38 
THERMODYNAMICS OF RADIATION 
Stefan’s Law. 
30 . Consider radiation in an enclosure of adjustable volume, and 
change the volume from v x to v 2 and the temperature of the walls (and 
therefore of the enclosed radiation) from T 1 to T 2 ; the temperature at 
intermediate volumes is arbitrary. If E is the energy-density of the 
radiation, the heat added in a change dv, dT is 
dQ = d ( Ev) + pdv (30-1), 
or since E = 3p 
dQ = 4 pdv + 3 vdp 
= ±p^d (log p^v) (30-2). 
Hence by (27-7) the change of entropy of the radiation is 
S 2 - S 1 = 4 d (log ph). 
By the same discussion as in § 27 it follows that the integral on the 
right must be independent of the intermediate stages and hence that p^/T 
cannot be altered by varying T. But by § 29 p is a function of T only. 
Hence p^/T is a constant. Thus we can write 
p = \aT\ E = aT 4 (30-3), 
where a is a universal constant. The experimental value of a is 7-64.10 -15 
for c.g.s. units and degrees Centigrade. 
The result oc T 4 is also obtained if we set e = 2 (the appropriate 
value for radiation) in (28-3). But it has a more general meaning for 
radiation than for a gas since it is not now confined to adiabatic changes. 
(The above argument if applied to a gas would break down at the state 
ment “p is a function of T only.”) 
The result that the energy-density of radiation is proportional to the 
fourth power of the absolute temperature is known as Stefan’s Law. 
31 . Subject to certain reservations the equilibrium distribution of 
radiation in an enclosure will not be upset by admitting molecules into 
the enclosure. The waves permeate freely the spaces between the molecules 
and in the interior of the molecules, and the reduction of the volume 
occupied by radiation is insignificant. The matter in the enclosure must 
take up the same temperature as the walls and the radiation. Thus radia 
tion of density aT x will fill any region occupied by matter maintained at 
temperature T —except near the edges where the radiation is not properly 
“enclosed.” 
The reservation is necessary when the matter in the enclosure has an 
appreciable refractive index for radiation of the wave-lengths concerned. 
In this case the internal energy of the molecules and the energy of aether 
waves is so linked that the present argument is scarcely adequate. There