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 = U/V ∝ T⁴ and pressure p = 1/3(U/V). If the shell now undergoes an adiabatic expansion the relation between T and R is: (a) T ∝ 1/R (b) T ∝ 1R³ (c) T ∝ e-R (d) T ∝ e-3R

Answer: (a). Adiabatic: dU = -p dV, with U = aT⁴V and p = 1/3aT⁴. 4aT³V dT + aT⁴ dV = -1/3aT⁴ dV ⇒ dT/T = -1/3dV/V So T ∝ V-1/3, and V ∝ R³ gives T ∝ 1/R. So the answer is (a).

Consider a spherical shell of radius R at temperature T. The black… | JEE Main 2015

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Thermodynamic Processes

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:

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Answer: (a)

Adiabatic: $dU = -p\,dV$ , with $U = aT^4V$ and $p = \frac{1}{3}aT^4$ .
$4aT^3V\,dT + aT^4\,dV = -\frac{1}{3}aT^4\,dV \Rightarrow \frac{dT}{T} = -\frac{1}{3}\frac{dV}{V}$
So $T \propto V^{-1/3}$ , and $V \propto R^3$ gives $T \propto \frac{1}{R}$ .
So the answer is (a).

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