The substances `A(t_((1)/(2))=5" min") and B(t_((1)/(2))="15 min")` follow first order kinetics, are taken is such a way that initially `[A]=4[B]`. The time after which the concentration of both substances will be equal is :

Options

A. 5 min
B. 10 min
C. 15 min
D. 20 min

Answer

To solve the problem, we need to determine the time at which the concentrations of substances A and B become equal, given that they follow first-order kinetics and have different half-lives.

### Step-by-Step Solution:

1. **Identify Initial Concentrations**:
- Let the concentration of substance B be \( [B] = x \).
- Since \( [A] = 4[B] \), we have \( [A] = 4x \).

2. **Determine Half-Lives**:
- The half-life of substance A, \( t_{1/2}^A = 5 \) minutes.
- The half-life of substance B, \( t_{1/2}^B = 15 \) minutes.

3. **Calculate Concentration of A Over Time**:
- The concentration of A after \( t \) minutes can be calculated using the first-order kinetics formula:
\[ [A] = [A]_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}^A}} = 4x \left( \frac{1}{2} \right)^{\frac{t}{5}} \]

4. **Calculate Concentration of B Over Time**:
- Similarly, the concentration of B after \( t \) minutes is:
\[ [B] = [B]_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}^B}} = x \left( \frac{1}{2} \right)^{\frac{t}{15}} \]

5. **Set Concentrations Equal**:
- We want to find the time \( t \) when \( [A] = [B] \):
\[ 4x \left( \frac{1}{2} \right)^{\frac{t}{5}} = x \left( \frac{1}{2} \right)^{\frac{t}{15}} \]
- Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 4 \left( \frac{1}{2} \right)^{\frac{t}{5}} = \left( \frac{1}{2} \right)^{\frac{t}{15}} \]

6. **Simplify the Equation**:
- Taking logarithm base \( \frac{1}{2} \) on both sides:
\[ \log_{\frac{1}{2}}(4) + \frac{t}{5} = \frac{t}{15} \]
- Since \( \log_{\frac{1}{2}}(4) = 2 \) (because \( 4 = 2^2 \)):
\[ 2 + \frac{t}{5} = \frac{t}{15} \]

7. **Solve for \( t \)**:
- Rearranging gives:
\[ 2 = \frac{t}{15} - \frac{t}{5} \]
- Finding a common denominator (15):
\[ 2 = \frac{t}{15} - \frac{3t}{15} = -\frac{2t}{15} \]
- Multiplying both sides by -15:
\[ -30 = 2t \implies t = 15 \text{ minutes} \]

### Final Answer:
The time after which the concentration of both substances will be equal is **15 minutes**.

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