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 and pressure . If the shell now undergoes an adiabatic expansion the relation between T and R is :

Engineering

Physics

Thermodynamics

Radiation Pressure due to Photon

de Broglie Hypothesis and Dual Nature

Question

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 $P = \frac{1}{3} (\frac{U}{V})$ . If the shell now undergoes an adiabatic expansion the relation between T and R is :

T µ e^–3R

T $\propto$ e^–R

$T \propto \frac{1}{R}$

$T \propto \frac{1}{R^{3}}$

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dU + dW = 0

dU + PdV = 0

$dU + \frac{U}{3 V} dV = 0$

$\frac{dU}{U} + \frac{dV}{3 V} = 0$

UV^1/3 = constant

V^4/3T⁴ = constant

V^1/3 T = constant

⇒ TR = constant