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:

$A$ 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

One mole of an ideal monoatomic gas at temperature $T_0$ expands slowly according to the law $P/V = \text{constant}$. If the final temperature is $2 T_0$, heat supplied to the gas is:

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

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 $V_2 = 2V_1$.

$2$ 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?