The half-life is related to the decay constant by the relation.
`T_(1//2) = (0.693)/lambda`
The average life of a sample is expressed by the relation
`tau = 1/lambda`
The average life and half-life are related as

Options

A. `T_(1//2)=1.44 tau`
B. `tau =1.44 T_(1//2)`
C. `T_(1//2)=1/2tau`
D. `tau=1/2T_(1//2)`

Answer

To find the relationship between the average life (τ) and the half-life (T₁/₂) of a radioactive sample, we start with the given formulas:

1. The half-life is related to the decay constant (λ) by the equation:
\[ T_{1/2} = \frac{0.693}{\lambda} \]

2. The average life of a sample is expressed by the relation:
\[ \tau = \frac{1}{\lambda} \]

Now, we want to express the average life in terms of the half-life.

### Step 1: Express λ in terms of T₁/₂
From the half-life equation, we can express the decay constant (λ):
\[ \lambda = \frac{0.693}{T_{1/2}} \]

### Step 2: Substitute λ into the average life equation
Now, substitute this expression for λ into the average life equation:
\[ \tau = \frac{1}{\lambda} = \frac{1}{\frac{0.693}{T_{1/2}}} \]

### Step 3: Simplify the equation
This simplifies to:
\[ \tau = \frac{T_{1/2}}{0.693} \]

### Step 4: Relate τ to T₁/₂
Now, we can rearrange this to find the relationship between τ and T₁/₂:
\[ \tau = 1.4427 \cdot T_{1/2} \]
This can be approximated to:
\[ \tau \approx 1.44 \cdot T_{1/2} \]

### Conclusion
Thus, the relationship between the average life (τ) and the half-life (T₁/₂) is:
\[ \tau = 1.44 \cdot T_{1/2} \]