Consider two containers A and B containing identical gases at the same pressure, volume and temperature. The gas in container A is compressed to half of its original volume isothermally while the gas in container B is compressed to half of its original value adiabatically. The ratio of final pressure of gas in B to that of gas in A is

The correct option is 2^(γ-1).

Let the initial pressure, volume, and temperature of the identical gases in both containers A and B be $P_0$, $V_0$, and $T_0$ respectively.

Both gases are compressed to half of their original volume, so the final volume for both containers is:
$V_f=\frac{V_0}{2}$

1. Isothermal Compression of Gas in Container A:
For an isothermal process, the temperature remains constant, and the gas obeys Boyle's law:
$P_i{V}_i=P_f{V}_f$

Let $P_A$ be the final pressure in container A. Substituting the values:
$P_0{V}_0=P_A\left(\frac{V_0}{2}\right)$

Solving for $P_A$:
$P_A=2P_0$

2. Adiabatic Compression of Gas in Container B:
For an adiabatic process, the gas obeys the relation:
$P_i{V}_i^{\gamma }=P_f{V}_f^{\gamma }$

Let $P_B$ be the final pressure in container B. Substituting the values:
$P_0{V}_0^{\gamma }=P_B{\left(\frac{V_0}{2}\right)}^{\gamma }$

Solving for $P_B$:
$P_B=P_0{\left(\frac{V_0}{V_0/2}\right)}^{\gamma }=2^{\gamma }{P}_0$

3. Ratio of Final Pressures:
Now, we find the ratio of the final pressure in container B to that in container A:
$\frac{P_B}{P_A}=\frac{2^{\gamma }{P}_0}{2P_0}$

Simplifying the ratio:
$\frac{P_B}{P_A}=2^{\gamma -1}$