In a first-order reaction of the type: `A(g)to2B(g)`, the initial and final pressures are `P_(1)` and p respectively. The rate constant can be expressed by

Options

A. `k=1/t"ln"(p_(1))/(2p_(1)-p)`
B. `k=1/t"ln"(p_(1))/(p_(1)-p)`
C. `k=1/t"ln"(p_(1))/(p-p_(1))`
D. `k=1/t"ln"(p_(1))/(p)`

Answer

To determine the rate constant for the first-order reaction \( A(g) \to 2B(g) \) in terms of the initial and final pressures, we can follow these steps:

### Step 1: Understand the Reaction
In this reaction, one mole of gas A produces two moles of gas B. This means that if we start with \( P_1 \) as the initial pressure of A, the change in pressure will affect the total pressure of the system.

### Step 2: Define the Changes in Pressure
Let \( x \) be the amount of A that reacts. Therefore, the change in pressure due to the reaction can be expressed as:
- The pressure of A decreases by \( x \) (since 1 mole of A produces 2 moles of B).
- The pressure of B increases by \( 2x \).

### Step 3: Express the Final Pressure
The final pressure \( P \) can be expressed in terms of the initial pressure \( P_1 \) and the change \( x \):
\[ P = P_1 - x + 2x = P_1 + x \]
This simplifies to:
\[ x = P - P_1 \]

### Step 4: Relate the Change to the Rate Constant
For a first-order reaction, the rate equation is given by:
\[ \frac{d[A]}{dt} = -k[A] \]
Integrating this for a first-order reaction gives us:
\[ \ln\left(\frac{[A]_0}{[A]}\right) = kt \]
Where \( [A]_0 \) is the initial concentration and \( [A] \) is the concentration at time \( t \).

### Step 5: Substitute Pressure for Concentration
Since pressure is directly proportional to concentration for gases, we can substitute pressures into the equation:
\[ \ln\left(\frac{P_1}{P - P_1}\right) = kt \]
Where \( P - P_1 \) represents the pressure of A at time \( t \).

### Step 6: Solve for the Rate Constant \( k \)
Rearranging the equation gives us:
\[ k = \frac{1}{t} \ln\left(\frac{P_1}{P - P_1}\right) \]

### Final Expression
Thus, the rate constant \( k \) for the reaction can be expressed as:
\[ k = \frac{1}{t} \ln\left(\frac{P_1}{P - P_1}\right) \]