A(g) arrow B(g)+C(g) is a first order reaction. Time T ∞ P_ system...

\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})

is a first order reaction.

<table class="qx-match-list qx-match-table" border="1" cellpadding="6" cellspacing="0"><tr><td>Time

T\infty

\mathbf{P}_{system }

P_{t}P_{\infty}
The reaction was started with reactant A only. Which of the following expression is correct for rate constant k ?

Options

A $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{2\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}{\mathrm{P}_{\mathrm{t}}}$
B $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{\mathrm{P}_{\mathrm{t}}}$
C\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{2 $\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$
D\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{ $\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$

C. \mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{\mathrm{p}_{\infty}}{2 $\left(\mathrm{p}_{\infty}-\mathrm{P}_{\mathrm{t}}\right)}$

\begin{aligned}<br />& \begin{array}{lllll} <br />& A_{(g)} & \rightarrow & B_{(g)} & + <br />C_{(g)} \\<br>t=0 & P^o & & 0 & 0 \\<br>t=t & P^o-x & & x & x \\<br>t=\infty & 0 & & P^o & P^o<br />\end{array} \\<br>& \mathrm{P}_{\mathrm{t}}=\mathrm{P}^{\mathrm{o}}+\mathrm{x} \Rightarrow \mathrm{x}=\mathrm{P}_{\mathrm{t}}-\mathrm{P}^{\mathrm{o}}=\mathrm{P}_{\mathrm{t}}-\frac{\mathrm{P}_{\infty}}{2} \\<br>& \mathrm{P}_{\infty}=2 \mathrm{P}^{\mathrm{o}} \Rightarrow \mathrm{P}^0=\frac{\mathrm{P} \infty}{2} \\<br>& \mathrm{k}=\frac{1}{\mathrm{t}} \ell \ln \frac{\mathrm{P}^{\mathrm{o}}}{\mathrm{P}^{\mathrm{o}}-\mathrm{x}} \\<br>& k=\frac{1}{t} \ell n \frac{P_{\infty}}{2\left(P_{\infty}-P_t\right)}<br />\end{aligned}