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 use the first-order reaction kinetics formula and the information given in the question.

### Step 1: Understand the given information
We know that the concentration of the reactant decreases to nine-tenths of its initial value in 10 minutes. This means:
- Initial concentration, \( [A_0] = A_0 \)
- Concentration after 10 minutes, \( [A] = \frac{9}{10} A_0 \)
- Time, \( t = 10 \) minutes

### Step 2: Use the first-order rate constant formula
For a first-order reaction, the rate constant \( k \) can be calculated using the formula:
\[ k = \frac{2.303}{t} \log \left( \frac{[A_0]}{[A]} \right) \]
Substituting the values we have:
\[ k = \frac{2.303}{10} \log \left( \frac{A_0}{\frac{9}{10} A_0} \right) \]
This simplifies to:
\[ k = \frac{2.303}{10} \log \left( \frac{10}{9} \right) \]

### Step 3: Calculate the value of \( k \)
Now, we can calculate \( k \):
\[ k = \frac{2.303}{10} \log \left( \frac{10}{9} \right) \]
Using a calculator, we find:
\[ \log \left( \frac{10}{9} \right) \approx 0.04576 \]
Thus:
\[ k \approx \frac{2.303}{10} \times 0.04576 \approx 0.01053 \text{ min}^{-1} \]

### Step 4: Find the concentration after 30 minutes
Now we want to find the concentration of the reactant after 30 minutes. We will use the same first-order formula:
\[ k = \frac{2.303}{t} \log \left( \frac{[A_0]}{[A]} \right) \]
Where \( t = 30 \) minutes. Rearranging gives:
\[ \log \left( \frac{[A_0]}{[A]} \right) = k \cdot t \cdot \frac{2.303}{1} \]
Substituting the known values:
\[ \log \left( \frac{[A_0]}{[A]} \right) = 0.01053 \times 30 \times 2.303 \]
Calculating this:
\[ \log \left( \frac{[A_0]}{[A]} \right) \approx 0.01053 \times 30 \times 2.303 \approx 0.726 \]

### Step 5: Calculate the ratio of concentrations
Taking the antilog:
\[ \frac{[A_0]}{[A]} = 10^{0.726} \approx 5.32 \]
Thus:
\[ [A] = \frac{[A_0]}{5.32} \]

### Step 6: Calculate the concentration as a fraction of the initial concentration
To find the fraction of the initial concentration:
\[ \frac{[A]}{[A_0]} = \frac{1}{5.32} \approx 0.188 \]

### Step 7: Convert to percentage
To express this as a percentage:
\[ \text{Percentage} = 0.188 \times 100 \approx 18.8\% \]

### Final Answer
The concentration of the reactant after 30 minutes from the start will be approximately **18.8%** of its initial value.

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