Two moles of helium gas $\left(\gamma = \frac{5}{3}\right)$ at $27^{\circ} C$ is expanded at constant pressure until its volume is doubled. Then it undergoes an adiabatic change until the temperature returns to its initial value. The work done during the adiabatic process is (universal gas constant $R = 8.3 \ J \ mol^{-1} \ K^{-1}$) (in $J$)

$A$ gas obeying the equation of state $pV = RT$ undergoes a hypothetical reversible process described by the equation $pV^{5/3} \exp \left(-\frac{pV}{E_0}\right) = C_1$,where $C_1$ and $E_0$ are dimensioned constants. Then,for this process,the thermal compressibility at high temperature

An ideal gas is taken from state $1$ to state $2$ through optional paths $A, B, C$ and $D$ as shown in the $P-V$ diagram. Let $Q, W$ and $\Delta U$ represent the heat supplied,work done,and change in internal energy of the gas,respectively. Then:

Consider two containers $A$ and $B$ containing monoatomic gases at the same Pressure $(P)$,Volume $(V)$ and Temperature $(T)$. The gas in $A$ is compressed isothermally to $\frac{1}{8}$ of its original volume while the gas $B$ is compressed adiabatically to $\frac{1}{8}$ of its original volume. The ratio of final pressure of gas in $B$ to that of gas in $A$ is ...........

One mole of an ideal gas is taken through a cyclic process with alternating isothermal and adiabatic curves. In the $P-V$ diagram, $AB, CD, EF$ are isothermal curves at absolute temperatures $T_1, T_2,$ and $T_3$ respectively, and $BC, DE,$ and $FA$ are adiabatic curves. If $\frac{V_B}{V_A} = 2$ and $\frac{V_D}{V_C} = 2$, then for the cycle shown in the figure, four statements are made below. (Figure is not drawn to scale)
Statement $1$: Ratio of volumes $\frac{V_E}{V_F} = 4$
Statement $2$: Magnitude of work done in isothermal compression $EF$ is $2RT_3 \ln(2)$
Statement $3$: Ratio of heat supplied to the gas in process $AB$ to heat rejected by the gas in process $EF$ is $\frac{T_1}{T_3}$
Statement $4$: Net work done by the gas in the cycle $ABCDEFA$ is $(T_1 + T_2 - 2T_3) R \ln(2)$
Find the number of correct statements given for the cyclic process followed by the gas.

An ideal monoatomic gas with pressure $P$, volume $V$ and temperature $T$ is expanded isothermally to a volume $2V$ and a final pressure $P_i$. If the same gas is expanded adiabatically to a volume $2V$, the final pressure is $P_a$. The ratio $\frac{P_a}{P_i}$ is