Two gases $A$ and $B$ have the same initial state $(P, V, n, T)$. Gas $A$ is compressed to $V/8$ by an isothermal process,and gas $B$ is compressed to $V/8$ by an adiabatic process. The ratio of the final pressure of gas $A$ to that of gas $B$ is (Both gases are monoatomic,$\gamma = 5/3$).

$A$ 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 $\gamma = 5/3$)

Three samples $X, Y$,and $Z$ of the same gas have equal volumes and temperatures. The volume of each sample is doubled. The process is isothermal for $X$,adiabatic for $Y$,and isobaric for $Z$. If the final pressures are equal for the three samples,find the ratio of the initial pressures. (Take adiabatic exponent $\gamma = 3/2$)

An ideal gas at pressure $P_0$ undergoes an isothermal expansion until its volume is $8.0$ times its initial volume. The gas is slowly and adiabatically compressed back to its original volume. If the adiabatic constant of the gas is $\gamma = 4/3$, then the ratio of the average kinetic energy per molecule in this final state to that in the initial state is

The relation between efficiency $(\eta)$ of a Carnot engine and the coefficient of performance $(\beta)$ of a refrigerator working between the same temperatures is:

For an ideal gas,which of the following statements is correct?