Consider a first order gas phase decomposition reaction given below : $A (g) \to B (g) + C (g)$
The initial pressure of the system before decomposition of A was Pi· After lapse of time 't', total pressure of the system increased by x units and became 'p/ The rate constant k for the reaction is given as:

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$k = \frac{2.303}{t} \log \frac{p_{i}}{2 p_{i} - p_{t}}$

$k = \frac{2.303}{t} \log \frac{p_{i}}{p_{i} + x}$

$k = \frac{2.303}{t} \log \frac{p_{i}}{2 p_{i} + p_{t}}$

$k = \frac{2.303}{t} \log \frac{p_{i}}{p_{i} - 2 x}$

answer is B.

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Detailed Solution

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

At $t = 0, P_{i}^{atm} o atm o atm$

At time $t, {(P_{i} - X)}_{zm} X atm X atm$

Now,

$P_{t} = {(P_{i} - X)}_{em} + X + X = P_{i} + X$

or $X = (P_{t} - P_{i})$

$P_{A} = (P_{1} - X)$ on substituting the value of X

$\begin{aligned} P_{A} = P_{i} - (P_{t} - P_{i}) = 2 P_{i} - P_{t} \\ ∴ K = \frac{2.303}{t} \log \frac{P_{i}}{2 P_{i} - P_{t}} \end{aligned}$

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