102.

Let us make one more simple application of the laws just found to the special case of black radiation. For this, according to [eqn:(81)] (81), the total space density of entropy is $s=\frac{4}{3}{a}^{3}T.\mathrm{\backslash Label}[eqn](138)\text{upshape (138)}$ Hence, according to [eqn:(132)] (132), the specific intensity of the total entropy radiation in any direction is $L=\frac{c}{3\pi }a{T}^3,\mathrm{\backslash Label}[eqn](139)\text{upshape (139)}$ and the total entropy radiation through an element of area $d\sigma$ toward one side is, according to [eqn:(128)] (128), $\frac{c}{3}a{T}^{3}d\sigma dt.\mathrm{\backslash Label}[eqn](140)\text{upshape (140)}$ As a special example we shall now apply the two principles of thermodynamics to the case in which the surface of a black body of temperature $T$ and of infinitely large heat capacity is struck by black radiation of temperature $T^{\prime }$ coming from all directions. Then, according to [eqn:(7)] (7) and [eqn:(76)] (76), the black body emits per unit area and unit time the energy $\pi K=\frac{ac}{4}{T}^4,$ and, according to [eqn:(140)] (140), the entropy $\frac{ac}{3}{T}^3.$ On the other hand, it absorbs the energy $\frac{ac}{4}{T}^{\prime 4}$ and the entropy $\frac{ac}{3}{T}^{\prime 3}.$ Hence, according to the first principle, the total heat added to the body, positive or negative according as $T^{\prime }$ is larger or smaller than $T$, is $Q=\frac{ac}{4}{T}^{\prime 4}-\frac{ac}{4}{T}^4=\frac{ac}{4}(T^{\prime 4}-T^4),$

and, according to the second principle, the change of the entire entropy is positive or zero. Now the entropy of the body changes by ${\displaystyle \frac{Q}{T}}$, the entropy of the radiation in the vacuum by $\frac{ac}{3}(T^3-T^{\prime 3}).$ Hence the change per unit time and unit area of the entire entropy of the system considered is $\frac{ac}{4}\frac{T^{\prime 4}-T^4}{T}+\frac{ac}{3}(T^3-T^{\prime 3})\geqq 0.$ In fact this relation is satisfied for all values of $T$ and $T^{\prime }$. The minimum value of the expression on the left side is zero; this value is reached when $T=T^{\prime }$. In that case the process is reversible. If, however, $T$ differs from $T^{\prime }$, we have an appreciable increase of entropy; hence the process is irreversible. In particular we find that if $T=0$ the increase in entropy is $\infty$, i.e., the absorption of heat radiation by a black body of vanishingly small temperature is accompanied by an infinite increase in entropy and cannot therefore be reversed by any finite compensation. On the other hand for $T^{\prime }=0$, the increase in entropy is only equal to ${\displaystyle \frac{ac}{12}}{T}^3$, i.e., the emission of a black body of temperature $T$ without simultaneous absorption of heat radiation is irreversible without compensation, but can be reversed by a compensation of at least the stated finite amount. For example, if we let the rays emitted by the body fall back on it, say by suitable reflection, the body, while again absorbing these rays, will necessarily be at the same time emitting new rays, and this is the compensation required by the second principle.

Generally we may say: Emission without simultaneous absorption is irreversible, while the opposite process, absorption without emission, is impossible in nature.