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(C) For the first order reaction $A_{(g)} \rightarrow B_{(g)} + C_{(g)}$:At $t = 0$,$P_A = P_0$,$P_B = 0$,$P_C = 0$At $t = t$,$P_A = P_0 - x$,$P_B = x

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$k = \frac{1}{t} \ln \frac{P_{\infty}}{2(P_{\infty} - P_t)}$

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(C) For the first order reaction $A_{(g)} \rightarrow B_{(g)} + C_{(g)}$:At $t = 0$,$P_A = P_0$,$P_B = 0$,$P_C = 0$At $t = t$,$P_A = P_0 - x$,$P_B = x$,$P_C = x$Total pressure $P_t = (P_0 - x) + x + x = P_0 + x \Rightarrow x = P_t - P_0$At $t = \infty$,$P_A = 0$,$P_B = P_0$,$P_C = P_0$Total pressure $P_{\infty} = 2P_0 \Rightarrow P_0 = \frac{P_{\infty}}{2}$Pressure of $A$ at time $t$: $P_A = P_0 - x = P_0 - (P_t - P_0) = 2P_0 - P_t$Substituting $P_0 = \frac{P_{\infty}}{2}$: $P_A = 2(\frac{P_{\infty}}{2}) - P_t = P_{\infty} - P_t$Rate constant $k = \frac{1}{t} \ln \frac{P_0}{P_A} = \frac{1}{t} \ln \frac{P_{\infty}/2}{P_{\infty} - P_t} = \frac{1}{t} \ln \frac{P_{\infty}}{2(P_{\infty} - P_t)}$.

Time	$t$	$\infty$
$P_{\text{system}}$	$P_t$	$P_{\infty}$

$A_{(g)} \rightarrow B_{(g)} + C_{(g)}$ is a first order reaction.

Time $t$ $\infty$

$P_{\text{system}}$ $P_t$ $P_{\infty}$

The reaction was started with reactant $A$ only. Which of the following expressions is correct for the rate constant $k$?

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$A_{(g)} \rightarrow B_{(g)} + C_{(g)}$ is a first order reaction. Time $t$ $\infty$ $P_{\text{system}}$ $P_t$ $P_{\infty}$ The reaction was started with reactant $A$ only. Which of the following expressions is correct for the rate constant $k$?