For the `1^("st")` order reacton: `A(g)rarr 2B(g)+C(s)`, the `t_((1)/(2))="24 min"`. The reaction is carried out taking certain mass of A enclosed in a vessel in which it exerts a pressure of 400 mm Hg. The pressure of the reaction mixture in mm Hg after expiry of 48 min will be

Answer

To solve the problem, we need to determine the pressure of the reaction mixture after 48 minutes for the first-order reaction:

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

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- The initial pressure of gas A, \( P_0 = 400 \, \text{mm Hg} \).
- The half-life of the reaction, \( t_{1/2} = 24 \, \text{min} \).

2. **Determine the Rate Constant (k):**
- For a first-order reaction, the relationship between the half-life and the rate constant is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
- Rearranging gives:
\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{24 \, \text{min}} \approx 0.028875 \, \text{min}^{-1} \]

3. **Calculate the Amount of A Remaining After 48 Minutes:**
- The time elapsed, \( t = 48 \, \text{min} \).
- The first-order reaction equation is:
\[ P = P_0 e^{-kt} \]
- Substituting the values:
\[ P = 400 \, \text{mm Hg} \cdot e^{-0.028875 \cdot 48} \]
- Calculate \( e^{-0.028875 \cdot 48} \):
\[ e^{-1.3842} \approx 0.250 \]
- Therefore, the pressure of A remaining is:
\[ P_A = 400 \cdot 0.250 \approx 100 \, \text{mm Hg} \]

4. **Calculate the Pressure of B Produced:**
- From the stoichiometry of the reaction, for every mole of A that reacts, 2 moles of B are produced.
- The decrease in pressure of A is:
\[ \Delta P_A = P_0 - P_A = 400 - 100 = 300 \, \text{mm Hg} \]
- The pressure of B produced is:
\[ P_B = 2 \times \Delta P_A = 2 \times 300 = 600 \, \text{mm Hg} \]

5. **Calculate Total Pressure of the Reaction Mixture:**
- The total pressure of the reaction mixture after 48 minutes is the sum of the pressures of A and B:
\[ P_{\text{total}} = P_A + P_B = 100 + 600 = 700 \, \text{mm Hg} \]

### Final Answer:
The pressure of the reaction mixture after 48 minutes is **700 mm Hg**.