For a certain first order reaction, the concentration of the reactant decreases to nine-tenth of its initial value in ten minutes. Concentration of the reactant after 30 minutes from start will be :

Options

A. 64% of its initial value
B. 50% of its initial value
C. 72.9% of its initial value
D. 36% of its initial value

Answer

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

### Step 1: Understand the given information
We know that for a first-order reaction, the concentration of the reactant decreases to nine-tenth (9/10) of its initial value in 10 minutes.

### Step 2: Use the first-order rate equation
For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
\[ k = \frac{2.303}{t} \log \left( \frac{[A_i]}{[A_f]} \right) \]
where:
- \( [A_i] \) = initial concentration
- \( [A_f] \) = final concentration after time \( t \)
- \( t \) = time in minutes

### Step 3: Substitute the known values
In this case, after 10 minutes, the concentration decreases to \( \frac{9}{10} \) of its initial value. Thus, we can write:
\[ [A_f] = \frac{9}{10} [A_i] \]
Substituting this into the equation gives:
\[ k = \frac{2.303}{10} \log \left( \frac{[A_i]}{\frac{9}{10} [A_i]} \right) \]
This simplifies to:
\[ k = \frac{2.303}{10} \log \left( \frac{10}{9} \right) \]

### Step 4: Calculate \( k \)
Calculating \( k \):
\[ k = \frac{2.303}{10} \log \left( \frac{10}{9} \right) \]
Using a calculator, we find:
\[ \log \left( \frac{10}{9} \right) \approx 0.04576 \]
Now substituting this value:
\[ k \approx \frac{2.303}{10} \times 0.04576 \approx 0.01053 \, \text{min}^{-1} \]

### Step 5: Find concentration after 30 minutes
Now we need to find the concentration after 30 minutes using the same first-order rate equation:
\[ [A_f] = [A_i] \cdot 10^{-kt} \]
Substituting \( k \) and \( t = 30 \):
\[ [A_f] = [A_i] \cdot 10^{-0.01053 \times 30} \]
Calculating \( 0.01053 \times 30 \):
\[ 0.01053 \times 30 \approx 0.3159 \]
Thus:
\[ [A_f] = [A_i] \cdot 10^{-0.3159} \]
Calculating \( 10^{-0.3159} \):
\[ 10^{-0.3159} \approx 0.729 \]
So:
\[ [A_f] \approx [A_i] \cdot 0.729 \]

### Step 6: Calculate the percentage of the initial concentration
To find the percentage of the initial concentration:
\[ \text{Percentage} = 0.729 \times 100 \approx 72.9\% \]

### Final Answer
The concentration of the reactant after 30 minutes is approximately **72.9% of its initial value**.

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