A clock that starts at death
High in the atmosphere, cosmic rays knock neutrons out of nuclei. A slow neutron that meets a nitrogen-14 nucleus can turn it into carbon-14 and a proton: \[ n + {}^{14}_{7}\text{N} \;\to\; {}^{14}_{6}\text{C} + p \] The carbon-14 joins oxygen as carbon dioxide and mixes through the air. It is radioactive, with a half-life of 5730 years, and it decays back to nitrogen-14 by beta-minus decay.
The air is a case of production and decay together, as in the last lesson. Carbon-14 has been made at a nearly steady rate for far longer than its half-life. So its amount has reached saturation, where it decays as fast as it is made. The air holds a steady share of carbon-14, about 1.3 atoms in every \(10^{12}\) atoms of carbon.
A plant takes in carbon dioxide as it grows, and animals eat plants. So every living thing holds the same share of carbon-14 as the air. It shows as an activity of 15.3 decays per minute for every gram of carbon. When the plant or animal dies, the intake stops. No fresh carbon-14 comes in, and what is there decays. The activity per gram of carbon then halves every 5730 years.
Why does the carbon-14 activity per gram of carbon stay steady in a living tree but fall in a wooden beam cut from it?
Reading the clock
Measure the carbon-14 activity \(A\) per gram of carbon in a sample, and compare it with the living value \(A_0 = 15.3\) per minute per gram. The decay law gives \(A/A_0 = e^{-\lambda t}\). Taking logarithms, the time since death is \[ t = \dfrac{1}{\lambda}\ln\dfrac{A_0}{A} = \dfrac{T_{1/2}}{\ln 2}\ln\dfrac{A_0}{A} \] For carbon-14, \(T_{1/2}/\ln 2 = 5730/0.693 = 8270\) years, the mean life.
When the fraction is a power of one half, no logarithm is needed. A sample with a quarter of the living activity has lived through two half-lives, \(2 \times 5730 = 11\,460\) years. One with an eighth has lived through three, \(17\,190\) years.
At Rakhigarhi in Haryana, archaeologists dig out a layer that holds a steatite seal of the Harappan civilisation. Charcoal from a hearth in the same layer gives \(8.9\) decays per minute per gram of carbon. Living carbon gives \(15.3\). How old is the layer? Why is the charcoal measured, not the seal?
The fraction left is \(A/A_0 = 8.9/15.3 = 0.58\). The charcoal was wood that stopped taking in carbon when the tree was cut and burnt.
\(t = 8270\,\text{y} \times \ln(15.3/8.9)\).
\(\ln(15.3/8.9) = \ln 1.72 = 0.54\), so \(t = 8270 \times 0.54 = 4500\) years, to two figures.
The hearth was last used about 4500 years ago, and the seal, lying in the same layer, is about that old. The seal itself is steatite, a soft stone. Stone never took in carbon from the air, so it has no carbon-14 clock to read. Values used: \(T_{1/2} = 5730\,\text{y}\), \(\ln 2 = 0.693\), living carbon \(15.3\) per minute per gram.
The curve in the figure above flattens as it goes. After 50 000 years, nearly nine half-lives, only \(2^{-8.7} = 0.24\,\%\) of the carbon-14 is left. A gram of carbon then gives about two decays an hour, hard to tell from the background of cosmic rays and the rocks around. So carbon-14 dates things up to about 50 000 years old and no further.
The method rests on two assumptions. The first is that the air's share of carbon-14 was the same when the sample died as it is for living matter now. It has varied a little, so laboratories correct their ages with tree rings of known age. The second is that nothing but decay has changed the sample's carbon-14 since death. Modern carbon, from roots or from handling, makes a sample look younger.
A bone has \(12.5\,\%\) of the carbon-14 activity of living bone. Its half-life is 5730 years. How old is it, in years?
Fresh wood gives \(15.3\) decays per minute per gram of carbon, and an old wooden bowl gives \(5.1\). Find the age of the bowl, in years. Take \(\ln 3 = 1.099\) and \(T_{1/2}/\ln 2 = 8270\) years.
Which of these could be dated by carbon-14?
A clock for rocks
Carbon-14 is far too short-lived to date the Earth. For rocks a slower clock is needed, and uranium-238 is one. As lesson 2 showed, it reaches lead-206 through fourteen decays. Every step after the first is fast compared with the first, whose half-life is \(4.47 \times 10^9\) years. So in effect uranium-238 turns straight into lead-206, with that half-life.
When a rock crystallises from molten magma, some minerals take in uranium but shut lead out. The rock then starts with uranium-238 and no lead-206. From then on, every lead-206 atom in it was once a uranium-238 atom. After a time \(t\), \(N_U = N_0e^{-\lambda t}\) are left, and \(N_{Pb} = N_0 - N_U\) have become lead. Divide one by the other: \[ \dfrac{N_{Pb}}{N_U} = e^{\lambda t} - 1 = 2^{t/T_{1/2}} - 1 \]
Fig 6.1 shows the clock running. Equal numbers of lead-206 and uranium-238 atoms mean that half the uranium has gone: the rock is one half-life old, \(4.47 \times 10^9\) years. Three lead atoms for every uranium atom mean two half-lives, \(8.94 \times 10^9\) years, older than the Earth, so no rock shows it. The Earth itself is about \(4.54 \times 10^9\) years old. So just under half of the uranium-238 it began with is still here.
A rock holds three uranium-238 atoms for every lead-206 atom. Assuming it formed with no lead-206, find its age. Take \(T_{1/2} = 4.47 \times 10^9\) years, \(\ln 2 = 0.693\) and \(\ln(4/3) = 0.288\).
Of every four uranium atoms the rock began with, three remain and one has become lead. So \(N_{Pb}/N_U = 1/3\).
\(t = \dfrac{T_{1/2}}{\ln 2}\ln\left(1 + \tfrac13\right) = \dfrac{4.47 \times 10^9}{0.693} \times \ln\tfrac43\).
\(t = 6.45 \times 10^9 \times 0.288 = 1.86 \times 10^9\) years.
The rock is about \(1.86 \times 10^9\) years old. Values used: \(T_{1/2} = 4.47 \times 10^9\) years for uranium-238, \(\ln 2 = 0.693\).
A mineral holds equal numbers of uranium-238 and lead-206 atoms. Assuming it formed with no lead-206, how old is it, in units of \(10^9\) years? The half-life of uranium-238 is \(4.47 \times 10^9\) years.
Why is carbon-14 used to date charcoal from a Harappan site, while uranium-238 is used to date a rock of the Deccan?
A clock must have a half-life comparable to the age being measured. Carbon-14, with 5730 years, has changed measurably in a few thousand years, and it is in anything that once took in carbon from the air. Uranium-238, with 4.47 billion years, would hardly have changed in 4500 years, but in a rock tens of millions of years old it has made a measurable amount of lead. A rock also holds no carbon from the air, so carbon-14 cannot date it.
Radioactive dating in general reads the same law in two ways. Carbon dating measures how much of a nuclide is left against a known starting amount. Uranium-lead dating counts the daughter atoms that have piled up. Some of the oldest rocks of India lie in the Singhbhum region of Jharkhand and Odisha. They have been dated in these ways to more than three billion years.
- Dating a stone object, such as a seal, by carbon-14.Only once-living matter took in carbon-14. Date the charcoal or grain found with the object.
- Taking the living activity as the activity of the whole sample.Compare activities per gram of carbon, 15.3 per minute for living matter.
- Using carbon-14 for ages beyond about 50 000 years.After nine half-lives too little is left to measure. Use a longer-lived clock.
- Forgetting the assumptions of uranium-lead dating.The rock must have started with no lead-206, and none may have entered or left since.