Problem

A radioactive isotope has a half-life of 8 hours. A solution containing 500 million atoms of this isotope is prepared. Determine how many atoms remain after:
(a) 8 hours
(b) 24 hours

Data

• Initial number of atoms ($N_0$): $500$ million
• Half-life ($T_{1/2}$): $8$ hours
• Number of half-lives after 8 hours ($n$): $1$
• Number of half-lives after 24 hours ($n$): $3$

Prerequisite Concepts

1. Decay Formula:
   The number of remaining atoms after $n$ half-lives is given by:

$$N = \left(\frac{1}{2}\right)^n N_0$$

Where:

• $N$: Number of remaining atoms
• $N_0$: Initial number of atoms
• $n$: Number of half-lives

2. Relationship Between Time and Half-Lives:
   The number of half-lives is calculated as:

$$n = \frac{\text{Elapsed Time}}{\text{Half-Life}}$$

Solution

(a) After 8 hours:

1. Determine the number of half-lives:

$$n = \frac{8}{8} = 1$$

2. Apply the decay formula:

$$N = \left(\frac{1}{2}\right)^n N_0$$

Substitute $n = 1$ and $N_0 = 500$:

$$N = \left(\frac{1}{2}\right)^1 \cdot 500$$ $$N = \frac{500}{2} = 250 \, \text{million atoms}$$

(b) After 24 hours:

1. Determine the number of half-lives:

$$n = \frac{24}{8} = 3$$

2. Apply the decay formula:

$$N = \left(\frac{1}{2}\right)^n N_0$$

Substitute $n = 3$ and $N_0 = 500$:

$$N = \left(\frac{1}{2}\right)^3 \cdot 500$$ $$N = \frac{500}{8} = 62.5 \, \text{million atoms}$$

Answer

(a) After 8 hours, 250 million atoms remain.
(b) After 24 hours, 62.5 million atoms remain.