67.

If, on raising the piston, the temperature of the black body forming the bottom is kept constant by a corresponding addition of heat from the heat reservoir, the process takes place isothermally.

Then, along with the temperature $T$ of the black body, the energy density $u$, the radiation pressure $p$, and the density of the entropy $s$ also remain constant; hence the total energy of radiation increases from $U=uV$ to $U^{\prime }=u{V}^{\prime }$, the entropy from $S=sV$ to $S^{\prime }=s{V}^{\prime }$ and the heat supplied from the heat reservoir is obtained by integrating [eqn:(72)] (72) at constant $T$,

$Q=T(S^{\prime }-S)=Ts(V^{\prime }-V)$

or, according to [eqn:(81)] (81) and [eqn:(75)] (75),

$Q=\frac{4}{3}a{T}^4(V^{\prime }-V)=\frac{4}{3}(U^{\prime }-U).$

Thus it is seen that the heat furnished from the outside exceeds the increase in energy of radiation $(U^{\prime }-U)$ by $\frac{1}{3}(U^{\prime }-U)$. This excess in the added heat is necessary to do the external work accompanying the increase in the volume of radiation.