An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. To reduce the root mean square velocity of the molecules by a factor of $2$,the gas should be expanded by how many times?

$A$ vessel containing nitrogen gas is supplied a heat of $498 \,J$, so as to raise the temperature of the gas by $40^{\circ} C$ at constant pressure. The mass of nitrogen gas in the vessel is (Molecular mass of nitrogen $= 28 \,g/mol$; Universal gas constant $= 8.3 \,J \,mol^{-1} \,K^{-1}$) (in $\,g$)

$0.056 \, kg$ of Nitrogen is enclosed in a vessel at a temperature of $127 \, ^{\circ}C$. The amount of heat required to double the speed of its molecules is $k \, cal$. (Take $R = 2 \, cal \, mole^{-1} K^{-1}$)

For two different gases $X$ and $Y$, having degrees of freedom $f_1$ and $f_2$ and molar heat capacities at constant volume $C_{V1}$ and $C_{V2}$ respectively, the $\ln P$ versus $\ln V$ graph is plotted for an adiabatic process, as shown.

Moon has no atmosphere because

One mole of a monoatomic ideal gas undergoes a quasistatic process, which is depicted by a straight line joining points $(V_{0}, T_{0})$ and $(2 V_{0}, 3 T_{0})$ in a $V-T$ diagram. What is the value of the heat capacity of the gas at the point $(V_{0}, T_{0})$?