One mole of an ideal monatomic gas undergoes the process $p=\alpha T^{\frac{1}{2}}$, where $\alpha$ is a constant.
$\left(\mathrm{a}\right)$ Find the work done by the gas if the temperature increases by $50 \mathrm{K}$.
$\left(\mathrm{b}\right)$ Also, find the molar specific heat of the gas.

Two moles of a monatomic ideal gas undergoes a cyclic process $ABCDA$ as shown in the figure. $BCD$ is a semicircle. Find the efficiency of the cycle.

Pressure $p$, volume $V$ and temperature $T$ for a certain gas are related by, $p=\frac{\mathrm{\alpha }T\mathit{-}\mathrm{\beta }{T}^{\mathit{2}}}{V}$.

where, $\mathrm{\alpha }$ and $\mathrm{\beta }$ are constants. Find the work done by the gas if the temperature changes from $T_1$ to $T_2$ while the pressure remains constant.

The pressure versus temperature $(p\mathit{-}T)$ graph of $n \mathrm{moles}$ of an ideal gas is shown in the figure. Plot the corresponding,
$\left(\mathrm{a}\right)$ density versus volume $(\rho -V)$ graph
$\left(\mathrm{b}\right)$ pressure versus volume $(p-V)$ graph and
$\left(\mathrm{c}\right)$ density versus pressure $(\rho -p)$ graph.