The half-life of a radioactive substance is 30 min. The time for disintegration \(\frac{7}{8}\)th part of its original mass will be:
- T
- 2T
- 3T
- 8T
The Correct Option is C
Solution and Explanation
Step 1: Understanding the decay process. Radioactive decay follows the equation: \[ N = N_0 \times \left( \frac{1}{2} \right)^n \] Where \( N_0 \) is the initial number of nuclei, \( N \) is the number of nuclei left after \( n \) half-lives, and \( n \) is the number of half-lives elapsed. If \( \frac{7}{8} \) of the substance has disintegrated, then \( \frac{1}{8} \) of the substance remains. This means the number of remaining radioactive nuclei is \( \frac{N_0}{8} \).
Step 2: Finding the number of half-lives. Using the decay equation: \[ \frac{N_0}{2^n} = \frac{N_0}{8} \] Simplifying: \[ 2^n = 8 \] \[ n = 3 \] So, 3 half-lives have elapsed.
Step 3: Calculating the time. Since the half-life is \( T \), the time taken for the disintegration of \( \frac{7}{8} \) of the substance is: \[ \text{Time} = 3 \times T = 3T \]
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