The reaction `X to ` Product follows first order kinetics. In 40 minutes the concentration of X changes from 0.1 M to 0.025 M. Then the rate of reaction when concentration of X is 0.01 M will be

Options

A. `1.73 xx 10^(-4) min""^(-1)`
B. `3.47 xx 10^(-5) M min""^(-1)`
C. `3.47 xx 10^(-4) M min""^(-1)`
D. `1.73 xx 10^(-5) M min""^(-1)`

Explanation

To solve the problem step by step, we will follow the first-order kinetics principles and the given data.

### Step 1: Understand the given data
The reaction is a first-order reaction where the concentration of reactant X changes from 0.1 M to 0.025 M in 40 minutes. We need to find the rate of the reaction when the concentration of X is 0.01 M.

### Step 2: Use the first-order rate equation
For a first-order reaction, the rate constant \( k \) can be calculated using the equation:
\[ k = \frac{1}{t} \ln \left( \frac{[A_0]}{[A_t]} \right) \]
where:
- \( [A_0] \) = initial concentration of X = 0.1 M
- \( [A_t] \) = concentration of X after time \( t \) = 0.025 M
- \( t \) = time = 40 minutes

### Step 3: Calculate the rate constant \( k \)
Substituting the values into the equation:
\[ k = \frac{1}{40} \ln \left( \frac{0.1}{0.025} \right) \]
Calculating the ratio:
\[ \frac{0.1}{0.025} = 4 \]
Now, substituting this back into the equation:
\[ k = \frac{1}{40} \ln(4) \]
Using the logarithmic identity \( \ln(4) = 2 \ln(2) \):
\[ \ln(4) \approx 1.386 \]
Thus,
\[ k = \frac{1.386}{40} \approx 0.03465 \text{ min}^{-1} \]

### Step 4: Find the time for concentration to change from 0.1 M to 0.01 M
Now we need to find the time taken for the concentration to decrease from 0.1 M to 0.01 M using the same formula:
\[ k = \frac{1}{t} \ln \left( \frac{0.1}{0.01} \right) \]
Calculating the ratio:
\[ \frac{0.1}{0.01} = 10 \]
Now substituting this back into the equation:
\[ k = \frac{1}{t} \ln(10) \]
Using \( \ln(10) \approx 2.303 \):
\[ 0.03465 = \frac{2.303}{t} \]
Rearranging to find \( t \):
\[ t = \frac{2.303}{0.03465} \approx 66.44 \text{ minutes} \]

### Step 5: Calculate the rate of reaction when concentration is 0.01 M
The rate of reaction can be calculated using the formula:
\[ \text{Rate} = -\frac{d[A]}{dt} = k[A] \]
Substituting \( [A] = 0.01 \) M:
\[ \text{Rate} = 0.03465 \times 0.01 = 0.0003465 \text{ M/min} \]
Converting to scientific notation:
\[ \text{Rate} = 3.465 \times 10^{-4} \text{ M/min} \]

### Final Answer
The rate of reaction when the concentration of X is 0.01 M is approximately \( 3.465 \times 10^{-4} \text{ M/min} \).

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