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 ∝ T 4 and pressure p = 1 3 U V . If the shell now undergoes an adiabatic expansion the relation between T and R is:

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 = U V ∝ T 4 and pressure p = 1 3 U V . If the shell now undergoes an adiabatic expansion the relation between T and R is:, T ∝ 1 R 3, T ∝ e - R, T ∝ e - 3 R, T ∝ 1 R

MARKS App · Accepted Answer

∵ in an adiabatic process. Q = 0 So by first law of thermodynamics Q = d U + W ⇒ 0 = d U + W ⇒ W = - d U ∴ W = P d V ⇒ P d V = - d U ....(i) Given that U V ∝ T 4 ⇒ U = k V T 4 ⇒ d U = k d V T 4 = K T 4 d V + 4 T 3 V d T Also, P = 1 3 U V = 1 3 k V T 4 V = K T 4 3 Putting these values in equation (i) ⇒ K T 4 3 d V = - k T 4 d V + 4 T 3 V d T ⇒ T d V 3 = - T d V - 4 V d T ⇒ 4 T 3 d V = - 4 V d T ⇒ 1 3 d V V = - d T T ⇒ 1 3 ln ⁡ V = - ln ⁡ T ⇒ ln ⁡ V = ln ⁡ T - 3 ⇒ V T 3 = constant 4 3 π R 3 T 3 = constant R T = constant ⇒ T ∝ 1 R

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