$A$ perfect gas is found to obey the relation $PV^{3/2} =$ constant,during an adiabatic process. If such a gas,initially at a temperature $T$,is compressed adiabatically to half its initial volume,then its final temperature will be

Work done to increase the temperature of one mole of an ideal gas by $30^{\circ} C$, if it is expanding under the condition $V \propto T^{2/3}$ is, $(R = 8.314 \ J/mol \cdot K)$ (in $J$)

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

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 $V_2 = 2V_1$.

$A$ diatomic gas follows the equation $PV^m =$ constant during a process. What should be the value of $m$ such that its molar heat capacity during the process is equal to $R$?