The relation between absolute temperature $(T)$ and pressure $(P)$ of a gas in an adiabatic process is

Consider a cycle tyre being filled with air by a pump. Let $V$ be the volume of the tyre (fixed) and at each stroke of the pump $\Delta V$ ( < < V) of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from $P_1$ to $P_2$?

Five moles of hydrogen initially at $STP$ is compressed adiabatically so that its temperature becomes $673 \, K$. The increase in internal energy of the gas, in kilo joule is $(R=8.3 \, J/mol-K; \gamma=1.4$ for diatomic gas$)$

The initial pressure and volume of a given mass of an ideal gas (with $\frac{C_{p}}{C_{V}}=\gamma$), taken in a cylinder fitted with a piston, are $p_{0}$ and $V_{0}$ respectively. At this stage, the gas has the same temperature as that of the surrounding medium, which is $T_{0}$. It is adiabatically compressed to a volume equal to $\frac{V_{0}}{2}$. Subsequently, the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surrounding?

$200 \ cc$ of an ideal gas $(\gamma = 1.5)$ expands adiabatically. If the rms speed of the gas molecules becomes half of the initial value,the final volume of the gas is (in $cc$)

$5$ moles of Hydrogen $\left(\gamma=\frac{7}{5}\right)$ initially at $S.T.P.$ are compressed adiabatically so that its temperature becomes $400^{\circ} C$. The increase in the internal energy of the gas in kilo-joules is $\left(R=8.30 \ J \ mol^{-1} \ K^{-1}\right)$.