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 $E = \frac{U}{V} \propto T^4$ 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:

An ideal gas undergoes a process $A \rightarrow B \rightarrow C \rightarrow A$ cycle. The process $A \rightarrow B$ is adiabatic. Calculate the work done in the process $A \rightarrow B$.

An ideal gas undergoes an adiabatic process from state $(P_1, V_1, T_1)$ to $(P_2, V_2, T_2)$. The work done during the process is ..... (where $\mu$ = number of moles,$C_P$ and $C_V$ = molar specific heats).

$A$ gas is suddenly compressed to one fourth of its original volume. What will be its final pressure,if its initial pressure is $P$?

Identify the characteristics of an adiabatic process in a monoatomic gas.
$(A)$ Internal energy is constant.
$(B)$ Work done in the process is equal to the change in internal energy (in magnitude).
$(C)$ The product of temperature and volume is a constant.
$(D)$ The product of pressure and volume is a constant.
$(E)$ The work done to change the temperature from $T_1$ to $T_2$ is proportional to $(T_2 - T_1)$.
Choose the correct answer from the options given below.

$Assertion :$ Air quickly leaking out of a balloon becomes cooler.
$Reason :$ The leaking air undergoes adiabatic expansion.