An ideal gas is contained in an isolated closed chamber. During an adiabatic expansion, the average time between molecular collisions increases as V^q, where V is the volume. Determine the value of q in terms of gamma = C_p/C_v.

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EEJ MAIN Physics QUESTION #7070

Question 1

An ideal gas is contained in an isolated closed chamber. During an adiabatic expansion, the average time between molecular collisions increases as $V^q$, where $V$ is the volume. Determine the value of $q$ in terms of $\gamma = C_p/C_v$.

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

Correct Answer Explanation

Average time between collisions $\tau$ is given by $\tau = \frac{\lambda}{v_{rms}}$, where $\lambda$ is the mean free path.

1. $\lambda \propto \frac{1}{n} \propto V$ (where $n$ is number density).
2. $v_{rms} \propto \sqrt{T}$.
3. For an adiabatic process, $TV^{\gamma-1} = \text{constant}$, so $T \propto V^{1-\gamma}$ and $\sqrt{T} \propto V^{\frac{1-\gamma}{2}}$.

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

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