$A$ gas is found to obey the law $P^2V =$ constant. The initial temperature and volume are $T_0$ and $V_0$. If the gas expands to a volume $3V_0$,its final temperature becomes

In a thermodynamic process, $2 \, \text{moles}$ of a monatomic ideal gas obey $P \propto V^{-2}$. If the temperature of the gas increases from $300 \, K$ to $400 \, K$, find the work done by the gas in terms of $R$ (where $R$ is the universal gas constant).

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 $R$?

An ideal gas expands according to the law $P^2 V = \text{constant}$. The internal energy of the gas

Suppose an ideal gas follows the process $VP^3 = \text{constant}$. The initial temperature and volume of the gas are $T$ and $V$ respectively. If the gas expands to $27V$, then its final temperature will be: