• # question_answer For a first-order homogeneous gaseous reaction $A\to 2B+C,$ if the total pressure after time t was ${{P}_{t}}$ and after long time $(t\to \infty )$ was ${{P}_{\infty }}$ then k in terms of ${{P}_{t}},\,{{P}_{\infty }}$ and t is: A) $k=\frac{2.303}{t}\log \left( \frac{{{P}_{\infty }}}{{{P}_{\infty }}-{{P}_{t}}} \right)$ B) $k=\frac{2.303}{t}\log \left( \frac{2{{P}_{\infty }}}{{{P}_{\infty }}-{{P}_{t}}} \right)$ C) $k=\frac{2.303}{t}\log \left( \frac{2{{P}_{\infty }}}{3\left( {{P}_{\infty }}-{{P}_{t}} \right)} \right)$ D) None of these

 $A$ $\to$ $2B$ $+$ $C$ $t=0$ ${{p}_{i}}$ 0 0 $t$ ${{p}_{i}}-x$ $2x$ $x$ $t\to \infty$ 0 $2{{p}_{i}}$ ${{p}_{i}}$
${{P}_{\infty }}=3{{P}_{i}}$ or ${{P}_{i}}=\frac{{{P}_{\infty }}}{3};{{P}_{i}}+2x={{P}_{i}}$ As we know $k=\frac{2.303}{t}\log \left( \frac{{{P}_{i}}}{{{P}_{i}}-x} \right)$ So $k=\frac{2.303}{t}\log \left( \frac{2{{P}_{\infty }}}{3\left( {{P}_{\infty }}-{{P}_{t}} \right)} \right)$