The reaction `A(g)+2B(g)rarrC(g)+D(g)` is an elementary process. In an experimetn, the initial partial pressure of `A` and `B` are `P_(A)= 0.60` and `P_(B)= 0.80 atm`. When `P_(C )=0.2 atm`, the rate of reaction relative to the initial rate is

Options

A. `1//48`
B. `1//24`
C. `9//16`
D. `1//6`

Explanation

To solve the problem, we need to determine the rate of reaction relative to the initial rate when the partial pressure of \( C \) is \( 0.2 \, \text{atm} \).

### Step-by-Step Solution:

1. **Write the Rate Law for the Reaction:**
The given reaction is:
\[ A(g) + 2B(g) \rightarrow C(g) + D(g) \]
Since it is an elementary process, the rate law can be expressed as:
\[ R = k [A]^1 [B]^2 \]
Here, \( k \) is the rate constant.

2. **Convert Partial Pressures to Concentrations:**
The initial partial pressures are given as:
\[ P_A = 0.60 \, \text{atm}, \quad P_B = 0.80 \, \text{atm} \]
We can directly use these pressures in the rate law since pressure is proportional to concentration for gases.

3. **Calculate the Initial Rate \( R_1 \):**
Substitute the initial pressures into the rate law:
\[ R_1 = k (0.60)^1 (0.80)^2 \]
\[ R_1 = k (0.60) (0.64) = k (0.384) \]

4. **Determine the Change in Pressures After Reaction:**
When \( P_C = 0.2 \, \text{atm} \), we can find the changes in the pressures of \( A \) and \( B \):
- For every mole of \( C \) produced, 1 mole of \( A \) and 2 moles of \( B \) are consumed.
- Thus, if \( P_C = 0.2 \, \text{atm} \), then:
\[ P_A = 0.60 - 0.20 = 0.40 \, \text{atm} \]
\[ P_B = 0.80 - 2 \times 0.20 = 0.40 \, \text{atm} \]

5. **Calculate the Rate at This Point \( R_2 \):**
Substitute the new pressures into the rate law:
\[ R_2 = k (0.40)^1 (0.40)^2 \]
\[ R_2 = k (0.40) (0.16) = k (0.064) \]

6. **Calculate the Relative Rate \( \frac{R_2}{R_1} \):**
Now, we can find the ratio of the rates:
\[ \frac{R_2}{R_1} = \frac{k (0.064)}{k (0.384)} = \frac{0.064}{0.384} \]
\[ \frac{R_2}{R_1} = \frac{1}{6} \]

### Final Answer:
The rate of reaction relative to the initial rate is:
\[ \frac{R_2}{R_1} = \frac{1}{6} \]