The life-life of `Bi^210` is `5` days. What time is taken by `(7//8)^th` part of the sample of decay ?

Options

A. 7.5 days
B. 15 days
C. 10 days
D. 20 days

Answer

To solve the problem of how long it takes for \( \frac{7}{8} \) of a sample of \( Bi^{210} \) to decay, we can follow these steps:

### Step 1: Understand the half-life concept
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. For \( Bi^{210} \), the half-life is given as 5 days.

### Step 2: Determine the remaining fraction after decay
If \( \frac{7}{8} \) of the sample has decayed, then the remaining fraction of the sample is:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]
This means that after the decay, \( \frac{1}{8} \) of the original sample remains.

### Step 3: Relate the remaining fraction to half-lives
We can express the remaining quantity of the sample in terms of half-lives. The remaining quantity after \( n \) half-lives can be expressed as:
\[ \text{Remaining quantity} = N_0 \left( \frac{1}{2} \right)^n \]
where \( N_0 \) is the initial quantity. We need to find \( n \) such that:
\[ N_0 \left( \frac{1}{2} \right)^n = N_0 \left( \frac{1}{8} \right) \]
Dividing both sides by \( N_0 \) (assuming \( N_0 \) is not zero), we get:
\[ \left( \frac{1}{2} \right)^n = \frac{1}{8} \]

### Step 4: Solve for \( n \)
We know that:
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^3 \]
Thus, we can equate the exponents:
\[ n = 3 \]
This means it takes 3 half-lives for \( \frac{7}{8} \) of the sample to decay.

### Step 5: Calculate the total time taken
Since each half-life is 5 days, the total time taken for 3 half-lives is:
\[ \text{Total time} = n \times \text{half-life} = 3 \times 5 \text{ days} = 15 \text{ days} \]

### Final Answer
The time taken by \( \frac{7}{8} \) part of the sample to decay is **15 days**.

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