A radioactive substance takes 20 min to decay 25%. How much time will be taken to decay 75% :

Options

A. 96.4 min
B. 68 min
C. 964 min
D. 680 min

Answer

To solve the problem of how long it takes for a radioactive substance to decay by 75%, given that it takes 20 minutes to decay by 25%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Decay Process**:
- The decay of a radioactive substance is a first-order reaction. This means that the rate of decay is proportional to the amount of substance remaining.

2. **Determine the Half-Life**:
- If a substance decays by 25% in 20 minutes, it means that 75% remains.
- To find the half-life, we can observe that decaying by 50% (which is equivalent to one half-life) would take twice the time it takes to decay by 25%.
- Therefore, if it takes 20 minutes to decay by 25%, it will take 40 minutes to decay by 50%.
- Thus, the half-life (t₁/₂) is 40 minutes.

3. **Use the First-Order Kinetics Formula**:
- The formula for first-order kinetics is:
\[ k = \frac{2.303}{t} \log \left( \frac{[A]_0}{[A]} \right) \]
- Where:
- \( k \) is the rate constant,
- \( t \) is the time,
- \([A]_0\) is the initial concentration,
- \([A]\) is the concentration at time \( t \).

4. **Calculate the Rate Constant (k)**:
- Initially, let’s assume the initial amount of the substance is 100 units.
- After 20 minutes, the amount remaining is 75 units (since it decayed by 25%).
- Plugging these values into the formula:
\[ k = \frac{2.303}{20} \log \left( \frac{100}{75} \right) \]
- Calculate the logarithm:
\[ \log \left( \frac{100}{75} \right) = \log(1.3333) \approx 0.1249 \]
- Now substitute back into the equation:
\[ k = \frac{2.303}{20} \times 0.1249 \approx 0.1438 \, \text{min}^{-1} \]

5. **Calculate Time for 75% Decay**:
- To find the time it takes to decay by 75%, we need to find the time when only 25% remains.
- Using the same formula:
\[ t = \frac{2.303}{k} \log \left( \frac{[A]_0}{[A]} \right) \]
- Here, \([A]_0 = 100\) and \([A] = 25\) (since 75% has decayed):
\[ t = \frac{2.303}{0.1438} \log \left( \frac{100}{25} \right) \]
- Calculate the logarithm:
\[ \log \left( \frac{100}{25} \right) = \log(4) \approx 0.6021 \]
- Now substitute back into the equation:
\[ t = \frac{2.303}{0.1438} \times 0.6021 \approx 96.4 \, \text{minutes} \]

### Final Answer:
The time taken to decay 75% of the radioactive substance is approximately **96.4 minutes**.

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