Rate Constant Expression for First Order Gas Phase Reaction

To determine the rate constant expression for the given first-order gas phase reaction:

$\text{A(g)} \rightarrow \text{B(g)} + \text{C(g)}$

Let $p_i$ be the initial pressure of gas $\text{A}$ at time $t = 0$, and let $x$ be the decrease in partial pressure of gas $\text{A}$ at time $t$.

The partial pressures of the components at different time intervals can be expressed as:

$$\begin{array}{lcccc} & \text{A(g)} & \rightarrow & \text{B(g)} & + & \text{C(g)} \\ \text{At } t = 0: & p_i & & 0 & & 0 \\ \text{At time } t: & p_i - x & & x & & x \end{array}$$

The total pressure of the reaction mixture at time $t$, denoted as $p_t$, is the sum of the partial pressures of all gaseous species present at time $t$:

$p_t = (p_i - x) + x + x$ $p_t = p_i + x$

Solving for $x$: $x = p_t - p_i$

Now, substituting $x$ into the expression for the partial pressure of $\text{A}$ at time $t$ ($p_A$):

$p_A = p_i - x = p_i - (p_t - p_i) = 2p_i - p_t$

For a first-order reaction, the rate constant $k$ in terms of partial pressures is given by:

$k = \frac{1}{t} \ln \left( \frac{p_i}{p_A} \right)$

Substituting the expression for $p_A$:

$k = \frac{1}{t} \ln \left( \frac{p_i}{2p_i - p_t} \right)$

This matches Option A.