Question

Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V}\propto T^4$ and pressure $P=\frac{1}{3}\left(\frac{U}{V}\right)$ . If the shell now undergoes an adiabatic expansion the relation between $T$ and $R$ is

• A. $T\propto e^{-R}$
• B. $T\propto e^{-3R}$
• C. $T\propto \frac{1}{R}$
• D. $T\propto \frac{1}{R^3}$

Answer 1

The correct option is C $T\propto \frac{1}{R}$

$\frac{U}{V}\propto T^4$ $\Rightarrow U=CV{T}^4$ where C is constant.
$P=\frac{1}{3}\left(\frac{U}{V}\right)=\frac{1}{3}\left(\frac{CV{T}^4}{V}\right)=\left(\frac{C{T}^4}{3}\right)$

For adiabatic expansion, $dQ=0\Rightarrow dU=-dW$
$\Rightarrow d\left(CV{T}^4\right)=-PdV$
$\Rightarrow 4CV{T}^{3}dT+C{T}^{4}dV=\frac{-C{T}^4}{3}dV$
$\Rightarrow 4VdT=-TdV-\frac{T}{3}dV4VdT=-\frac{4}{3}TdV\frac{dT}{T}=-\frac{dV}{3V}\ln T=-\frac{1}{3}\ln V+\ln C^{{}^{\prime }}\ln T{V}^{1/3}=\ln C^{{}^{\prime }}T{V}^{\frac{1}{3}}=C^{{}^{\prime }}T{\left(\frac{4}{3}\pi R^3\right)}^{\frac{1}{3}}=C^{{}^{\prime }}$
$\Rightarrow TR=$ constant
$\Rightarrow T\propto \frac{1}{R}$