were different according to the intermediate
values of v and T, we could by taking the gas out by one route and back
by another increase its entropy. Owing to the reversibility no entropy
is created; hence the increase of entropy requires a decrease of entropy
of our reservoirs of heat. The cycle could be repeated any number of times
so that a small mass of gas would be able to furnish an infinite decrease
of S (i.e. increase of organisation) to the rest of the universe.
The expression on the right of (27-5) must therefore depend only on
the initial and final stages. The first term evidently satisfies this; and
therefore the second integral must in spite of appearances be independent
of the intermediate stages. This requires that Td ( T) shall be a constant.
To prove this, the second integral can be written
r log v 2
| Td ( T ) d (log v).
• log Vi
Consider any elementary step d (log v). During this change of volume the
gas can have any temperature we please; so that if it is possible to vary
Td (T) by varying T, we can vary the contribution made to the integral
by this step. Thus the integral cannot be independent of the intermediate
conditions of the gas unless
Td (T) = const.
By suitably connecting the units of entropy and temperature the
constant may be set equal to unity so that
d{T) = HT (27-6).
Here T is identified with the temperature on the scale of a perfect-gas
thermometer; but, of course, the value of d (T) here found is applicable
to “ordinary” heat transferred from or to any kind of material in ac
cordance with (26T).
Equation (27-4) can now be written
S 2 - S, = f dQ/T
(27-7).
