The figure shows a polytropic process for an ideal gas. The work done by the gas in process $AB$ is

$A$ monatomic ideal gas undergoes a process in which the ratio of $P$ to $V$ at any instant is constant and equals to $1$. What is the molar heat capacity of the gas?

$2$ moles of a diatomic gas undergoes the process: $PT^2/V = \text{constant}$. The molar heat capacity of the gas during the process is:

Consider a $P - V$ diagram in which the path followed by one mole of perfect gas in a cylindrical container is shown in the figure.
$(a)$ Find the work done when the gas is taken from state $1$ to state $2$.
$(b)$ What is the ratio of temperature $\frac{T_1}{T_2}$ if $V_2 = 2V_1$?
$(c)$ Given the internal energy for one mole of gas at temperature $T$ is $\frac{3}{2}RT$,find the heat supplied to the gas when it is taken from state $1$ to $2$,with $V_2 = 2V_1$.

$A$ gas expands with temperature according to the relation $V = k T^{2/3}$. What is the work done when the temperature changes by $30^{\circ}C$ (in $R$)?

The variation of pressure $P$ with volume $V$ for an ideal diatomic gas is parabolic as shown in the figure. The molar specific heat of the gas during this process is