Radioactive decay follows first-order kinetic. The mean life and half-life of nuclear decay process are `tau = 1// lambda` and `t_(1//2) = 0.693//lambda`. Therefore are a number of radioactive elements in nature, their abundance is directly proportional to half life. The amount remaining after `n` half lives of radioactive elements can be calculated using the relation:
`N = N_(0) ((1)/(2))^(n)`
Amount of radioactive elements (activity) decreases with passage of time as

Options

A. Linearly
B. Exponentially
C. Parabolically
D. All of these

Answer

To solve the problem regarding the decay of radioactive elements and their activity over time, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Half-Life**:
- The half-life (\(t_{1/2}\)) is the time required for half of the radioactive substance to decay. The relationship between the initial amount (\(N_0\)) and the remaining amount (\(N\)) after \(n\) half-lives is given by:
\[ N = N_0 \left(\frac{1}{2}\right)^n \]

2. **Setting Up a Table**:
- We can create a table to illustrate how the amount of radioactive substance decreases over time. The table will show the time elapsed (in half-lives) and the corresponding remaining amount.

| Time (n) | Remaining Amount (N) |
|----------|-------------------------------|
| 0 | \(N_0\) |
| 1 | \(\frac{N_0}{2}\) |
| 2 | \(\frac{N_0}{4}\) |
| 3 | \(\frac{N_0}{8}\) |

3. **Calculating Remaining Amounts**:
- For \(n = 0\):
\[ N = N_0 \left(\frac{1}{2}\right)^0 = N_0 \]
- For \(n = 1\):
\[ N = N_0 \left(\frac{1}{2}\right)^1 = \frac{N_0}{2} \]
- For \(n = 2\):
\[ N = N_0 \left(\frac{1}{2}\right)^2 = \frac{N_0}{4} \]
- For \(n = 3\):
\[ N = N_0 \left(\frac{1}{2}\right)^3 = \frac{N_0}{8} \]

4. **Observing the Trend**:
- From the table and calculations, we can observe that the remaining amount of the radioactive substance decreases exponentially as time progresses. This indicates that the activity of the radioactive elements decreases over time.

5. **Conclusion**:
- The amount of radioactive elements (activity) decreases with the passage of time in an exponential manner, confirming that the activity diminishes as more half-lives pass.