7/8 of the original mass of a radioactive substance decays in 30 min. What is the half life of the radioactive substance ?

Options

A. a) 5 min
B. b) 7.5 min
C. c) 10 min
D. d) 15 min

Explanation

To solve the problem of finding the half-life of a radioactive substance given that 7/8 of its original mass decays in 30 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Decay Information**:
- If 7/8 of the original mass has decayed, then the remaining mass is:
\[ \text{Remaining mass} = 1 - \frac{7}{8} = \frac{1}{8} \]

2. **Use the Decay Formula**:
- The relationship between the remaining mass (\(N\)) and the original mass (\(N_0\)) can be expressed using the formula:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \]
- Here, \(n\) is the number of half-lives that have passed.

3. **Set Up the Equation**:
- From the previous step, we have:
\[ \frac{1}{8} = \left(\frac{1}{2}\right)^n \]

4. **Express \(\frac{1}{8}\) as a Power of \(\frac{1}{2}\)**:
- We know that:
\[ \frac{1}{8} = \left(\frac{1}{2}\right)^3 \]
- Therefore, we can equate the powers:
\[ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 \]
- This implies:
\[ n = 3 \]

5. **Calculate the Half-Life**:
- We know that \(n\) represents the number of half-lives that have passed in the given time. Since 3 half-lives have occurred in 30 minutes, we can find the half-life (\(T_{1/2}\)) using the formula:
\[ T_{1/2} = \frac{\text{Total time}}{n} \]
- Substituting the values:
\[ T_{1/2} = \frac{30 \text{ minutes}}{3} = 10 \text{ minutes} \]

### Conclusion:
The half-life of the radioactive substance is **10 minutes**.

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