$A$ gas obeys the relation $P^2V = \text{constant}$. The initial temperature and volume are $T_0$ and $V_0$. If the gas expands to a volume $2V_0$, what will be its final temperature?

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 $V_2 = 2V_1$.

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

$A$ 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

Find the amount of work done to increase the temperature of one mole of an ideal gas by $30^o\ C$ if it is expanding under the condition $V \propto T^{2/3}$. $[R = 1.99 \ cal/mol-K]$

$A$ monoatomic gas is taken through a process $TP^{-1/3} = \text{constant}$. If heat is given to the gas, what happens to its temperature?