$A$ monoatomic gas performs a work of $\frac{Q}{4}$,where $Q$ is the heat supplied to it. The molar heat capacity of the gas during this transformation will be $xR$,where $R$ is the gas constant. Find the value of $x$.

The initial pressure and volume of a gas are $P$ and $V$ respectively. First,the gas is expanded to a volume of $9V$ by an isothermal process,and then it is compressed to a volume of $V$ by an adiabatic process. What is its final pressure (in $P$)? (Ratio of specific heat at constant pressure to constant volume $\gamma = \frac{3}{2}$)

The figure below shows two paths that may be taken by a gas to go from a state $A$ to a state $C.$ In process $AB,$ $400 \, J$ of heat is added to the system and in process $BC,$ $100 \, J$ of heat is added to the system. The heat absorbed by the system in the process $AC$ will be ...... $J$

Two cylinders of equal size are filled with an equal amount of ideal diatomic gas at room temperature. Both cylinders are fitted with pistons. In cylinder $A$,the piston is free to move (isobaric process),while in cylinder $B$,the piston is fixed (isochoric process). When the same amount of heat is supplied to both cylinders,the temperature of the gas in cylinder $A$ rises by $30\, K$. What will be the rise in temperature of the gas in cylinder $B$ (in $;K$)?

An ideal gas goes through a reversible cycle $a \to b \to c \to d$ and has the $V - T$ diagram shown below. Processes $d \to a$ and $b \to c$ are adiabatic. The corresponding $P - V$ diagram for the process is (all figures are schematic and not drawn to scale):

Given $P_A = 3 \times 10^4 \, Pa$,$P_B = 8 \times 10^4 \, Pa$,$V_A = 2 \times 10^{-3} \, m^3$,and $V_D = 5 \times 10^{-3} \, m^3$. An ideal gas absorbs $600 \, J$ of heat in the process $AB$ and $200 \, J$ of heat in the process $BC$. Find the change in internal energy between $A$ and $C$ in $J$.