The half life of a radioactive material is 12 days. Calculate the decay constant.

1. Analyze the question

We are given the half-life ($T_{1/2}$) of a radioactive material as 12 days, and we are asked to calculate the decay constant ($\lambda$).

2. Explain the concept

In radioactivity, the half-life ($T_{1/2}$) is the time required for half of the radioactive nuclei in a sample to decay. The relationship between the half-life and the decay constant ($\lambda$) is given by the formula:

Where:

• $T_{1/2} = 12 \text{ days}$
• $\ln(2) \approx 0.693$

To find the decay constant ($\lambda$), we rearrange the formula:

3. Show the solution

• Step 1: Identify the given values. $T_{1/2} = 12 \text{ days}$
• Step 2: Use the formula for the decay constant. $\lambda = \frac{0.693}{12}$
• Step 3: Perform the division. $\lambda = 0.05775 \text{ day}^{-1}$

4. Identify the answer

Comparing our result with the provided options: A. 0.8663 B. 0.04331 C. 0.17325 D. 0.05775

The calculated value matches option D.

5. Tip

When dealing with radioactivity problems in JAMB, always remember that $\ln(2)$ is approximately 0.693. If you ever forget the formula, just remember that the decay constant and half-life are inversely proportional—as one increases, the other decreases. Also, ensure your units are consistent (if the half-life is in days, your decay constant must be in $day^{-1}$).