A certain gas A polymerizes to a very small extent at a given temperature as: nA(g) ⇌ An(g). The rea — Chemical Equilibrium Chemistry Question

💡 Solution & Explanation

Step 1 - Set up the ICE table for moles For the polymerization equilibrium: \[\ce{nA(g) <=> A_n(g)}\] Starting with 1 mol of A, let \(\alpha\) = moles of A that polymerize: | Species | A(g) | An(g) | |---------|------|-------| | Initial | 1 | 0 | | Change | -α | +α/n | | Equilibrium | 1-α | α/n | Step 2 - Express total moles and PV/RT By the ideal gas law: \[\frac{PV}{RT} = n_{\text{total}} = (1 - \alpha) + \frac{\alpha}{n} = 1 - \alpha\left(\frac{n-1}{n}\right)\] Step 3 - Express Kc using the small-α approximation Since \(\alpha \ll 1\), we approximate \((1-\alpha) \approx 1\), so \([\ce{A}] \approx \frac{1}{V}\) and \([\ce{A_n}] = \frac{\alpha}{nV}\): \[K_c = \frac{[\ce{A_n}]}{[\ce{A}]^n} \approx \frac{\frac{\alpha}{nV}}{\left(\frac{1}{V}\right)^n} = \frac{\alpha V^{n-1}}{n}\] Solving for \(\alpha\): \[\alpha \approx \frac{nK_c}{V^{n-1}}\] Step 4 - Substitute α to get PV/RT \[\frac{PV}{RT} = 1 - \left(\frac{nK_c}{V^{n-1}}\right)\left(\frac{n-1}{n}\right) = 1 - \frac{(n-1)K_c}{V^{n-1}}\] \[\boxed{\frac{PV}{RT} = 1 - \frac{(n-1)K_c}{V^{n-1}}}\] Step 5 - Explain each option * **Option (A)**: Correct. Derived above using the small-α approximation. * **Option (B)**: Incorrect. This is just the subtracted term, not the total moles expression. * **Option (C)**: Incorrect. Uses n×Kc/V^(n-1) without the (n-1)/n stoichiometric factor. * **Option (D)**: Incorrect. Wrong exponent — uses V^n instead of V^(n-1).