The reaction A(g) + 2B(g) `to` C(g) is an elementary reaction. In an experiment involving this reaction, the initial pressures of A and B are ​`P_A` = 0.40 atm and `P_B` = 1.0 atm respectively. When `P_C`=0.3 atm, the rate of the reaction relative to the initial rate is:-

Options

A. `(1)/(12)`
B. `(1)/(50)`
C. `(1)/(25)`
D. none of these

Answer

To solve the problem, we need to determine the rate of the reaction at a certain point compared to its initial rate. The reaction in question is:

\[ A(g) + 2B(g) \rightarrow C(g) \]

### Step-by-Step Solution:

1. **Identify Initial Pressures**:
- The initial pressure of A, \( P_A = 0.40 \, \text{atm} \)
- The initial pressure of B, \( P_B = 1.0 \, \text{atm} \)

2. **Determine Change in Pressures**:
- When \( P_C = 0.3 \, \text{atm} \), we can determine how much of A and B have reacted.
- Since the stoichiometry of the reaction is 1:2:1, for every 1 mole of A that reacts, 2 moles of B react.
- The change in pressure of C, \( \Delta P_C = 0.3 \, \text{atm} \) means that:
- \( \Delta P_A = 0.3 \, \text{atm} \) (for A)
- \( \Delta P_B = 2 \times 0.3 \, \text{atm} = 0.6 \, \text{atm} \) (for B)

3. **Calculate Remaining Pressures**:
- Remaining pressure of A:
\[ P_A = 0.40 \, \text{atm} - 0.3 \, \text{atm} = 0.10 \, \text{atm} \]
- Remaining pressure of B:
\[ P_B = 1.0 \, \text{atm} - 0.6 \, \text{atm} = 0.40 \, \text{atm} \]

4. **Write the Rate Expressions**:
- The rate of the reaction is given by:
\[ \text{Rate} = k [A]^1 [B]^2 \]
- Initial rate (Rate 1):
\[ \text{Rate}_1 = k (0.40) (1.0)^2 = k (0.40) (1.0) = 0.40k \]
- Rate at the point when \( P_C = 0.3 \, \text{atm} \) (Rate 2):
\[ \text{Rate}_2 = k (0.10) (0.40)^2 = k (0.10) (0.16) = 0.016k \]

5. **Calculate the Relative Rate**:
- To find the relative rate:
\[ \frac{\text{Rate}_2}{\text{Rate}_1} = \frac{0.016k}{0.40k} = \frac{0.016}{0.40} = 0.04 \]
- This can be expressed as:
\[ \frac{\text{Rate}_2}{\text{Rate}_1} = \frac{1}{25} \]

### Final Answer:
The rate of the reaction relative to the initial rate is \( \frac{1}{25} \).