A radioactive element undergoes two different types of radioactive disintegration, one with disintegration constant `lambda_(1)` and the other with `lambda_(2)`. The half-life of the element is

Options

A. `(0.693)/(lambda_(1)+lambda_(2))`
B. `(0.693)/(lambda_(1)+lambda_(2)//2)`
C. `(0.693)(lambda_(1)+lambda_(2))/(lambda_(1)+lambda_(2))`
D. `(0.693)/(2)(lambda_(1)+lambda_(2))/(lambda_(1)+lambda_(2))`

Answer

To find the half-life of a radioactive element that undergoes two different types of radioactive disintegration, we can follow these steps:

### Step 1: Understand the Disintegration Process
The radioactive element (let's call it A) disintegrates into two different products (B and C) through two different decay processes characterized by their respective disintegration constants, \( \lambda_1 \) and \( \lambda_2 \).

### Step 2: Write the Rate of Disintegration
The total rate of disintegration of A can be expressed as the sum of the rates of formation of B and C:
\[ \text{Rate of disintegration of A} = \text{Rate of formation of B} + \text{Rate of formation of C} \]
This can be mathematically represented as:
\[ -\frac{dN_A}{dt} = \lambda_1 N_A + \lambda_2 N_A \]
where \( N_A \) is the number of radioactive nuclei of A at time \( t \).

### Step 3: Combine the Disintegration Constants
Since both processes contribute to the disintegration of A, we can combine the disintegration constants:
\[ -\frac{dN_A}{dt} = (\lambda_1 + \lambda_2) N_A \]
This means the effective disintegration constant \( \lambda \) for the element A is:
\[ \lambda = \lambda_1 + \lambda_2 \]

### Step 4: Use the Half-Life Formula
The half-life \( T_{1/2} \) of a radioactive substance is related to its disintegration constant by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
Substituting the effective disintegration constant we found:
\[ T_{1/2} = \frac{\ln(2)}{\lambda_1 + \lambda_2} \]

### Step 5: Calculate the Half-Life
We know that \( \ln(2) \approx 0.693 \), so we can write:
\[ T_{1/2} = \frac{0.693}{\lambda_1 + \lambda_2} \]

### Final Result
Thus, the half-life of the radioactive element is:
\[ T_{1/2} = \frac{0.693}{\lambda_1 + \lambda_2} \]

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