Two cylinders $A$ and $B$ are fitted with pistons and contain equal amounts of a diatomic gas at $300 \ K$. The piston of cylinder $A$ is free to move,while the piston of cylinder $B$ is fixed. If the same amount of heat is supplied to each cylinder,the temperature of gas in $A$ increases by $30 \ K$. What is the increase in the temperature of gas in $B$ (in $K$)?

An engine (consisting of one mole of an ideal gas in a cylinder with a piston) follows the cycle shown in the figure. Find the heat exchanged by the engine with the surroundings in each part of the cycle. Given $C_V = \frac{3}{2}R$.
$(a)$ $A$ to $B$: Constant volume
$(b)$ $B$ to $C$: Constant pressure
$(c)$ $C$ to $D$: Adiabatic expansion
$(d)$ $D$ to $A$: Constant pressure

If a heat engine and a refrigerator are working between the same two temperatures $T_1$ and $T_2$ $(T_1 > T_2)$,then the ratio of the efficiency of the heat engine to the coefficient of performance of the refrigerator is:

Suppose an ideal gas ($n$ moles) undergoes an expansion process $P = f(V)$ which passes through the point $(V_0, P_0)$. If the slope of the curve $P = f(V)$ is greater than the slope of the adiabatic curve passing through $(V_0, P_0)$,show that the gas absorbs heat at $(V_0, P_0)$.

The figure shows a cylindrical adiabatic container of total volume $2V_0$ divided into two equal parts by a conducting piston (which is free to move). Each part contains an identical gas at pressure $P_0$. Initially,the temperature of the left and right parts is $4T_0$ and $T_0$ respectively. An external force is applied on the piston to keep it at rest. Find the value of the external force required when thermal equilibrium is reached. ($A =$ Area of the piston)

Two samples of gas $A$ and $B$ are initially at the same pressure and temperature. They are compressed from volume $V$ to $V/2$. If $A$ is compressed isothermally and $B$ is compressed adiabatically,then the final pressure of $A$ is: