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For the reaction : $2 A (g) \to 3 B (g) + C (l),$ the rate law is $r = - \frac{1}{2} . \frac{d P_{A}}{d t} = K . P_{A}$ The reaction is performed at constant volume and temperature, starting with only pure A(g). If PT and P∞ are the total pressure of system at t = T min and $t = \infty$ and P0 is the vapour pressure of $C (l)$ , t hen the rate constant of reaction may be expressed as

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$k = \frac{1}{2 T_{\min}} . \ln \frac{P_{\infty} - P_{0}}{P_{\infty} - P_{T}}$

$k = \frac{1}{2 T_{\min}} . \ln \frac{P_{\infty}}{P_{\infty} - P_{T}}$

$k = \frac{1}{T_{\min}} . \ln \frac{P_{\infty}}{P_{\infty} - P_{T}}$

$k = \frac{1}{2 T_{\min}} . \ln \frac{P_{\infty} - P_{0}}{3 (P_{\infty} - P_{T})}$

answer is C.

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Detailed Solution

$- \frac{d P}{d t} = 2 K P$

$\Rightarrow - \int \frac{d P}{P} = 2 K \int d t$

$\Rightarrow K = \frac{1}{2 t} \ln \frac{P_{A (in i t i a l)}}{P_{A} (f i n a l)} - - - - 1$

$2 A \to 3 B + C$

t = 0 $P_{1}$ O O

$t = T P_{1} - 2 x 3 x P_{o} P_{T} = P_{1} + x + P_{0} - - - 2$

$t = α O \frac{3 P_{1}}{2} P_{0} \frac{3 P_{1}}{2} + P_{0} = P_{α}$

$\Rightarrow P_{1} = \frac{2 (P_{α} - P_{0})}{3} - - - - 3$

From 2 - - $x = P_{T} - P_{0} - P_{1} = P_{T} - P_{0} - \frac{2 (P_{α} - P_{0})}{3}$

$P_{in i t i a l} = P_{1} = \frac{2 (P_{α} - P_{0})}{3}$

$P_{f i n a l}$ $= P_{1} - 2 x = 2 (P_{a} - P_{T})$

From -1

$k = \frac{1}{2 T} \ln \frac{P_{α} - P_{0}}{3 (P_{α} - P_{T})}$

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