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

Answer: 25

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