For a homogeneous gaseous reaction $A \to 3 B$ if pressure after time t was P_T and after completion of reaction, pressure was $P_{\infty}$ then select correct relation

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$k = \frac{1}{t} In (\frac{3 P_{\infty}}{2 P_{\infty} - P_{t}})$

$k = \frac{1}{t} In (\frac{P_{\infty}}{3 (P_{\infty} - P_{t})})$

$k = \frac{1}{t} In (\frac{2 P_{\infty}}{3 (P_{\infty} + P_{T})})$

$k = \frac{1}{t} In (\frac{2 P_{\infty}}{3 (P_{\infty} - P_{T})})$

answer is D.

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Detailed Solution

$\begin{array}{l} \begin{array}{ll} A \to & 3 B \\ att = 0 p_{0} & 0 \end{array} p_{T} = p_{0} + 2 x, x = \frac{p_{T} - p_{0}}{2} \\ {tp}_{0} - x 3 x x = \frac{p_{T} - \frac{p_{\infty}}{3}}{2} x = \frac{3 p_{T} - p_{\infty}}{6} K = \frac{1}{t} \ln \frac{\frac{p_{\infty}}{3}}{\frac{p_{\infty}}{3} - \frac{(3 p_{T} - p_{\infty})}{6}} \\ at t = \infty - 3 p_{0} p_{\infty} = 3 p_{0} \\ = \frac{1}{t} \ln \frac{2 p_{\infty}}{3 (p_{\infty} - p_{T})} \end{array}$