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 propT^4` and pressure `P=1/3(U/V)`. If the shell now undergoes an adiabatic expansion the relation between T and R is :

Options

A. `T prop e^(-R)`
B. `T prop e^(-3R)`
C. `T prop 1/R`
D. `T prop 1/(R^(3))`

Answer

C

Explanation

Here, `P= 1/3(U/V) and u=U/V propT^(4)`
or `u=U/V=KT^(4)`
`:.P=1/3KT^(4), As PV=nRT, :. P=(nRT)/V`
or `(nRT)/V= KT^(4) or (nR)/V=KT^(3)`
`:. V prop T^(-3) or VT^(3)= constant`
As volume, `V= 4/3 piR^(3) :. 4/3piR^(3)T^(3)=constant`
or `R^(3)T^(3)= constant`
`RT= constant or T prop 1/R`