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 $$u = {U \over V}\, \propto \,{T^4}$$ and pressure $$p = {1 \over 3}\left( {{U \over V}} \right)$$ . If the shell now undergoes an adiabatic expansion the relation between $$T$$ and $$R$$ is:

Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as $${V^q},$$ where $$V$$ is the volume of the gas. The value of $$q$$ is: $$\left( {\gamma = {{{C_p}} \over {{C_v}}}} \right)$$

A solid body of constant heat capacity $$1$$ $$J/{}^ \circ C$$ is being heated by keeping it in contact with reservoirs in two ways:
$$(i)$$ Sequentially keeping in contact with $$2$$ reservoirs such that each reservoir
$$\,\,\,\,\,\,\,\,$$supplies same amount of heat.
$$(ii)$$ Sequentially keeping in contact with $$8$$ reservoirs such that each reservoir
$$\,\,\,\,\,\,\,\,\,\,$$supplies same amount of heat.
In both the cases body is brought from initial temperature $${100^ \circ }C$$ to final temperature $${200^ \circ }C$$. Entropy change of the body in the two cases respectively is :

One mole of a diatomic ideal gas undergoes a cyclic process $$ABC$$ as shown in figure. The process $$BC$$ is adiabatic. The temperatures at $$A, B$$ and $$C$$ are $$400$$ $$K$$, $$800$$ $$K$$ and $$600$$ $$K$$ respectively. Choose the correct statement :