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:

$0.02 \, mol$ of an ideal diatomic gas with initial temperature $20^{\circ} C$ is compressed from $1500 \, cm^3$ to $500 \, cm^3$. The thermodynamic process is such that $p V^2 = \beta$,where $\beta$ is a constant. Then,the value of $\beta$ is close to (the gas constant,$R = 8.31 \, J / K / mol$).

$2$ 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?

$Q$ amount of heat is given to $0.5 \text{ mole}$ of an ideal mono-atomic gas by a process $TV^n = \text{constant}$. The following graph shows the variation of temperature with $Q$. Find the value of $n$.

The specific heat of a gas in a polytropic process is given by

$1 \, \text{mole}$ 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 $, R$)