One mole of an ideal gas at temperature $T_1$ expands according to the law $\frac{P}{V^2} = a$ (constant). The work done by the gas until the temperature of the gas becomes $T_2$ is:

Consider one mole of an ideal gas in a cylindrical container following the path shown in the $P-V$ diagram. The process follows $PV^{1/2} = \text{constant}$.
$(a)$ Find the work done in taking the gas from state $1$ to state $2$.
$(b)$ If $V_2 = 2V_1$, what is the ratio of temperatures $\frac{T_1}{T_2}$?
$(c)$ The internal energy of one mole of gas at temperature $T$ is $\frac{3}{2}RT$. Find the heat supplied to the gas in taking it from state $1$ to state $2$.

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:

$2$ moles of a monoatomic gas are expanded to double its initial volume, through a process $P/V = \text{constant}$. If its initial temperature is $300\, K$, then which of the following is not true?

An ideal gas is found to obey $Pv^{\frac{3}{2}} = \text{constant}$ during an adiabatic process. If such a gas initially at a temperature $T$ is adiabatically compressed to $\frac{1}{4}$th of its volume,then its final temperature is

$1 \, \text{mole}$ of gas expands with temperature according to the relation $V = KT^{2/3}$. When the temperature changes by $30 \, ^\circ\text{C}$, the work done will be: (in $, R$)