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 much time it takes for a radioactive substance to decay 75% when it takes 20 minutes to decay 25%, we can use the principles of first-order kinetics. Here’s a step-by-step solution:

### Step 1: Understand the decay process
Radioactive decay follows first-order kinetics, which means the rate of decay is proportional to the amount of substance that remains. The formula for first-order kinetics is:

\[ k = \frac{2.303}{t} \log \left( \frac{A_0}{A_t} \right) \]

Where:
- \( k \) is the rate constant,
- \( A_0 \) is the initial amount,
- \( A_t \) is the amount remaining after time \( t \).

### Step 2: Calculate the rate constant \( k \)
Given that the substance decays 25% in 20 minutes, we can set:
- \( A_0 = 100 \) (initial amount),
- \( A_t = 100 - 25 = 75 \) (amount remaining after 25% decay),
- \( t = 20 \) minutes.

Now substituting these values into the formula:

\[ k = \frac{2.303}{20} \log \left( \frac{100}{75} \right) \]

Calculating the logarithm:

\[ \log \left( \frac{100}{75} \right) = \log(1.3333) \approx 0.1249 \]

Now substituting this back into the equation for \( k \):

\[ k = \frac{2.303}{20} \times 0.1249 \]

Calculating \( k \):

\[ k \approx \frac{2.303 \times 0.1249}{20} \approx 0.01438 \, \text{min}^{-1} \]

### Step 3: Calculate the time for 75% decay
Now we want to find out how long it takes to decay 75%. If 75% has decayed, then 25% remains:

- \( A_t = 100 - 75 = 25 \).

Using the same formula for \( t \):

\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{A_t} \right) \]

Substituting the values:

\[ t = \frac{2.303}{0.01438} \log \left( \frac{100}{25} \right) \]

Calculating the logarithm:

\[ \log \left( \frac{100}{25} \right) = \log(4) \approx 0.6021 \]

Now substituting this back into the equation for \( t \):

\[ t = \frac{2.303}{0.01438} \times 0.6021 \]

Calculating \( t \):

\[ t \approx \frac{2.303 \times 0.6021}{0.01438} \approx 96.4 \, \text{minutes} \]

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

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