For a 1 st order gaseous reaction A → 3 B , if total pressure after time ' t ' was P T and after completion of reaction, total pressure was P ∞ , then select correct relation

Question

For a 1 st order gaseous reaction A → 3 B , if total pressure after time ' t ' was P T ​ and after completion of reaction, total pressure was P ∞ ​ , then select correct relation

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Explaining the Question:

Choose the correct relation.

Concept:

This question is based on the experimental determination of the rate constant using pressure .

Solution:

A → 3B

At $t = 0$ :

$P_{A} = P_{0}$

$P_{B} = 0$

At $t = t$ :

$P_{A} = P_{0} - x$

$P_{B} = 3 x$

At $t = \infty$ :

$P_{A} = 0$

$P_{B} = 3 P_{0}$

At $t = \infty$ , the total pressure $P_{\infty} = P_{A} + P_{B} = 0 + 3 P_{0} = 3 P_{0}$ .

At $t = t$ , the total pressure $P_{t} = P_{A} + P_{B} = (P_{0} - x) + 3 x = P_{0} + 2 x$ .

From $P_{T} = P_{0} + 2 x$

⇒

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

$x = \frac{3 P _{T} - \frac{P _{_{\infty}}}{3}}{6}$

For a first-order reaction, the rate constant $k$ is given by:

$k t = ln (\frac{P _{0}}{P _{0} - x})$

k t = ln (\frac{\frac{P _{_{\infty}}}{3}}{\frac{P _{_{\infty}}}{3} - \frac{3 P _{T}}{6} + \frac{3 P _{_{\infty}}}{6}})

$k t = ln (\frac{2 P _{\infty}}{3 P _{\infty} - 3 P _{T}})$

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

Final Answer: (B)

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