If the total pressure of the gases is found to be 200 torr after 23 sec. and 300 torr upon the complete decomposition of A after a very long time, then the rate constant of the given reaction is

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Solution

Q.
Consider the following first order gas phase reaction at constant temperature

$A (g) \to 2 B (g) + C (g)$

If the total pressure of the gases is found to be 200 torr after 23 sec. and 300 torr upon the complete decomposition of A after a very long time, then the rate constant of the given reaction is ______ $\times 10^{- 2} s^{- 1}$ (nearest integer).

[Given: $\log_{10} (2) = 0.301$ ] [2024]

Ans.
(3)

$A(g)$ $\to$ $2 B(g)$    $+$ $C(g)$

t = 0 $P^{o}$ 0 0

(press.)

t = 23 s    $P^{o} - P$    $2 P$    $P$    $Total = P^{o} + 2 P = 200$ ...(i)

(press.)

t = $\infty$ 0 $2 P^{o}$ $P^{o}$ $Total = 3 P^{o} = 300$ ...(ii)

(press.)

From (i) and (ii)

$P^{o} = 100$ torr, $P = 50$ torr

For first order reaction, rate constant k is given as:

$k = \frac{2.303}{t} \log_{10} \frac{P^{o}}{P^{o} - P}$

   $k = \frac{2.303}{23} \log_{10} \frac{100}{100 - 50}$

   $k = \frac{2.303}{23} \log_{10} 2 s^{- 1} = \frac{2.303}{23} \times 0.301 s^{- 1} = 0.0301 s^{- 1} \approx 3 \times 10^{- 2} s^{- 1}$

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