An ideal gas with adiabatic exponent $(\gamma = 1.5)$ undergoes a process in which the work done by the gas is equal to the increase in the internal energy of the gas. The molar heat capacity of the gas for this process is:

The $P-V$ diagram of a certain process (Carnot cycle) is as shown. The process is also represented as:

An ideal gas at initial temperature $T_0$ and initial volume $V_0$ is expanded adiabatically to a volume $2 V_0$. The gas is then expanded isothermally to a volume $5 V_0$ and thereafter compressed adiabatically so that the temperature of the gas becomes again $T_0$. If the final volume of the gas is $\alpha V_0$, then the value of constant $\alpha$ is

For an ideal gas with initial pressure and volume $p_{i}$ and $V_{i}$ respectively, a reversible isothermal expansion occurs until its volume becomes $V_{0}$. Then, it is compressed to its original volume $V_{i}$ by a reversible adiabatic process. If the final pressure is $p_{f}$, then which of the following statement(s) is/are true?

One mole of an ideal gas undergoes two different cyclic processes $I$ and $II$,as shown in the $P-V$ diagrams below. In cycle $I$,processes $a, b, c$ and $d$ are isobaric,isothermal,isobaric and isochoric,respectively. In cycle $II$,processes $a^{\prime}, b^{\prime}, c^{\prime}$ and $d^{\prime}$ are isothermal,isochoric,isobaric and isochoric,respectively. The total work done during cycle $I$ is $W_I$ and that during cycle $II$ is $W_{II}$. The ratio $W_I / W_{II}$ is . . . .

$A$ thermodynamic cycle takes in heat energy at a high temperature and rejects energy at a lower temperature. If the amount of energy rejected at the low temperature is $3$ times the amount of work done by the cycle,the efficiency of the cycle is