A gaseous reaction `A_(2)(g) rarr B(g) + (1)/(2) C(g)` shows increase in pressure form `100 mm` to `120 mm` in `5 min`. What is the rate of disappearance of `A_(2)` ?

Options

A. 4
B. 8
C. 16
D. 2

Explanation

To find the rate of disappearance of \( A_2 \) in the reaction \( A_2(g) \rightarrow B(g) + \frac{1}{2} C(g) \), we can follow these steps:

### Step 1: Understand the Initial and Final Conditions
Initially, the pressure of \( A_2 \) is given as 100 mm. At the end of the reaction, the pressure increases to 120 mm.

### Step 2: Calculate the Change in Pressure
The change in pressure can be calculated as:
\[ \Delta P = P_{\text{final}} - P_{\text{initial}} = 120 \, \text{mm} - 100 \, \text{mm} = 20 \, \text{mm} \]

### Step 3: Relate Pressure Change to Moles of Gases
In the reaction, for every 1 mole of \( A_2 \) that reacts, 1 mole of \( B \) and 0.5 moles of \( C \) are produced. This means that the total change in pressure due to the formation of products can be expressed as:
\[ \Delta P = P_B + \frac{1}{2} P_C \]
Let \( x \) be the change in pressure due to the formation of \( B \) and \( C \). Then:
\[ P_B = x \quad \text{and} \quad P_C = \frac{1}{2}x \]
The total pressure change is then:
\[ \Delta P = x + \frac{1}{2}x = \frac{3}{2}x \]

### Step 4: Set Up the Equation
From the previous steps, we know:
\[ \frac{3}{2}x = 20 \, \text{mm} \]
Now, we can solve for \( x \):
\[ x = \frac{20 \times 2}{3} = \frac{40}{3} \, \text{mm} \approx 13.33 \, \text{mm} \]

### Step 5: Relate \( x \) to the Disappearance of \( A_2 \)
Since \( A_2 \) disappears as it forms \( B \) and \( C \), the change in pressure of \( A_2 \) is equal to the pressure change \( x \):
\[ \Delta P_{A_2} = -x = -\frac{40}{3} \, \text{mm} \]

### Step 6: Calculate the Rate of Disappearance
The rate of disappearance of \( A_2 \) can be calculated using the formula:
\[ \text{Rate of disappearance of } A_2 = -\frac{d[A_2]}{dt} = \frac{\Delta P_{A_2}}{\Delta t} \]
Given that \( \Delta t = 5 \, \text{min} \):
\[ \text{Rate} = -\left(-\frac{40}{3} \, \text{mm}\right) \div 5 \, \text{min} = \frac{40}{15} \, \text{mm/min} \approx 2.67 \, \text{mm/min} \]

### Final Answer
The rate of disappearance of \( A_2 \) is approximately \( 2.67 \, \text{mm/min} \).