31.5 Half-Life and Activity

Radioactive Decay and Half-Life

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. Half-life tells you how quickly a given isotope decays, and it's the foundation for techniques like radioactive dating that let scientists determine the age of fossils, rocks, and artifacts.

Half-Life in Radioactive Decay

Half-life ($t_{1/2}$) is the time required for half of the radioactive atoms in a sample to decay. Two things to remember about it:

• It's constant for each isotope. Carbon-14 always has a half-life of 5,730 years; uranium-238 always has a half-life of 4.47 billion years.
• It doesn't depend on how much material you start with. Whether you have 10 grams or 10 kilograms, half will decay in the same amount of time.

Types of radioactive decay:

• Alpha decay: the nucleus emits an alpha particle (a helium-4 nucleus, with 2 protons and 2 neutrons)
• Beta decay: the nucleus emits a beta particle (an electron or a positron)
• Gamma decay: the nucleus emits a gamma ray (a high-energy photon), usually after an alpha or beta decay leaves the nucleus in an excited state

The math behind decay. The rate of decay is proportional to the number of radioactive atoms present, which produces exponential behavior:

$N(t) = N_0 e^{-\lambda t}$

• $N(t)$ = number of radioactive atoms remaining at time $t$
• $N_0$ = initial number of radioactive atoms
• $\lambda$ = decay constant (a measure of how quickly the isotope decays)

Half-life and the decay constant are related by:

$t_{1/2} = \frac{\ln(2)}{\lambda}$

This comes from setting $N(t) = \frac{N_0}{2}$ in the exponential decay equation and solving for $t$.

Concept of Radioactive Dating

Radioactive dating determines the age of an object by comparing the amount of a radioactive isotope remaining in a sample to the amount of its decay products.

Carbon-14 dating is used for organic materials like fossils, wood, and bone. Here's how it works:

1. Cosmic rays hitting the atmosphere constantly produce carbon-14.
2. Living organisms absorb carbon from the environment, maintaining a roughly constant ratio of carbon-14 to carbon-12.
3. When the organism dies, it stops taking in new carbon-14. The carbon-14 already present begins to decay.
4. By measuring how much carbon-14 remains relative to carbon-12, you can calculate how long ago the organism died.

Carbon-14 dating is effective for samples up to about 50,000 years old. Beyond that, too little carbon-14 remains to measure accurately.

Uranium-lead dating is used for inorganic materials like rocks and minerals, and it works on much longer timescales:

• Uranium-238 decays to lead-206 (half-life: 4.47 billion years)
• Uranium-235 decays to lead-207 (half-life: 704 million years)

By measuring the ratio of lead isotopes to uranium isotopes in a rock sample, geologists can calculate its age. This method has been used to determine the age of the Earth itself.

Age Calculation with Decay Rates

To find the age of a sample, you rearrange the exponential decay equation to solve for time:

$t = -\frac{1}{\lambda} \ln\left(\frac{N(t)}{N_0}\right)$

Steps to calculate age:

1. Determine $N_0$ (the initial amount of the radioactive isotope). For carbon-14 dating, you assume the sample started with the same carbon-14 ratio as the atmosphere.
2. Measure $N(t)$ (the current amount of the radioactive isotope in the sample).
3. Look up the decay constant $\lambda$, or calculate it from the half-life using $\lambda = \frac{\ln(2)}{t_{1/2}}$.
4. Plug the values into the equation and solve for $t$.

Worked example: A fossil contains 25% of its original carbon-14. The half-life of carbon-14 is 5,730 years. How old is the fossil?

• The fraction remaining: $\frac{N(t)}{N_0} = 0.25$
• Calculate the decay constant: $\lambda = \frac{\ln(2)}{5730} \approx 1.21 \times 10^{-4} \text{ yr}^{-1}$
• Solve for age: $t = -\frac{1}{1.21 \times 10^{-4}} \ln(0.25) \approx 11{,}460 \text{ years}$

Quick sanity check: 25% means two half-lives have passed (100% → 50% → 25%), and $2 \times 5{,}730 = 11{,}460$ years. The math checks out.

Radioactive Series and Activity

Radioactive series (also called a decay chain) is a sequence of decays from an initial unstable isotope (the parent nuclide) through one or more intermediate isotopes until a stable isotope (the daughter nuclide) is reached. For example, uranium-238 goes through 14 decay steps before finally becoming stable lead-206.

Activity is the rate at which decays occur in a sample. It's calculated as:

$A = \lambda N$

where $N$ is the number of radioactive atoms present. Activity is measured in becquerels (Bq), where 1 Bq equals one decay per second. Since $N$ decreases over time, activity also decreases. After one half-life, the activity drops to half its original value.