One mole of an ideal gas $\left( \frac{C_p}{C_v} = \gamma \right)$ is heated according to the law $P = \alpha V$,where $P$ is the pressure of the gas,$V$ is the volume,and $\alpha$ is a constant. What is the molar heat capacity of the gas in this process?

One mole of a monoatomic ideal gas $\left(c_{V} = \frac{3}{2} R\right)$ undergoes a cycle where it first goes isochorically from the state $\left(\frac{3}{2} P_{0}, V_{0}\right)$ to $\left(P_{0}, V_{0}\right)$,and then is isobarically contracted to the volume $\frac{1}{2} V_{0}$. It is then taken back to the initial state by a path which is a quarter ellipse on the $P-V$ diagram. The efficiency of this cycle is

One mole of an ideal monoatomic gas is heated at a constant pressure of one atmosphere from $0^{\circ}C$ to $100^{\circ}C$. Then the change in the internal energy is

Suppose an ideal gas ($n$ moles) undergoes an expansion process $P = f(V)$ which passes through the point $(V_0, P_0)$. If the slope of the curve $P = f(V)$ is greater than the slope of the adiabatic curve passing through $(V_0, P_0)$,show that the gas absorbs heat at $(V_0, P_0)$.

Which of the following processes will release the maximum amount of heat to the surroundings when the volume is reduced to half of its initial value?

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.