The half-life of a certain radioactive nucleus is 3.2 days. Calculate (i) decay constant (ii) average life of radioactive nucleus.

Answer

To solve the problem, we need to calculate two things: the decay constant (λ) and the average life (τ) of the radioactive nucleus given its half-life (T_half) of 3.2 days.

### Step 1: Calculate the Decay Constant (λ)

The relationship between the half-life (T_half) and the decay constant (λ) is given by the formula:

\[ \lambda = \frac{0.693}{T_{half}} \]

Substituting the given half-life:

\[ \lambda = \frac{0.693}{3.2 \text{ days}} \]

Now, performing the calculation:

\[ \lambda = 0.2166 \text{ per day} \]

### Step 2: Calculate the Average Life (τ)

The average life (τ) of a radioactive nucleus is related to the decay constant (λ) by the formula:

\[ \tau = \frac{1}{\lambda} \]

Now substituting the value of λ we found:

\[ \tau = \frac{1}{0.2166 \text{ per day}} \]

Calculating this gives:

\[ \tau \approx 4.617 \text{ days} \]

### Final Answers

1. Decay constant (λ) = 0.2166 per day
2. Average life (τ) = 4.617 days

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