`(7)/(8)th` of nuclei have been disintegrated in 2 minutes after fresh sample was prepared. The half life of the sample is

Options

A. 30 seconds
B. 1 minute
C. 40 seconds
D. 20 seconds

Explanation

To find the half-life of the sample given that \( \frac{7}{8} \) of the nuclei have disintegrated in 2 minutes, we can follow these steps:

### Step 1: Determine the remaining fraction of nuclei
If \( \frac{7}{8} \) of the nuclei have disintegrated, then the fraction of nuclei remaining is:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]

### Step 2: Relate the remaining nuclei to half-lives
The relationship between the remaining number of nuclei and the number of half-lives can be expressed as:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
where \( N \) is the remaining number of nuclei, \( N_0 \) is the initial number of nuclei, and \( n \) is the number of half-lives.

From Step 1, we have:
\[ \frac{N}{N_0} = \frac{1}{8} \]
This implies:
\[ \left(\frac{1}{2}\right)^n = \frac{1}{8} \]

### Step 3: Solve for \( n \)
We know that \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 \). Therefore, we can equate the exponents:
\[ n = 3 \]

### Step 4: Relate the total time to half-lives
We are given that this disintegration occurs over a period of 2 minutes. Since \( n = 3 \) half-lives occur in this time, we can find the duration of one half-life:
\[ \text{Total time} = n \times \text{half-life} \]
\[ 2 \text{ minutes} = 3 \times \text{half-life} \]

### Step 5: Calculate the half-life
Convert 2 minutes to seconds:
\[ 2 \text{ minutes} = 2 \times 60 = 120 \text{ seconds} \]
Now, substituting into the equation:
\[ 120 \text{ seconds} = 3 \times \text{half-life} \]
\[ \text{half-life} = \frac{120 \text{ seconds}}{3} = 40 \text{ seconds} \]

### Final Answer
The half-life of the sample is \( 40 \text{ seconds} \).

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