$A$ monatomic gas of volume $V$ and pressure $P$ expands isothermally to a volume $27 V$ and then is compressed adiabatically to a volume $V$. The final pressure of the gas is: (in $P$)

The ratio of the specific heat capacities of a gas is $1.5$. When the gas undergoes an adiabatic process, its volume is doubled and its pressure becomes $P_1$. When the gas undergoes an isothermal process, its volume is doubled and its pressure becomes $P_2$. If $P_1 = P_2$, what is the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes?

One mole of a gas expands such that its volume $V$ changes with absolute temperature $T$ in accordance with the relation $V = K T^2$,where $K$ is a constant. If the temperature of the gas changes by $60 \text{ K}$,then the work done by the gas is ($R$ is the universal gas constant).

When a system is taken from a state $i$ to $f$ along the path $iaf$ (as shown in the figure),$Q = 50 \, cal$ and $W = 20 \, cal$. Along path $ibf$,$Q = 36 \, cal$.
$(i)$ What is $W$ along path $ibf$?
$(ii)$ If $W = -13 \, cal$ for path $fi$,what is $Q$ for the path $fi$?
$(iii)$ Take $E_{int,i} = 10 \, cal$,then what is $E_{int,f}$?

The initial state of a certain gas is $(P_i, V_i, T_i)$. It undergoes expansion until its volume becomes $V_f$. Consider the following two cases:
$(a)$ The expansion takes place at constant temperature.
$(b)$ The expansion takes place at constant pressure.
Plot the $P-V$ diagram for each case. In which of the two cases is the work done by the gas more?

$A$ cyclic process $ABCD$ is shown in the $p-V$ diagram. Which of the following curves represents the same process if $BC$ and $DA$ are isothermal processes?