Consider two containers and containing monoatomic gases at the same pressure , volume and temperature . The gas in is compressed isothermally to of its original volume, while the gas in is compressed adiabatically to of its original volume. [202

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Solution

Q.
Consider two containers $A$ and $B$ containing monoatomic gases at the same pressure $(P)$ , volume $(V)$ and temperature $(T)$ . The gas in $A$ is compressed isothermally to $\frac{1}{8}$ of its original volume, while the gas in $B$ is compressed adiabatically to $\frac{1}{8}$ of its original volume. The ratio of final pressure of gas in $B$ to that of gas in $A$ is: [2023]

1 $8$

2 $8^{1 / 2}$

3 $\frac{1}{8}$

4 $4$

Ans.
(4)

$Isothermal process, T = constant$

$P V = n R T = constant$

$P_{1} V_{1} = P_{2} V_{2}$

$P V = P_{A} (\frac{V}{8})$

$P_{A} = 8 P$

$Adiabatic process, P V^{γ} = constant$

$γ for monoatomic gas is \frac{5}{3}$

$P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}$

$\frac{P_{B}}{P} = {(\frac{V_{1}}{V_{2}})}^{γ} = {(\frac{V}{V / 8})}^{5 / 3}$

$P_{B} = 32 P$

$\frac{P_{B}}{P_{A}} = \frac{32 P}{8 P} = 4$

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