One mole of a diatomic gas does a work $\frac{Q}{3}$,when the amount of heat supplied is $Q$. In this process,the molar heat capacity of the gas is:

When heat $Q$ is supplied to a monoatomic gas at constant pressure,the work done by the gas is:

One mole of a monoatomic ideal gas goes through a thermodynamic cycle,as shown in the volume versus temperature $(V-T)$ diagram. The correct statement$(s)$ is/are :
[$R$ is the gas constant]
$(1)$ Work done in this thermodynamic cycle $(1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 1)$ is $|W| = \frac{1}{2} RT_0$
$(2)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $2 \rightarrow 3$ is $\left|\frac{Q_{1 \rightarrow 2}}{Q_{2 \rightarrow 3}}\right| = \frac{5}{3}$
$(3)$ The above thermodynamic cycle exhibits only isochoric and adiabatic processes.
$(4)$ The ratio of heat transfer during processes $1 \rightarrow 2$ and $3 \rightarrow 4$ is $\left|\frac{Q_{1 \rightarrow 2}}{Q_{3 \rightarrow 4}}\right| = \frac{1}{2}$

Three samples of the same gas $A, B$ and $C$ $(\gamma = 3/2)$ have initially equal volume. Now the volume of each sample is doubled. The process is adiabatic for $A$,isobaric for $B$,and isothermal for $C$. If the final pressures are equal for all three samples,the ratio of their initial pressures is:

One mole of an ideal gas $\left( \frac{C_p}{C_v} = \gamma \right)$ is heated according to the law $P = \alpha V$,where $P$ is the pressure of the gas,$V$ is the volume,and $\alpha$ is a constant. What is the molar heat capacity of the gas in this process?

Two cylinders $A$ and $B$ fitted with pistons contain equal number of moles of an ideal monoatomic gas at $400 \,K$. The piston of $A$ is free to move while that of $B$ is held fixed. The same amount of heat energy is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $42 \,K$, what is the rise in temperature of the gas in $B$ (in $\,K$)? (Given $\gamma = 5/3$)