An ideal gas is found to obey $Pv^{\frac{3}{2}} = \text{constant}$ during an adiabatic process. If such a gas initially at a temperature $T$ is adiabatically compressed to $\frac{1}{4}$th of its volume,then its final temperature is

An ideal gas with heat capacity at constant volume $C_V$ undergoes a quasistatic process described by $p V^{\alpha} = \text{constant}$ in a $p-V$ diagram,where $\alpha$ is a constant. The heat capacity of the gas during this process is given by

One mole of an ideal gas undergoes a process $PV^3 = \text{constant}$, where $P$ and $V$ are pressure and volume respectively. Let $W$ be the work done by the gas as its temperature is increased by $\Delta T$. The value of $|W|$ is ($R$ is the universal gas constant).

Consider a $P - V$ diagram in which the path followed by one mole of perfect gas in a cylindrical container is shown in the figure.
$(a)$ Find the work done when the gas is taken from state $1$ to state $2$.
$(b)$ What is the ratio of temperature $\frac{T_1}{T_2}$ if $V_2 = 2V_1$?
$(c)$ Given the internal energy for one mole of gas at temperature $T$ is $\frac{3}{2}RT$,find the heat supplied to the gas when it is taken from state $1$ to $2$,with $V_2 = 2V_1$.

One mole of an ideal gas at temperature $T_1$ expands according to the law $\frac{P}{V^2} = a$ (constant). The work done by the gas until the temperature of the gas becomes $T_2$ is:

$1 \, \text{mole}$ of gas expands with temperature according to the relation $V = KT^{2/3}$. When the temperature changes by $30 \, ^\circ\text{C}$, the work done will be: (in $, R$)