The half life of a given radioactive nuclide is 138.6 days. What is the mean life of this nuclide? After how much time will a given sample of this nuclide get reduced to only 12.5% of its initial value?

Explanation

To solve the problem, we need to find two things: the mean life of the radioactive nuclide and the time it takes for the sample to reduce to 12.5% of its initial value.

### Step 1: Calculate the Mean Life (τ)
The mean life (τ) of a radioactive nuclide is related to its half-life (T) by the formula:
\[ \tau = 1.44 \times T \]
Given that the half-life \( T = 138.6 \) days, we can substitute this value into the formula:
\[ \tau = 1.44 \times 138.6 \]
Calculating this gives:
\[ \tau = 199.584 \text{ days} \approx 199.58 \text{ days} \]

### Step 2: Determine the Time for the Sample to Reduce to 12.5%
To find out how long it takes for the sample to reduce to 12.5% of its initial value, we first express 12.5% in fraction form:
\[ \frac{N}{N_0} = 12.5\% = \frac{12.5}{100} = \frac{1}{8} \]
This means that the sample has gone through a certain number of half-lives (N). Since each half-life reduces the quantity of the substance by half, we can express this as:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^N \]
Setting this equal to \(\frac{1}{8}\):
\[ \left(\frac{1}{2}\right)^N = \frac{1}{8} \]
We know that \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 \), so:
\[ N = 3 \]

Now, we can find the time \( t \) it takes for the sample to reduce to 12.5%:
\[ t = N \times T \]
Substituting the values:
\[ t = 3 \times 138.6 \]
Calculating this gives:
\[ t = 415.8 \text{ days} \]

### Final Answers:
1. Mean life (τ): 199.58 days
2. Time to reduce to 12.5%: 415.8 days