Radioactive Half-Life: Concepts, Calculations, and Applications

Radioactive Half-Life

Concept of Radioactive Half-Life

The radioactive half-life (t1/2) is the amount of time required for half of a radioactive isotope to decay. This concept is fundamental in nuclear chemistry and is used to describe the rate at which unstable nuclei lose their radioactivity.

• Radioisotope: An isotope with an unstable nucleus that emits radiation as it decays.

• Half-life: The time required for half of the radioactive atoms in a sample to decay.

Example: What is the half-life of the radioisotope that shows the following data of remaining percentage vs. time?

Time (days)	% Remaining
0	100
1	50
2	25
3	12.5
4	6.25

Answer: $t_{1/2} = 1 \text{ day}$

Method 1: Direct Calculation of Half-Life or Rate Constant

When dealing with only the half-life and the decay constant, use the radioactive half-life equation:

• Equation: $t_{1/2} = \frac{0.693}{k}$

• k: Decay constant (rate constant for radioactive decay)

• Half-life does not depend on initial concentration and is constant throughout the reaction.

Example: If the decay constant of plutonium-244 is $6.84 \times 10^{-7} \text{ years}^{-1}$ at 25°C, what is its half-life?

Answer: $t_{1/2} = \frac{0.693}{6.84 \times 10^{-7}} = 1.01 \times 10^{6} \text{ yr}$

Method 2: Radioactive Nuclei Concentrations

This method uses the radioactive half-life equation and the radioactive integrated rate law to solve for the half-life, time, initial, and final radioactive nuclei concentrations.

• Integrated Rate Law: $\ln\left(\frac{N_0}{N}\right) = kt$

• N0: Initial amount of radioactive nuclei

• N: Remaining amount after time t

Example: A sample of radon-222 has an initial particle activity of $8.5 \times 10^4$ disintegrations per minute. After 7 days, its activity is $4.7 \times 10^4$ dpm. What is the half-life of radon-222?

Answer: $t_{1/2} \approx 3.8 \text{ days}$

Method 3: Fractions and Percentages

This method uses the radioactive half-life equation and the radioactive integrated rate law to determine the fraction or percentage of radioactive nuclei remaining after a given time.

• Fraction remaining: $\left(\frac{1}{2}\right)^{n}$, where n = number of half-lives elapsed

• Percentage remaining: $\left(\frac{1}{2}\right)^{n} \times 100\%$

Example: The half-life of iodine-131, used in thyroid therapy, is 8.025 days. What fraction of iodine-131 remains in a sample that is estimated to be 40.125 days old?

Answer: $n = \frac{40.125}{8.025} = 5$ half-lives Fraction remaining: $\left(\frac{1}{2}\right)^5 = 0.03125$ Percentage remaining: $0.03125 \times 100\% = 3.125\%$

Practice Problems

• Problem 1: The half-life of arsenic-74 is about 18 days. If a sample initially contains $5.1 \times 10^3$ mg arsenic-74, what mass (in mg) would be left after 60 days?

• Solution: $n = \frac{60}{18} \approx 3.33$ Mass remaining: $5.1 \times 10^3 \times \left(\frac{1}{2}\right)^{3.33} \approx 775.52$ mg

• Problem 2: What percentage of carbon-14 ($t_{1/2} = 5715$ years) remains in a sample estimated to be 16,115 years old?

• Solution: $n = \frac{16,115}{5,715} \approx 2.82$ Percentage remaining: $\left(\frac{1}{2}\right)^{2.82} \times 100\% \approx 13.75\%$

Summary Table: Radioactive Decay Calculations

Method	Equation	Application
Direct Half-Life Calculation	$t_{1/2} = \frac{0.693}{k}$	Find half-life from decay constant
Integrated Rate Law	$\ln\left(\frac{N_0}{N}\right) = kt$	Find remaining nuclei after time t
Fraction/Percentage Remaining	$\left(\frac{1}{2}\right)^n$	Find fraction or percentage remaining after n half-lives

Additional info: Radioactive half-life calculations are essential in nuclear chemistry, radiometric dating, medical diagnostics, and environmental science. Understanding these methods allows students to solve problems involving decay rates, sample ages, and remaining quantities of radioactive substances.