The decay constant of a radioactive sample is 1. The half life and mean life of the sample, respectively, are given by

Options

A. `1/(lamda)` and `(log_(e)2)/(lamda)`
B. `(log_(e)2)/(lamda)` and `1/(lamda)`
C. `lamdalog_(e)2` and `1/(lamda)`
D. `(lamda)/(log_(e)2)` and `1/(lamda)`

Answer

To solve the problem, we need to find the half-life and mean life of a radioactive sample given that the decay constant (λ) is 1.

### Step 1: Understand the formulas
The half-life (T₁/₂) and mean life (T_avg) of a radioactive sample are related to the decay constant (λ) by the following formulas:
- Half-life: \( T_{1/2} = \frac{\ln(2)}{\lambda} \)
- Mean life: \( T_{avg} = \frac{1}{\lambda} \)

### Step 2: Calculate the half-life
Given that the decay constant \( \lambda = 1 \):
\[ T_{1/2} = \frac{\ln(2)}{1} = \ln(2) \]

### Step 3: Calculate the mean life
Using the same decay constant:
\[ T_{avg} = \frac{1}{1} = 1 \]

### Step 4: State the results
Thus, the half-life and mean life of the sample are:
- Half-life (T₁/₂) = \( \ln(2) \)
- Mean life (T_avg) = 1

### Final Answer
The half-life and mean life of the sample, respectively, are \( \ln(2) \) and 1.

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