Question

A monoatomic ideal gas is expanded adiabatically to n times of its initial volume. The ratio of final rate of collision of molecules with unit area of container wall to the initial rate will be

Moderate

Solution

We know that the rate of collision of molecules with unit area of walls of container is proportional to pressure. Therefore
$P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}$
Given that $V_{2} = {nV}_{1}$
$∴ P_{1} V_{1}^{γ} = P_{2} {({nV}_{1})}^{γ}$
Now $\frac{P_{2}}{P_{1}} = {(\frac{V_{1}}{{nV}_{1}})}^{γ} = \frac{1}{n^{γ}}$
or $\frac{P_{2}}{P_{1}} = n^{- γ}$
For monoatomic gas, $γ = \frac{5}{3}$
$∴ \frac{P_{2}}{P_{1}} = n^{- 5 / 3}$

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