Consider two containers A and B containing identical gases at the… - Vidyadip

MCQ

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

✓
$2^{\gamma-1}$
B
$\Big(\frac{1}{2}\Big)^{\gamma-1}$
C
$\Big(\frac{1}{1-\gamma}\Big)^2$
D
$\Big(\frac{1}{\gamma-1}\Big)^2$

✓

Answer

Correct option: A.

$2^{\gamma-1}$

According to the P-V diagram shown for the container A (which is going through isothermal process) and for container B (which is going through adiabatic process).

Both the process involves compression of the gas.
(I) Isothermal compression (gas A) (during 1 → 2)
$\text{P}_{1}\text{V}_{1}=\text{P}_{2}\text{V}_{2}$
$\Rightarrow\text{P}_{0}(2\text{V}_{0})^\gamma=\text{P}_{2}(\text{V}_{0})^\gamma$
$\Rightarrow\text{P}_{0}(2\text{V}_{0})=\text{P}_{2}(\text{V}_{0})$
(ii) Adiabatic compression, (gas B) (during 1 → 2)
$\text{P}_{1}\text{V}_{1} ^\gamma=\text{P}_{2}\text{V}_{2}^\gamma$
$\text{P}_{0}(2\text{V}_{0})^\gamma=\text{P}_{2}(\text{V}_{0})^\gamma$
$\text{P}_{2}=\Big(\frac{2\text{V}_{0}}{\text{V}_{0}}\Big)^\gamma\text{P}_{0}=(2)^\gamma\text{P}_{0}$
Hence $\frac{(\text{P}_{2})_\text{B}}{(\text{P}_{2})_\text{A}}=$ Ratio of final pressure $=\frac{(2)^\gamma\text{P}_{0}}{2\text{P}_{0}}=2^{\gamma-1}$
where, $\gamma$ is ratio of specific heat capacities for the gas.

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