The activity of a nucleus is inversely proportional to its half of average life. Thus, shorter the half life of an element, greater is its radioactivity, i.e., greater the number of atomsd disintegrating per second. The relation between half life and average life is `t_(1//2) = (0.693)/(lambda) = tau xx 0.693`
or `tau = 1.44 t_(1//2)`
The half life of a radioactive element is 10 years. What percentage of it will decay in 100 years?

Options

A. 0.999
B. 0.1
C. 0.5
D. 0.665

Answer

To solve the problem, we need to determine the percentage of a radioactive element that will decay over a period of 100 years, given its half-life of 10 years.

### Step-by-Step Solution:

1. **Identify the Half-Life and Calculate the Decay Constant (k)**:
The half-life (\(t_{1/2}\)) of the radioactive element is given as 10 years. The decay constant (\(k\)) can be calculated using the formula:
\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{10} = 0.0693 \text{ per year} \]

**Hint**: Remember that the decay constant is a measure of how quickly a radioactive substance decays.

2. **Use the Time-Decay Formula**:
We can use the formula for radioactive decay:
\[ t = \frac{2.303}{k} \log\left(\frac{N_0}{N_t}\right) \]
where:
- \(t\) is the time elapsed (100 years),
- \(N_0\) is the initial number of atoms,
- \(N_t\) is the number of atoms remaining after time \(t\).

3. **Substitute Known Values**:
Substitute \(t = 100\) years and \(k = 0.0693\) into the equation:
\[ 100 = \frac{2.303}{0.0693} \log\left(\frac{N_0}{N_t}\right) \]

4. **Calculate the Logarithmic Term**:
First, calculate \(\frac{2.303}{0.0693}\):
\[ \frac{2.303}{0.0693} \approx 33.24 \]
Now, we can rewrite the equation:
\[ 100 = 33.24 \log\left(\frac{N_0}{N_t}\right) \]
Dividing both sides by 33.24 gives:
\[ \log\left(\frac{N_0}{N_t}\right) = \frac{100}{33.24} \approx 3.00 \]

5. **Convert Logarithm to Exponential Form**:
Converting from logarithmic to exponential form:
\[ \frac{N_0}{N_t} = 10^3 = 1000 \]

6. **Determine Remaining Atoms**:
If we assume \(N_0 = 100\%\), then:
\[ N_t = \frac{N_0}{1000} = \frac{100}{1000} = 0.1 \text{ or } 10\% \]

7. **Calculate the Percentage Decayed**:
The percentage of the element that has decayed is:
\[ \text{Percentage Decayed} = 100\% - N_t = 100\% - 10\% = 90\% \]

### Final Answer:
Thus, the percentage of the radioactive element that will decay in 100 years is **90%**.