The figure shows a polytropic process for an ideal gas. The work done by the gas in process $AB$ is

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 $V_2 = 2V_1$.

An ideal gas undergoes a quasi-static,reversible process in which its molar heat capacity $C$ remains constant. If during this process the relation of pressure $P$ and volume $V$ is given by $PV^n = \text{constant}$,then $n$ is given by (Here $C_p$ and $C_v$ are molar specific heat at constant pressure and constant volume,respectively):

One mole of a monoatomic ideal gas is expanded by a process described by $p V^3 = C$,where $C$ is a constant. The heat capacity of the gas during the process is given by ($R$ is the gas constant).

$Q$ amount of heat is given to $0.5 \text{ mole}$ of an ideal mono-atomic gas by a process $TV^n = \text{constant}$. The following graph shows the variation of temperature with $Q$. Find the value of $n$.

The graph of pressure $(P)$ and volume $(V)$ according to $PV^n = C$,where $n = 1.4$. Identify the correct graph.