Radioactive Decay and Half-Life in GOB Chemistry

Radioactive Decay

Introduction to Radioactive Decay

Radioactive decay is a process by which unstable atomic nuclei lose energy by emitting radiation. This process is fundamental in understanding the behavior of radioisotopes in chemistry and medicine.

• Radioisotope: An isotope of an element that has an unstable nucleus and emits radiation as it decays.

• Half-Life (t1/2): The amount of time required for half of a radioactive isotope to decay.

Half-Life Concept

The half-life of a radioisotope is a constant value that describes how quickly the isotope decays. After each half-life, half of the original amount of the isotope remains.

• Key Point: After one half-life, 50% of the original radioisotope remains; after two half-lives, 25% remains; after three half-lives, 12.5% remains, and so on.

• Equation: $\text{Fraction Remaining} = \left(\frac{1}{2}\right)^{n}$ where n is the number of half-lives elapsed.

Example Calculation

• Example: If you start with 100 g of a radioisotope and 3 half-lives pass, the remaining amount is: $100 \times \left(\frac{1}{2}\right)^3 = 12.5 \text{ g}$

Radioisotope Remaining

Calculating Fraction and Final Amount

The fraction, percentage, and final amount of a radioisotope after each half-life can be calculated using specific formulas.

• Fraction Remaining Formula: $\text{Fraction Remaining} = \left(\frac{1}{2}\right)^{n}$

• Final Amount Formula: $\text{Final Amount Remaining} = \text{Initial Amount} \times \left(\frac{1}{2}\right)^{n}$

• Percentage Remaining: Multiply the fraction remaining by 100% to get the percentage of radioisotope left.

Table: Fraction and Percentage Remaining After Each Half-Life

Number of Half-Lives (n)	Fraction Remaining	Percentage Remaining (%)
0	1	100
1	0.5	50
2	0.25	25
3	0.125	12.5
4	0.0625	6.25

Application Example

• Example: Sodium-24, used in human circulation studies, has a half-life of 15 days. What percentage remains after 3 months (30 days)? Solution: Number of half-lives: $n = \frac{30}{15} = 2$ Fraction remaining: $\left(\frac{1}{2}\right)^2 = 0.25$ Percentage remaining: $0.25 \times 100\% = 25\%$

Practice Problems

Sample Questions

• Half-life of Arsenic-74: If a sample initially contains 100.00 mg arsenic-74 (half-life = 18 days), what mass remains after 72 days? Number of half-lives: $n = \frac{72}{18} = 4$ Final amount: $100.00 \times \left(\frac{1}{2}\right)^4 = 6.25 \text{ mg}$

• Half-life of Iodine-131: If the half-life is 8.021 days, what percentage remains after 40.11 days? Number of half-lives: $n = \frac{40.11}{8.021} = 5$ Fraction remaining: $\left(\frac{1}{2}\right)^5 = 0.03125$ Percentage remaining: $0.03125 \times 100\% = 3.125\%$

• Concentration after 4 half-lives: If 0.325 mol of CO2 is placed in a 5.0 L vessel, what is the concentration after 4 half-lives? Final amount: $0.325 \times \left(\frac{1}{2}\right)^4 = 0.0203 \text{ mol}$ Concentration: $\frac{0.0203}{5.0} = 0.00406 \text{ mol/L}$

Graphical Representation

Radioactive decay can be visualized as a plot of remaining percentage versus time, showing an exponential decrease. The half-life can be determined from the time interval corresponding to a drop from 100% to 50% remaining.

Summary Table: Key Terms and Definitions

Term	Definition
Radioisotope	Isotope with an unstable nucleus that emits radiation
Half-Life (t1/2)	Time required for half of a radioactive sample to decay
Fraction Remaining	Proportion of radioisotope left after a given number of half-lives
Final Amount	Mass or moles of radioisotope remaining after decay

Additional info: The notes include practice problems and graphical analysis, which are common in GOB Chemistry courses to reinforce understanding of radioactive decay and half-life calculations.