In an isolated chamber, an ideal gas undergoes adiabatic expansion. The average time between molecular collisions increases as V^q, where V is the gas volume. Calculate the value of q. (Let gamma = C_p/C_v)

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EEJ MAIN Physics QUESTION #7059

Question 1

In an isolated chamber, an ideal gas undergoes adiabatic expansion. The average time between molecular collisions increases as $V^q$, where $V$ is the gas volume. Calculate the value of $q$. (Let $\gamma = C_p/C_v$)

$\frac{3\gamma+5}{6}$
$\frac{3\gamma-5}{6}$
$\frac{\gamma+1}{2}$✔️
$\frac{\gamma-1}{2}$

Correct Answer Explanation

Average collision time $\tau = \frac{\lambda}{v_{rms}}$.

Mean free path $\lambda \propto \frac{1}{n} \propto V$.

Root mean square velocity $v_{rms} \propto \sqrt{T}$.

For an adiabatic process, $T \propto V^{-(\gamma-1)}$. So, $v_{rms} \propto V^{-\frac{\gamma-1}{2}}$.

Therefore, $\tau \propto \frac{V}{V^{-\frac{\gamma-1}{2}}} = V^{1 + \frac{\gamma-1}{2}} = V^{\frac{\gamma+1}{2}}$.

Thus, $q = \frac{\gamma+1}{2}$.

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