Kinetics of the reaction `A (g)to 2" B"(g)+C(g)` is followed by measuring the total pressure at different times. It is given that
Initial pressure of A = 0.5 atm.
Total pressure of A after 2 hours = 0.7 atm.
Rate constant of the reaction `=1xx10^(-3)s^(-1)`
What is the rate of reaction `-(d[A])/(dt)` when the total pressure is 0.7 atm ?

Options

A. `2.0xx10^(-4)" M s"^(-1)`
B. `4.0xx10^(-4)" M s"^(-1)`
C. `5.0xx10^(-4)" M s"^(-1)`
D. `7.0xx10^(-4)" M s"^(-1)`

Explanation

To solve the problem step by step, we will analyze the given reaction and the data provided.

### Step 1: Understand the Reaction
The reaction is given as:
\[ A(g) \rightarrow 2B(g) + C(g) \]

### Step 2: Initial Conditions
We are given:
- Initial pressure of A, \( P_{A0} = 0.5 \, \text{atm} \)
- Total pressure after 2 hours, \( P_{\text{total}} = 0.7 \, \text{atm} \)

### Step 3: Relate Changes in Pressure
Let \( x \) be the change in pressure of A that has reacted. The pressures of the products formed will be:
- Pressure of \( B = 2x \)
- Pressure of \( C = x \)

The total pressure at any time can be expressed as:
\[ P_{\text{total}} = P_{A0} - x + 2x + x \]
This simplifies to:
\[ P_{\text{total}} = P_{A0} + 2x \]

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ 0.7 = 0.5 + 2x \]
Now, solve for \( x \):
\[ 0.7 - 0.5 = 2x \]
\[ 0.2 = 2x \]
\[ x = 0.1 \]

### Step 5: Calculate the Pressure of A
Now, we can find the pressure of A at this time:
\[ P_A = P_{A0} - x = 0.5 - 0.1 = 0.4 \, \text{atm} \]

### Step 6: Use the Rate Constant
We are given the rate constant:
\[ k = 1 \times 10^{-3} \, \text{s}^{-1} \]

### Step 7: Calculate the Rate of Reaction
The rate of reaction for a first-order reaction can be expressed as:
\[ \text{Rate} = -\frac{d[A]}{dt} = k \cdot [A] \]

Since we are using pressure in place of concentration (for gases, pressure is proportional to concentration):
\[ \text{Rate} = k \cdot P_A \]
Substituting the values:
\[ \text{Rate} = (1 \times 10^{-3}) \cdot (0.4) \]
\[ \text{Rate} = 4 \times 10^{-4} \, \text{atm/s} \]

### Final Answer
The rate of reaction \( -\frac{d[A]}{dt} \) when the total pressure is 0.7 atm is:
\[ 4 \times 10^{-4} \, \text{atm/s} \]

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