103.

A further example of the application of the two principles of thermodynamics is afforded by the irreversible expansion of originally black radiation of volume $V$ and temperature $T$ to the larger volume $V^{\prime }$ as considered above in [sect:70.] Sec. 70, but in the absence of any absorbing or emitting substance whatever. Then not only the total energy but also the energy of every separate frequency $\nu$ remains constant; hence, when on account of diffuse reflection from the walls the radiation has again become uniform in all directions, ${𝗎}_{\nu }V={𝗎}_{\nu }^{\prime }{V}^{\prime }$; moreover by this relation, according to [eqn:(118)] (118), the temperature $T_{\nu }$' of the monochromatic radiation of frequency $\nu$ in the final state is determined. The actual calculation, however, can be performed only with the help of equation [eqn:(275)] (275) (see below). The total entropy of radiation, i.e., the sum of the entropies of the radiations of all frequencies, $V^{\prime }{\int }_0^{\infty }{𝗌}_{\nu }^{\prime }d\nu ,$ must, according to the second principle, be larger in the final state than in the original state. Since $T_{\nu }^{\prime }$ has different values for the different frequencies $\nu$, the final radiation is no longer black. Hence, on subsequent introduction of a carbon particle into the cavity, a finite change of the distribution of energy is obtained, and simultaneously the entropy increases further to the value $S^{\prime }$ calculated in [eqn:(82)] (82).