Maths IA · Modelling · topic 77 of 97
How old is it? Radiocarbon dating and its errors · IB Maths IA idea: guide, sample graph and step-by-step plan
the age is easy: the error is the IAA bone is a few thousand years old, or a few tens of thousands. The only clock is a tiny amount of carbon that fades by half every 5,730 years. How accurate is that clock, and where does it stop working?
- ✓ you like exponentials and logarithms and want to see them settle a real question
- ✓ you enjoy asking how much an answer can be trusted, not only what it is
- ✓ you are drawn to archaeology, geology or the history of science
The big idea
Living things take in carbon-14 and hold it at a steady level. After death it decays by a fixed fraction each year, so the amount left is an exponential function of time. Measure the fraction left and solve for the age with a logarithm. That is the routine part. The mathematics that matters is about error. A small error in the measured fraction becomes an error in the age, and the derivative tells you how much: for an old sample, little is left to measure, so the same instrument error gives a much larger age error. Add the uncertainty in the half-life itself and the choice between two published values. The marks come from tracing each source of error to the age it produces, then saying honestly when the method stops being useful.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Take a measured fraction of carbon-14 left, solve N = N₀e^(−λt) for t, and check by putting your answer back in.
→ A logarithm used to undo an exponential. Show the link between the half-life and λ, and say what the units are.Differentiate the age with respect to the measured amount and use it to turn an error in the amount into an error in the age.
→ A derivative used as an error tool. Compare a fixed percentage error with a fixed absolute error and say which fits a real instrument better.Find the age at which the amount left falls below the smallest amount you could measure, and the age where the age error passes 1,000 years.
→ An equation solved with logarithms that gives a limit, in years, for the method. Test how the limit moves when the instrument improves.Date the same sample with 5,730 years and with the older value 5,568 years, and compare.
→ A percentage difference that grows with age. Explain which value a published age uses and why.Compare ages from the decay formula with calibrated ages for the same samples from a published calibration table.
→ A model against a better model. Describe why carbon-14 in the air has not been constant, and where the curves differ most.Use published measurements for samples of known age, linearise with ln, and fit a line to estimate λ.
→ A regression with a clear meaning for the slope. Compare your λ and its uncertainty with the published half-life.Your Maths IA research question: run the check
which sample? what error?Write your own question first. Then make sure it ticks every box:
- Names the sample or set of samples, real or labelled made-up, and the quantity measured (the fraction of carbon-14 left).
- Names the quantity you will find: an age, an age uncertainty, or the age limit of the method.
- Says how the error is propagated and what kind of measurement error it assumes.
- Goes beyond a calculator: there is a derivation (inverting the decay, a derivative or a limit) and a comparison of two choices.
- States what you leave out: calibration, contamination, and variation in atmospheric carbon-14.
The model
After death, the carbon-14 left falls by half every half-life. Sketch the fraction left over the time range your samples cover, and mark where it gets too small to measure.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Why this method, which sample or samples, and what you expect before calculating: how old, and how sure?
Exponential decay from a half-life, every symbol defined, λ derived from the half-life and checked numerically.
The age formula derived with a logarithm, then used on your data and checked by substituting back.
The derivative of age with respect to the amount left, the two kinds of measurement error and the age error each one produces.
The age at which the error becomes too large to be useful, solved exactly, and how it changes with the instrument.
Your ages beside published calibrated ages. A number for the gap and a reason for it.
Back to your aim: the ages, their uncertainty, and the age limit, with numbers and units.
Constant half-life, no contamination, no change in the atmosphere. Say which assumption matters most and what a laboratory would also do.
Where every half-life, error value and calibration table came from.
What your Maths IA graph could look like
mark the age limit!A made-up set of readings to show the form of a strong graph, not the answer: the fraction of carbon-14 left against age, for two half-life values. Yours will look different. example only
- The fraction left against age, with the model drawn through your readings.
- Axis labels give units: thousands of years on the age axis, a fraction (no unit) on the other.
- Draw both half-life values on one graph and say how far apart they are at your sample's age.
- Plot your readings as points: agreement with the curve is your evidence, and the gaps are your evaluation.
- Another graph to consider: ln(N/N₀) against age, which should be a straight line with slope −λ.
The maths in action
worked through, not sketched!The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out how radiocarbon dating works. Before any calculation, guess how an error of one percent in the measured amount changes the age.
- Choose the samplesUse published measurements, or made-up ones you label as made-up. Note the source and the stated uncertainty of every value.
- Derive the decay and the ageWrite N = N₀e^(−λt), link λ to the half-life, and invert it with a logarithm. Define every symbol.
- Trial run (1 lesson)Date three samples with a spreadsheet and check each by substituting the age back. Fix any mismatch now: units, ln against log, half-life against mean life.
- Propagate the errorDifferentiate the age with respect to the amount left. Use it for a percentage error and for a fixed error, and compare.
- Find the limitSolve for the age where the error passes a size you choose. Describe how it moves as the instrument improves.
- Compare half-livesRedo your ages with a second published half-life. Say which one the published ages use, and why.
- Go further, with a reasonCompare with calibrated ages, or linearise and fit λ from data. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for how far a reader should trust the age and what it assumes. These lines become your evaluation.
- Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.
The maths you'll use
- Exponential decay – Write and link to the half-life by .
- Logarithms to invert it – Solve for the age and check by substitution.
- Derivative as an error tool – Differentiate with respect to to get and compare a fixed and a relative error.
- Solving for a limit – Solve for the age at which the uncertainty reaches a chosen size .
- Linearising data – Plot against for samples of known age and read from the gradient, with its uncertainty.
Key equations
Syllabus links
- 1.5Exponents and an introduction to logarithms
- 1.7Rational exponents and laws of logarithms
- 2.9Exponential and logarithmic functions
- 5.1Limits and the derivative as a rate of change
- 5.6Standard derivatives; chain, product and quotient rules
- 5.18First-order differential equations; Euler's methodHL
- 1.5Exponents and an introduction to logarithms
- 1.6Approximation, bounds and percentage error
- 2.9Logarithmic, logistic, piecewise and half-life modelsHL
- 5.1Limits and the derivative as a rate of change
- 5.14Setting up differential equations; separating variablesHL
Read up first
Data and sources to start from
Published radiocarbon measurements for a few samples (laboratory reports or a calibration table), or made-up readings clearly labelled.
- IntCal: the international radiocarbon calibration curves
- Oxford Radiocarbon Accelerator Unit: OxCal calibration software
References to cite
- Libby, W. F. (1955). Radiocarbon dating (2nd ed.). University of Chicago Press.
- Godwin, H. (1962). Half-life of radiocarbon. Nature, 195, 984.
- Reimer, P. J., Austin, W. E. N., Bard, E., Bayliss, A., Blackwell, P. G., Bronk Ramsey, C., et al. (2020). The IntCal20 Northern Hemisphere radiocarbon age calibration curve (0-55 cal kBP). Radiocarbon, 62(4), 725-757.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
ln, not log- Mixing the half-life and the mean life. λ = ln 2 divided by the half-life, and the units must match your time axis.
- Using log (base 10) where the formula needs ln, or the reverse.
- Treating an age from the decay formula as a calendar date. Radiocarbon ages need calibration to become calendar ages.
- Quoting an age without an uncertainty, or an uncertainty without saying where it comes from.
- Assuming a fixed percentage error at every age. Real instruments have a floor, and that is what limits old samples.
- Claiming the method works for any age. Say where it stops and show the calculation.
More Maths IA ideas like this
Medicine in the bloodstream: why are doses spaced out?●●○
The Keeling curve: modelling carbon dioxide●●○
Is Moore's law over? Fitting exponential growth●●○
Cooling drinks: does Newton's law of cooling fit?●○○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.