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 `p_(i)`. After lapse of time `t'`. Total pressure of the system increased by x units and became `P_(t)`. the rate constant k for the reaction is given as

Options

A. `k=2.303/t log(P_(i))/(P_(i)-x)`
B. `k=2.303/t log (P_(i))/(2P_(i)-P_(t))`
C. `k = 2.303/t log (P_(i))/(2P_(i) + P_(t))`
D. `k = 2.303/t log P_(i)/(2P_(i)+x)`

Explanation

To find the rate constant \( k \) for the first-order gas phase decomposition reaction \( A(g) \rightarrow B(g) + C(g) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial and Final Conditions**:
- The initial pressure of the system before decomposition of \( A \) is \( P_i \).
- After a time \( t' \), the total pressure of the system increases by \( x \) units, resulting in a total pressure \( P_t \).

2. **Relate Initial and Final Pressures**:
- The total pressure at time \( t' \) can be expressed as:
\[ P_t = P_i + x \]
- Rearranging gives:
\[ x = P_t - P_i \]

3. **Determine Changes in Pressure**:
- The pressure of the reactant \( A \) decreases by \( x \), and the pressures of products \( B \) and \( C \) increase by \( x \) each.
- Therefore, the pressure of \( A \) at time \( t' \) is:
\[ P_i - x \]
- The total pressure at time \( t' \) can also be expressed as:
\[ P_t = (P_i - x) + x + x = P_i + x \]
- This confirms our earlier expression for \( P_t \).

4. **Use the First-Order Kinetic Equation**:
- The integrated rate law for a first-order reaction is given by:
\[ t' = \frac{2.303}{k} \log \left( \frac{P_i}{P_i - x} \right) \]
- Substituting \( x = P_t - P_i \) into the equation gives:
\[ P_i - x = P_i - (P_t - P_i) = 2P_i - P_t \]

5. **Rearranging for the Rate Constant \( k \)**:
- Now substituting back into the rate law:
\[ t' = \frac{2.303}{k} \log \left( \frac{P_i}{2P_i - P_t} \right) \]
- Rearranging to solve for \( k \):
\[ k = \frac{2.303}{t'} \log \left( \frac{P_i}{2P_i - P_t} \right) \]

### Final Expression for the Rate Constant \( k \):
Thus, the rate constant \( k \) for the reaction is given by:
\[ k = \frac{2.303}{t'} \log \left( \frac{P_i}{2P_i - P_t} \right) \]