The decay constant of a radioactive substance for a and `beta` emission are `lambda_(a)` and `lambda_(beta)` respectively. It the substance emits a and `beta` simultaneously, the average half life of the material will be_______

Answer

To find the average half-life of a radioactive substance that emits alpha (α) and beta (β) particles simultaneously, we can follow these steps:

### Step 1: Understand the relationship between half-life and decay constant
The half-life (T½) of a radioactive substance is related to its decay constant (λ) by the formula:
\[ T_{1/2} = \frac{\ln 2}{\lambda} \]

### Step 2: Write down the decay constants for alpha and beta emissions
Let:
- \( \lambda_{α} \) = decay constant for alpha emission
- \( \lambda_{β} \) = decay constant for beta emission

### Step 3: Determine the total decay constant
When both alpha and beta emissions occur simultaneously, the total decay constant (λ_total) is the sum of the individual decay constants:
\[ \lambda_{total} = \lambda_{α} + \lambda_{β} \]

### Step 4: Relate the total decay constant to the average half-life
Using the relationship from Step 1, we can express the average half-life (T½_total) in terms of the total decay constant:
\[ T_{1/2,total} = \frac{\ln 2}{\lambda_{total}} \]

### Step 5: Substitute the total decay constant into the half-life formula
Substituting the expression for λ_total, we get:
\[ T_{1/2,total} = \frac{\ln 2}{\lambda_{α} + \lambda_{β}} \]

### Final Answer
Thus, the average half-life of the material that emits alpha and beta particles simultaneously is:
\[ T_{1/2,total} = \frac{\ln 2}{\lambda_{α} + \lambda_{β}} \]

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