The reaction was started with reactant A only. Which of the following expression is correct for rate constant ? [2025]

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Solution

Q.
$A (g) \to B (g) + C (g)$ is a first order reaction. [2025]

Time T $\infty$

$P_{system}$ $P_{t}$ $P_{\infty}$

The reaction was started with reactant A only.

Which of the following expression is correct for rate constant $k$

1 $k = \frac{1}{t} \ln \frac{2 (P_{\infty} - P_{t})}{P_{t}}$

2 $k = \frac{1}{t} \ln \frac{P_{\infty}}{P_{t}}$

3 $k = \frac{1}{t} \ln \frac{P_{\infty}}{2 (P_{\infty} - P_{t})}$

4 $k = \frac{1}{t} \ln \frac{P_{\infty}}{(P_{\infty} - P_{t})}$

Ans.
(3)

$\begin{matrix} A(g) & \to & B(g) & + & C(g) \\ t =0 (press) & P_{o} & 0 & 0 \\ t = t (press) & P_{o} - p & p & p & Total = P_{o} + p = P_{t} - I \\ t =∞ (press) & 0 & P_{o} & P_{o} & Total = 2 P_{o} = P_{\infty} - I I \end{matrix}$

From I and II:

$P_{o} = \frac{P_{\infty}}{2}, p = P_{t} - \frac{P_{\infty}}{2}$

For first order reaction, rate constant $k$ is:

$k = \frac{2.303}{t} \log \frac{P_{o}}{P_{o} - p}$

$= \frac{2.303}{t} \log \frac{\frac{P_{\infty}}{2}}{\frac{P_{\infty}}{2} - (P_{t} - \frac{P_{\infty}}{2})}$

$= \frac{2.303}{t} \log \frac{P_{\infty}}{2 (P_{\infty} - P_{t})}$

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