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Order of Reaction

The gas phase...

Question

The gas phase thermal decomposition of one mole of di-tert-butyl peroxide, in a constant volume apparatus, yields two moles of acetone and one mole of ethane. If the reaction obeys first order kinetics, the correct expression for the rate constant as a function of time $(t)$ , initial pressure $(p_{i})$ and total pressure $(p)$ .

\frac{1}{t} ln \frac{p_{i}}{(3 p_{i} - p)}

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\frac{1}{t} ln \frac{2 p_{i}}{(3 p_{i} - p)}

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\frac{1}{t} ln \frac{2 p_{i}}{(3 p_{i} - 2 p)}

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\frac{1}{t} ln \frac{p_{i}}{(3 p_{i} - 2 p)}

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Solution

The correct option is B $\frac{1}{t} ln \frac{2 p_{i}}{(3 p_{i} - p)}$
$\begin{matrix} (H_{3} C)_{3} C - O - O - C (C H_{3})_{3} & \to & 2 (C H_{3})_{2} C = O + & C_{2} H_{6} t = 0 & p_{i} & 0 & 0 t = t & p_{i} - x & 2 x & x \end{matrix}$
Total pressure of the reaction is given by,
$p = p_{i} - x + 2 x + x \Rightarrow p = p_{i} + 2 x \Rightarrow x = \frac{p - p_{i}}{2}$

The pressure of the peroxide after time $t$ is
$= p_{i} - x = p_{i} - \frac{p - p_{i}}{2} = \frac{(3 p_{i} - p)}{2}$

Hence the rate constant is
$k_{t} = \frac{1}{t} ln \frac{a}{a - x} = \frac{1}{t} ln \frac{p_{i}}{\frac{(3 p_{i} - p)}{2}} = \frac{1}{t} ln \frac{2 p_{i}}{(3 p_{i} - p)}$

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