During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its temperature. The ratio of  $\frac{{{C_P}}}{{{C_V}}}$  for the gas is

A diatomic ideal gas is compressed adiabatically to $\frac{1}{32}$ of its initial volume. If the initial temperature of the gas is $T_1$ (in Kelvin) and the final temperature is $a T_1$, the value of $a$ is

The pressure $P_{1}$ and density $d_{1}$ of diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes suddenly to $P _{2}\left(> P _{1}\right)$ and $d _{2}$ respectively during an adiabatic process. The temperature of the gas increases and becomes $……$ times of its initial temperature.$\left(\right.$ given $\left.\frac{ d _{2}}{ d _{1}}=32\right)$

$5.6$ $liter$ of helium gas at $STP$ is adiabatically compressed to $0.7$ $liter$. Taking the initial temperature to be $T_1$, the work done in the process is

Consider one mole of helium gas enclosed in a container at initial pressure $P_1$ and volume $V_1$. It expands isothermally to volume $4 V_1$. After this, the gas expands adiabatically and its volume becomes $32 V_1$. The work done by the gas during isothermal and adiabatic expansion processes are $W_{\text {iso }}$ and $W_{\text {adia, }}$ respectively. If the ratio $\frac{W_{\text {iso }}}{W_{\text {adia }}}=f \ln 2$, then $f$ is. . . . . . . .