Dating a Beam: Varnish, Logarithms and a Confident Curator
A museum sends a school laboratory a slice cut from the inner part of an old wooden beam and asks for a date. A technician counts the sample overnight, using the same geometry as for a modern wood standard, and obtains a specific activity of 0.190 Bq per gram of carbon. A second technician notices that the slice is coated in a modern varnish, removes it, and re-counts: 0.150 Bq per gram of carbon. Fresh wood gives 0.230 Bq per gram of carbon, the half-life of carbon-14 is 5730 years, and background has already been subtracted from every figure. The museum's label says the beam is 'about 2500 years old', and the curator is pleased that the first measurement supports it. A third student suggests that, because the two numbers differ by only 0.040 Bq/g, the age is uncertain by only a few decades.
1. Calculate the age implied by 0.150 Bq/g and by 0.190 Bq/g, and decide which is trustworthy. Explain the effect of the varnish and what it means for the museum's label.
Read the worked answer
Relation used. A = A₀e^(−λt) with λ = ln2/5730 = 1.21 × 10⁻⁴ y⁻¹, so
t = ln(A₀/A)/λ.
Untreated sample (0.190 Bq/g): t = ln(0.230/0.190)/1.21 × 10⁻⁴ = ln 1.2105/1.21 × 10⁻⁴ = 0.1911/1.21 × 10⁻⁴ ≈ 1.6 × 10³ y.
Cleaned sample (0.150 Bq/g): t = ln(0.230/0.150)/1.21 × 10⁻⁴ = ln 1.5333/1.21 × 10⁻⁴ = 0.4274/1.21 × 10⁻⁴ ≈ 3.5 × 10³ y.
Which to trust. The cleaned measurement. Varnish is made from modern plant or petrochemical material, so it carries carbon with a modern ¹⁴C activity. Any varnish left on the sample adds modern carbon to the count and pulls the measured activity towards 0.230 Bq/g, making the wood look younger than it is. That is precisely what happened: the untreated sample appears about 1600 years old, the cleaned one about 3500 years old. The wood, not its coating, is what is being dated, so the coating must be removed; a dating laboratory would also wash and pre-treat the surface, because the outer layers slowly exchange carbon with the air.
The label. About 3500 years is not consistent with 'about 2500 years old'. The beam is roughly a thousand years older than the label claims, and the first measurement agreed with the curator only because the contaminant pushed the number in the direction the curator wanted.
2. The third student says a 0.040 Bq/g difference means an age uncertainty of only a few decades. Show why this is wrong, and estimate a realistic uncertainty for the cleaned sample.
Read the worked answer
Age depends on the logarithm of activity. Differentiating A = A₀e^(−λt) gives
dt/dA = −1/(λA).
With A = 0.150 Bq/g and λ = 1.21 × 10⁻⁴ y⁻¹: dt/dA ≈ −1/(1.21 × 10⁻⁴ × 0.150) ≈ −5.5 × 10⁴ years per Bq/g.
So a difference of 0.040 Bq/g corresponds to roughly 0.040 × 5.5 × 10⁴ ≈ 2.2 × 10³ years - exactly the gap between the 1600-year and 3500-year answers. The student has treated activity and age as if they were proportional; they are related logarithmically, so small changes in a small activity mean large changes in age.
A realistic uncertainty. If counting statistics give A = 0.150 ± 0.010 Bq/g: at 0.160 Bq/g, t = ln(1.4375)/1.21 × 10⁻⁴ ≈ 3.0 × 10³ y; at 0.140 Bq/g, t = ln(1.6429)/1.21 × 10⁻⁴ ≈ 4.1 × 10³ y. The honest statement is therefore 'about 3500 years, give or take about 500 years', and the museum should be told that the uncertainty is centuries, not decades.
Practical consequence. Because the age is so sensitive to the count rate, the count must run long enough and the background must be subtracted carefully; a small error in a low count rate becomes a large error in the date.
3. State the assumptions behind the method and describe one further check that would make the result more reliable.
Read the worked answer
Assumptions. (i) The specific activity of carbon in the atmosphere, and hence in living wood, has stayed near 0.230 Bq/g for the whole period. This is roughly true over a few thousand years, but fossil-fuel burning has diluted atmospheric ¹⁴C and nuclear tests roughly doubled it in the 1960s, so a laboratory must use the appropriate value for the appropriate era, which is why calibration curves exist. (ii) The sample has neither gained nor lost carbon since the tree died: no contamination by modern material, no treatment with modern glues or preservatives, no exchange with groundwater. (iii) Background has been measured in the same geometry and subtracted correctly. (iv) The remaining activity is well above background but still measurable - beyond about nine half-lives, roughly 50 000 years, the method fails.
A further check. Date a second, physically separate piece of the same beam, taken from a different growth ring well inside the wood, not from the surface. If the two ages agree within their stated uncertainties, contamination of the first sample is unlikely and the date is much stronger. A second independent method - dendrochronology if a regional tree-ring sequence exists, or thermoluminescence dating of pottery from the same layer - would test the result from outside the radiocarbon method altogether.
What a single date cannot do. It cannot tell whether the beam is part of the original structure or a later repair inserted into an older building, so the archaeological context has to be considered alongside the number.