Which of the following represents integrated rate law equation for gas phase first order reaction,…

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?
  1. $\mathrm{k}=2.303 \times \log _{10} \frac{\mathrm{P}_{\mathrm{i}}}{2 \mathrm{P}_{\mathrm{i}}-\mathrm{P}}$
  2. $\mathrm{k}=\frac{2.303}{\mathrm{t}} \times \log _{10} \frac{\mathrm{P}_{\mathrm{i}}}{2 \mathrm{P}_{\mathrm{i}}-\mathrm{P}}$
  3. $\mathrm{k}=\frac{1}{\mathrm{t}} \ln \frac{2 \mathrm{P}_{\mathrm{i}}-\mathrm{P}}{\mathrm{P}_{\mathrm{i}}}$
  4. $\mathrm{k}=\frac{2.303}{\mathrm{t}} \times \log _{10} \frac{\mathrm{P}_{\mathrm{i}}-\mathrm{P}}{\mathrm{P}_{\mathrm{i}}}$

Solution

$\begin{array}{llcc} & \mathrm{A}_{(\mathrm{g})} \longrightarrow & \mathrm{B}_{(\mathrm{g})}+\mathrm{C}_{(\mathrm{g})} \\ \mathrm{t}=0 . & \mathrm{P}_{\mathrm{i}} & - & - \\ \mathrm{t}=\mathrm{t}, & \mathrm{P}_{\mathrm{i}}-\mathrm{x} & \mathrm{x} & \mathrm{x}\end{array}$ 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, $\begin{aligned} & \mathrm{K}=\frac{2.303}{\mathrm{t}} \cdot \log \frac{\mathrm{P}_{\mathrm{i}}}{\mathrm{P}_{\mathrm{i}}-\mathrm{x}} \\ & =\frac{2.303}{\mathrm{t}} \log \frac{\mathrm{P}_{\mathrm{i}}}{\mathrm{P}_{\mathrm{i}}-\left(\mathrm{P}-\mathrm{P}_{\mathrm{i}}\right)} \\ & \mathrm{K}=\frac{2.303}{\mathrm{t}} \cdot \log \frac{\mathrm{P}_{\mathrm{i}}}{2 \mathrm{P}_{\mathrm{i}}-\mathrm{P}} \end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 1)

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