Question

A spherical shell has radius $r$ and at temperature $T$ . Black body radiation inside, it can be considered as an ideal gas of photons. The internal energy per unit volume $U$ varying directly with $T^{\frac{1}{4}}$ and pressure $P=\frac{U}{3}$ .If the shell is expanded adiabatically. The relation between $T$ and $r$ is $r \propto(T)$ What is the value of $100x$ ?

Answer 1

$P=\left(\frac{1}{3}\right)u=\frac{1}{3}kT^{\frac{1}{4}}$
Where $k$ is proportionality constant.
$PV=nRT$
$\therefore \frac{nRT}{V}=\frac{1}{3}kT^{\frac{1}{4}}$
$\therefore V=\frac{3 n R T}{k T^{\frac{1}{4}}}$
$\therefore V \propto T^{\frac{3}{4}}$
As, volume $\left(\right.V\left.\right) \propto r^{3}$
$\therefore r^{3} \propto T^{\frac{3}{4}}$
$\therefore r \propto T^{\frac{1}{4}}$
$\therefore T^{x}=T^{\frac{1}{4}}\Rightarrow x=0.25$
Therefore, $100x=25$