Heat and Thermodynamics 6 Question 1
5. 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
(2015 Main)
(a) $T \propto e^{-R}$
(b) $T \propto \frac{1}{R}$
(c) $T \propto e^{-3 R}$
(d) $T \propto \frac{1}{R^{3}}$
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Answer:
Correct Answer: 5. (b)
Solution:
- Given, $\frac{U}{V} \propto T^{4}$
$$ \frac{U}{V}=\alpha T^{4} $$
It is also given that, $\quad P=\frac{1}{3}\left(\frac{U}{V}\right)$
$$ \begin{array}{ll} \Rightarrow & \frac{n R _0 T}{V}=\frac{1}{3}\left(\alpha T^{4}\right) \quad\left(R _0=\text { Gas constant }\right) \\ \text { or } & V T^{3}=\frac{3 n R _0}{\alpha}=\text { constant } \end{array} $$
$$ \begin{aligned} & \therefore & \left(\frac{4}{3} \pi R^{3}\right) T^{3} & =\text { constant or } R T=\text { constant } \\ & \therefore & T & \propto \frac{1}{R} \end{aligned} $$
