$A$ given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be $0.75$ of its initial pressure. The ratio of the specific heats of the gas is approximately:

Choose the incorrect statement from the following:
$S1$: The efficiency of a heat engine can be $1$,but the coefficient of performance of a refrigerator can never be infinity.
$S2$: The first law of thermodynamics is basically the principle of conservation of energy.
$S3$: The second law of thermodynamics does not allow several phenomena consistent with the first law.
$S4$: $A$ process,whose sole result is the transfer of heat from a colder to a hotter object,is impossible.

Given below are two statements:
Statement $I:$ If heat is added to a system,its temperature must increase.
Statement $II:$ If positive work is done by a system in a thermodynamic process,its volume must increase.
In the light of the above statements,choose the correct answer from the options given below.

The initial pressure and volume of a gas are $P$ and $V$ respectively. First,the gas is expanded to a volume of $9V$ by an isothermal process,and then it is compressed to a volume of $V$ by an adiabatic process. What is its final pressure (in $P$)? (Ratio of specific heat at constant pressure to constant volume $\gamma = \frac{3}{2}$)

Two identical samples of a gas are allowed to expand $(i)$ isothermally and $(ii)$ adiabatically. The work done is:

Two samples $A$ and $B$ of gas are initially at the same pressure and temperature. They are compressed from volume $V$ to $\frac{V}{2}$ ($A$ isothermally and $B$ adiabatically). The relation between the pressure of gas $A$ $(P_A)$ and the pressure of gas $B$ $(P_B)$ is: