Which of the following represents integrated rate law equation for gas phase first order reaction, \(\mathrm{A}_{(\mathrm{g})} \rightarrow \mathrm{B}_{(\mathrm{g})}+\mathrm{C}_{(\mathrm{g})}\)
If \(\mathrm{P}_{\mathrm{i}}=\) initial pressure of \(\mathrm{A}\)
\(\mathrm{P}=\) total pressure of reaction mixture at time?

\(\mathrm{A}_{(\mathrm{g})} \longrightarrow \mathrm{B}_{(\mathrm{g})}+\mathrm{C}_{(\mathrm{g})} \)
\( \mathrm{t}=0 . \mathrm{P}_{\mathrm{i}} - - \)
\( \mathrm{t}=\mathrm{t}, \mathrm{P}_{\mathrm{i}}-\mathrm{x} \mathrm{x} \mathrm{x}\)
Total pressure of reaction mixture at time \(\mathrm{t}(\mathrm{P})=\mathrm{P}_{\mathrm{i}}-\mathrm{x}+\mathrm{x}+\mathrm{x}\)
\(
\begin{aligned}
& \mathrm{P}=\mathrm{P}_{\mathrm{i}}+\mathrm{x} \\
& \mathrm{x}=\mathrm{P}-\mathrm{Pi}
\end{aligned}
\)
For first order reaction,
\(\mathrm{K}=\frac{2.303}{\mathrm{t}} \cdot \log \frac{\mathrm{P}_{\mathrm{i}}}{\mathrm{P}_{\mathrm{i}}-\mathrm{x}} \)
\( =\frac{2.303}{\mathrm{t}} \log \frac{\mathrm{P}_{\mathrm{i}}}{\mathrm{P}_{\mathrm{i}}-\left(\mathrm{P}-\mathrm{P}_{\mathrm{i}}\right)} \)
\( \mathrm{K}=\frac{2.303}{\mathrm{t}} \cdot \log \frac{\mathrm{P}_{\mathrm{i}}}{2 \mathrm{P}_{\mathrm{i}}-\mathrm{P}}\)