A certain radioactive substance has a half life of 5 years. Thus for a nucleus in a sample of the element, probability of decay in 10 years is

Options

A. `50%`
B. `75%`
C. `60%`
D. `100%`

Answer

To find the probability of decay of a nucleus in a radioactive substance over a period of 10 years, given that the half-life of the substance is 5 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life (t½) of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, the half-life is given as 5 years.

2. **Determine the Number of Half-Lives in 10 Years**:
Since we are interested in the decay over 10 years, we need to find out how many half-lives fit into this time frame:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{10 \text{ years}}{5 \text{ years}} = 2 \]
This means that in 10 years, the substance will go through 2 half-lives.

3. **Calculate the Remaining Fraction of Nuclei**:
After each half-life, the remaining fraction of the original amount of nuclei can be calculated using the formula:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]
where \(N_0\) is the initial amount, \(N\) is the remaining amount after \(n\) half-lives. For 2 half-lives:
\[ N = N_0 \left(\frac{1}{2}\right)^2 = N_0 \left(\frac{1}{4}\right) \]
This means that after 10 years, only \(\frac{1}{4}\) of the original nuclei remain.

4. **Calculate the Probability of Decay**:
The probability of decay is given by the fraction of the original nuclei that have decayed. Since \(\frac{1}{4}\) of the original nuclei remain, the fraction that has decayed is:
\[ \text{Fraction decayed} = 1 - \text{Remaining fraction} = 1 - \frac{1}{4} = \frac{3}{4} \]
Therefore, the probability of decay in 10 years is \(\frac{3}{4}\).

5. **Convert to Percentage**:
To express this probability as a percentage:
\[ \text{Probability of decay} = \frac{3}{4} \times 100\% = 75\% \]

### Final Answer:
The probability of decay of a nucleus in the sample over 10 years is **75%**.