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 $V \propto T^{2/3}$. $[R = 1.99 \ cal/mol-K]$

$A$ gas obeys $P^2V = \text{constant}$. The initial temperature and volume are $T_0$ and $V_0$. If the gas expands to a volume of $2V_0$,the final temperature is:

An ideal gas undergoes a quasi-static,reversible process in which its molar heat capacity $C$ remains constant. If during this process the relation of pressure $P$ and volume $V$ is given by $PV^n = \text{constant}$,then $n$ is given by (Here $C_p$ and $C_v$ are molar specific heat at constant pressure and constant volume,respectively):

An ideal gas expands in such a manner that its pressure and volume are related by the equation $PV^2 = \text{constant}$. During this process, the gas 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 $2$.

An ideal gas expands in such a way that $PV^2 = \text{constant}$ throughout the process.