During an experiment, an ideal gas is found to obey a condition $VP^2 = \text{constant}$. The gas is initially at a temperature $T$, pressure $P$, and volume $V$. The gas expands to volume $4V$. Which of the following statements is correct?

For a certain process, the pressure of a diatomic gas varies according to the relation $P = aV^2$, where $a$ is a constant. What is the molar heat capacity of the gas for this process?

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 $V_2 = 2V_1$.

An ideal gas follows a process described by $p V^2 = C$ from $(p_1, V_1, T_1)$ to $(p_2, V_2, T_2)$,where $C$ is a constant. Then,

$A$ 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 pressure and volume of an ideal gas are related as $PV^{3/2} = K$ (constant). The work done when the gas is taken from state $A(P_1, V_1, T_1)$ to state $B(P_2, V_2, T_2)$ is: