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  • 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&nbs

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 :

 

 

Option: 1

T\: \alpha\: e^{-R}


Option: 2

T\: \alpha\: e^{-3R}


Option: 3

T\: \alpha\: \frac{1}{R}


Option: 4

T\: \alpha\: \frac{1}{R^{3}}


Answers (1)

best_answer

\begin{array}{l} \frac{U}{V} \propto T^{4} \Rightarrow U=C V T^{4} \text { where } C \text { is constant. } \\ P=\frac{1}{3}\left(\frac{U}{V}\right)=\frac{1}{3}\left(\frac{C V T^{4}}{V}\right)=\left(\frac{C T^{4}}{3}\right) \end{array}

\text {For adiabatic expansion}, \\d Q=0 \Rightarrow d U=-d W$\\ $\Rightarrow d\left(C V T^{4}\right)=-P d V$\\ $\Rightarrow 4 C V T^{3} d T+C T^{4} d V=\frac{-C T^{4}}{3} d V$\\ $\Rightarrow 4 V d T=-T d V-\frac{T}{3} d V$ $4 V d T=-\frac{4}{3} T d V$\\ $ \Rightarrow \frac{d T}{T}=-\frac{d V}{3 V}

\begin{aligned} &\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 T R=\mathrm{constant}\\ &\Rightarrow T \propto \frac{1}{R} \end{aligned}

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