42-6 Radioactive Dating#
Prompts
How is \(^{14}\text{C}\) produced in the atmosphere? Why do living organisms maintain a constant \(^{14}\text{C}/^{12}\text{C}\) ratio?
After an organism dies, its \(^{14}\text{C}\) decays with half-life 5730 years. Derive the formula for the age \(t\) in terms of the current ratio \(N/N_0\) and the half-life.
A sample has \(^{14}\text{C}/^{12}\text{C}\) ratio 25% of the living ratio. How old is it? What is the practical upper limit for carbon-14 dating?
Why can’t carbon-14 date rocks or fossils older than ~50,000 years? What other isotopes are used for older materials?
What assumptions does carbon-14 dating rely on? What can cause errors (e.g., atmospheric variations, contamination)?
Lecture Notes#
Overview#
Radioactive dating uses the exponential decay law \(N = N_0 e^{-\lambda t}\) to determine the age of a sample. If we know \(N_0\), measure \(N\), and know \(\lambda\), we can solve for \(t\).
Carbon-14 dating applies to organic materials (wood, bone, cloth). Living organisms maintain a steady \(^{14}\text{C}/^{12}\text{C}\) ratio; after death, \(^{14}\text{C}\) decays (half-life 5730 y) and the ratio drops. Typical range: hundreds to ~50,000 years.
Other isotopes (e.g., uranium–lead, potassium–argon) date older materials such as rocks and minerals.
The Dating Principle#
From the decay law \(N = N_0 e^{-\lambda t}\) (see section 42-3), the age is
We need: (1) the initial amount \(N_0\) (or a ratio that reflects it), (2) the current amount \(N\) (or ratio), and (3) the half-life \(T_{1/2}\) (or decay constant \(\lambda\)).
Carbon-14 Dating#
Production: Cosmic-ray neutrons strike \(^{14}\text{N}\) in the atmosphere: \(n + {}^{14}\text{N} \to {}^{14}\text{C} + p\). The \(^{14}\text{C}\) oxidizes to \(\text{CO}_2\) and mixes into the carbon cycle.
Equilibrium in life: Living organisms exchange carbon with the environment (respiration, diet). They maintain a steady ratio \(^{14}\text{C}/^{12}\text{C} \approx 1.3\times 10^{-12}\) — the same as in the atmosphere.
After death: No new carbon is taken in. \(^{14}\text{C}\) decays by beta-minus to \(^{14}\text{N}\) with half-life \(T_{1/2} = 5730\) years. The ratio decreases exponentially.
Dating formula: If \(R = N(^{14}\text{C})/N(^{12}\text{C})\) and \(R_0\) is the ratio in living matter:
Quantity |
Value |
|---|---|
Half-life \(T_{1/2}\) |
5730 y |
Usable range |
~300 y – 50,000 y |
Best for |
Wood, bone, charcoal, cloth |
Why \(^{12}\text{C}\) as reference?
\(^{12}\text{C}\) is stable and abundant. Measuring the ratio \(^{14}\text{C}/^{12}\text{C}\) avoids needing the sample’s original mass — we compare to the known living ratio \(R_0\).
Assumptions and Limitations#
Constant \(R_0\): The atmospheric \(^{14}\text{C}\) ratio has varied (e.g., solar activity, nuclear tests). Calibration curves correct for historical variations.
No contamination: Adding or removing carbon (e.g., groundwater, handling) alters the ratio and biases the age.
Closed system: The sample must not have exchanged carbon with the environment since death.
Other Dating Methods#
For materials older than ~50,000 years, carbon-14 is too depleted to measure accurately. Other parent–daughter pairs are used:
Method |
Parent |
Daughter |
Half-life |
Typical range |
|---|---|---|---|---|
Carbon-14 |
\(^{14}\text{C}\) |
\(^{14}\text{N}\) |
5730 y |
300 y – 50 ky |
Uranium–lead |
\(^{238}\text{U}\) |
\(^{206}\text{Pb}\) |
4.5 Gy |
Rocks, billions of y |
Potassium–argon |
\(^{40}\text{K}\) |
\(^{40}\text{Ar}\) |
1.25 Gy |
Volcanic rocks |
The uranium series (\(^{238}\text{U} \to \cdots \to {}^{206}\text{Pb}\)) and similar long chains date minerals that lock in the parent at formation; the daughter accumulates over time.
Summary#
Dating: \(t = (T_{1/2}/\ln 2)\ln(N_0/N)\); need \(N_0\), \(N\), and \(T_{1/2}\).
Carbon-14: living ratio \(R_0\); after death \(R\) decreases; \(t = 5730\,\text{y} \times \ln(R_0/R)/0.693\).
Assumptions: constant \(R_0\) (calibrated), no contamination, closed system.
Older materials: uranium–lead, potassium–argon, etc.