9.2 Half-life and radioactive decay kinetics

Radioactive Decay and Half-Life

Radioactive decay follows first-order kinetics, meaning the rate of decay depends only on how much radioactive material is present at any moment. Half-life is the central concept here: it's the time required for exactly half of a radioactive sample to decay. Because the decay is first-order, the half-life stays constant regardless of how much material you started with.

These principles show up everywhere, from nuclear medicine dosing to determining the age of fossils and rocks.

Half-Life in Radioactive Decay

• Half-life ($t_{1/2}$) is the time required for half of a given quantity of a radioactive substance to decay into a daughter nuclide
  • Remains constant for each specific radioisotope (carbon-14: ~5,730 years; uranium-238: ~4.5 billion years)
  • Independent of the initial amount of substance present, temperature, or pressure
• Significance in radioactive decay:
  • Determines the rate at which a radioactive substance decays over time
  • Allows you to calculate how much of a substance remains after any given time period
  • Enables radioactive dating techniques for both organic materials (carbon-14 dating) and inorganic materials (uranium-lead dating)

Calculations with Half-Lives

The amount of a radioactive substance decreases exponentially. Two equivalent formulas let you calculate how much remains:

Formula 1 (using elapsed time):

$N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}}$

Formula 2 (using number of half-lives):

$N(t) = N_0 \cdot (0.5)^n$

where $n = t / t_{1/2}$ is the number of half-lives elapsed.

In both formulas:

• $N(t)$ = amount of substance remaining at time $t$
• $N_0$ = initial amount of substance
• $t_{1/2}$ = half-life of the isotope

Example: Suppose you start with 80.0 g of iodine-131 ($t_{1/2}$ = 8.02 days). How much remains after 24.06 days?

1. Find the number of half-lives: $n = 24.06 / 8.02 = 3$
2. Apply the formula: $N = 80.0 \cdot (0.5)^3 = 80.0 \cdot 0.125 = 10.0 \text{ g}$

After 3 half-lives, only 10.0 g of the original 80.0 g remains. The rest has decayed into xenon-131.

When $n$ isn't a whole number, the formula still works the same way. For instance, after 2.5 half-lives: $(0.5)^{2.5} \approx 0.177$, so about 17.7% of the original sample remains.

Half-Life and the Decay Constant

The decay constant ($\lambda$) connects half-life to the rate of decay mathematically:

$\lambda = \frac{\ln 2}{t_{1/2}} \approx \frac{0.693}{t_{1/2}}$

• $\lambda$ has units of inverse time (e.g., $\text{s}^{-1}$, $\text{yr}^{-1}$)
• A shorter half-life means a larger $\lambda$, which means faster decay
• A longer half-life means a smaller $\lambda$, which means slower decay

This also gives you the first-order integrated rate law form of the decay equation:

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

This version is equivalent to the half-life formula and is often more convenient when you're solving for time or when $\lambda$ is given directly.

Activity (the number of disintegrations per unit time) is directly proportional to the amount of substance present:

$A = \lambda N$

• Measured in becquerels (Bq): 1 Bq = 1 disintegration per second
• Or in curies (Ci): 1 Ci = $3.7 \times 10^{10}$ disintegrations per second

A substance with a short half-life (like radon-222, $t_{1/2}$ = 3.8 days) is highly active but depletes quickly. A substance with a long half-life (like potassium-40, $t_{1/2}$ = 1.3 billion years) has very low activity per gram but persists for an enormous span of time.

Radioactive Dating Techniques

Radioactive dating works by comparing the ratio of a radioactive parent isotope to its stable daughter product within a sample. The more daughter product has accumulated relative to the parent, the older the sample.

• Carbon-14 dating is used for organic materials (wood, bone, cloth). Carbon-14 has a half-life of ~5,730 years, making it useful for samples up to roughly 50,000 years old. Beyond that, too little carbon-14 remains to measure reliably.
• Uranium-lead dating is used for rocks and minerals. Uranium-238 has a half-life of ~4.5 billion years, making it suitable for dating geological samples billions of years old.

Steps in radioactive dating:

1. Measure the amounts of the radioactive parent isotope and its stable decay product in the sample.
2. Calculate the ratio of parent to daughter isotope.
3. Use the known half-life and decay equations to solve for the elapsed time since the sample formed.

For example, in carbon-14 dating, you compare the ratio of $^{14}\text{C}$ to $^{12}\text{C}$ in the sample against the known ratio in living organisms. A sample with half the expected $^{14}\text{C}/^{12}\text{C}$ ratio is approximately 5,730 years old (one half-life).