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?

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.

$A$ monoatomic gas at pressure $P$,having volume $V$ expands isothermally to a volume $2V$ and then adiabatically to a volume $16V$. The final pressure of the gas is (Take $\gamma = 5/3$)

For a monoatomic ideal gas following the cyclic process $ABCA$ shown in the $U$ vs $\rho$ plot,identify the incorrect option:

The molar heat capacity in a process of a diatomic gas,if it does a work of $\frac{Q}{4}$ when a heat of $Q$ is supplied to it,is

The $P-V$ diagram shown represents the thermodynamic cycle of an engine operating with an ideal monatomic gas. The amount of heat extracted from the source in a single cycle is: