A radioactive nucleus can decay by two different processes . The half life for the first process is `t_1` and that for the second process is `t_2`. If effective half life is t, then

Options

A. `t=t_1+t_2`
B. `1/t=1/(t_1)+1/(t_2)`
C. `t=(2t_1t_2)/(t_1+t_2)`
D. `t=(t_1+t_2)/2`

Answer

To solve the problem of finding the effective half-life \( t \) of a radioactive nucleus that can decay by two different processes with half-lives \( t_1 \) and \( t_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Decay Rate**:
The decay rate of a radioactive substance is proportional to the number of undecayed nuclei present. This can be expressed mathematically as:
\[ \frac{dN}{dt} = -\lambda N \]
where \( N \) is the number of nuclei and \( \lambda \) is the decay constant.

2. **Decay Constants for Each Process**:
For the two decay processes, we can denote their decay constants as \( \lambda_1 \) and \( \lambda_2 \). The relationship between half-life and decay constant is given by:
\[ \lambda_1 = \frac{\ln 2}{t_1} \quad \text{and} \quad \lambda_2 = \frac{\ln 2}{t_2} \]

3. **Total Decay Rate**:
The total decay rate for the nucleus considering both processes can be expressed as:
\[ \frac{dN}{dt} = -(\lambda_1 + \lambda_2) N \]
This implies that the effective decay constant \( \lambda_{\text{effective}} \) is:
\[ \lambda_{\text{effective}} = \lambda_1 + \lambda_2 \]

4. **Effective Half-Life**:
The effective half-life \( t \) can then be related to the effective decay constant:
\[ \lambda_{\text{effective}} = \frac{\ln 2}{t} \]
Therefore, we can write:
\[ \frac{\ln 2}{t} = \lambda_1 + \lambda_2 \]

5. **Substituting the Decay Constants**:
Substituting the expressions for \( \lambda_1 \) and \( \lambda_2 \):
\[ \frac{\ln 2}{t} = \frac{\ln 2}{t_1} + \frac{\ln 2}{t_2} \]

6. **Simplifying the Equation**:
Dividing through by \( \ln 2 \) (assuming \( \ln 2 \neq 0 \)):
\[ \frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2} \]

7. **Final Relation**:
Thus, the final relationship between the effective half-life \( t \) and the half-lives \( t_1 \) and \( t_2 \) is:
\[ \frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2} \]

### Final Answer:
The effective half-life \( t \) is given by:
\[ \frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2} \]