question_answer

For a homogeneous gaseous reaction \[\operatorname{A}\xrightarrow{{}}3B,\], if pressure after time t was \[{{\operatorname{P}}_{t}}\] and after completion of reaction, pressure was \[{{\operatorname{P}}_{\infty }}\] then select correct relation

A) \[\operatorname{k}=\frac{1}{t}ln\left( \frac{{{P}_{\infty }}}{3\left( {{P}_{\infty }}-{{P}_{t}} \right)} \right)\]

B) \[\operatorname{k}=\frac{1}{t}ln\left( \frac{2{{P}_{\infty }}}{\left( {{P}_{\infty }}-{{P}_{t}} \right)} \right)\]

C) \[\operatorname{k}=\frac{1}{t}ln\left( \frac{3{{P}_{\infty }}}{2{{P}_{\infty }}-{{P}_{\operatorname{t}}}} \right)\]

D) \[\operatorname{k}=\frac{1}{t}ln\left( \frac{2{{P}_{\infty }}}{3\left( {{P}_{\infty }}-{{P}_{\operatorname{t}}} \right)} \right)\]

Correct Answer: D

Solution :

\[{{P}_{T}}={{P}_{0}}+2x\]

\[x=\frac{{{P}_{T}}{{P}_{0}}}{2}\]

\[k=\frac{1}{t}\ln \left( \frac{{{P}_{0}}}{{{P}_{0}}-x} \right)\]

After long time,

\[x=\frac{{{P}_{T}}-\frac{{{P}_{\infty }}}{3}}{2}\]

\[k=\frac{1}{t}\ln \left( \frac{\frac{{{P}_{\infty }}}{3}}{\frac{{{P}_{\infty }}}{3}-\left( \frac{3{{P}_{T}}-{{P}_{\infty }}}{6} \right)} \right)\]

\[{{P}_{\infty }}=3{{P}_{O}},x=\frac{3{{P}_{T}}-{{P}_{\infty }}}{6}\]

\[k=\frac{1}{t}\ln \left( \frac{{{P}_{\infty }}/3}{\frac{{{P}_{\infty }}}{2}-\frac{{{P}_{T}}}{2}} \right)\]

\[k=\frac{1}{t}\ln \left( \frac{2{{P}_{\infty }}}{3\left( {{P}_{\infty }}-{{P}_{T}} \right)} \right)\]