Thermodynamics: Practice Problem & Solution
Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as \(V^q\), where V is the volume of the gas. The value of q is \(\left(\gamma = \frac{C_p}{C_v}\right)\)
Solution Explained:
To solve this problem, we apply the core principles of Thermodynamics. Understanding the underlying formula is key to arriving at the correct answer below:
Mean free path \(\lambda \propto V\) and \(v_{\text{rms}} \propto \sqrt{T}\). For an adiabatic process, \(T \propto V^{-(\gamma - 1)}\), so \(v_{\text{rms}} \propto V^{-(\gamma - 1)/2}\). Average time \(\tau = \frac{\lambda}{v_{\text{rms}}} \propto V^{1 + (\gamma - 1)/2} = V^{(\gamma+1)/2}\), so \(q = \frac{\gamma+1}{2}\).
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