42.5.4 Radioactive Dating

Many geological and archeological samples can be dated by measuring the decays of naturally occurring radioactive isotopes. Because we have no way to know N0, the initial number of radioactive nuclei, radioactive dating depends on the use of ratios.

A photograph of a researcher extracting a small sample of an ancient bone.

A researcher is extracting a small sample of an ancient bone. She will determine the age of the bone by measuring the ratio of 14C to 12C.

The most well-known dating technique is carbon dating. The carbon isotope 14C has a half-life of 5730 years, so any 14C present when the earth formed 4.5 billion years ago would long since have decayed away. Nonetheless, 14C is present in atmospheric carbon dioxide because high-energy cosmic rays collide with gas molecules high in the atmosphere. These cosmic rays are energetic enough to create 14C nuclei from nuclear reactions with nitrogen and oxygen nuclei. The creation and decay of 14C have reached a steady state in which the 14C/12C ratio is 1.3×10−12. That is, atmospheric carbon dioxide has 14C at the concentration of 1.3 parts per trillion. As small as this is, it’s easily measured by modern chemical techniques.

All living organisms constantly exchange carbon dioxide with the atmosphere, so the 14C/12C ratio in living organisms is also 1.3×10−12. When an organism dies, the 14C in its tissue begins to decay and no new 14C is added. Objects are dated by comparing the measured 14C/12C ratio to the 1.3×10−12 value of living material.

Carbon dating is used to date skeletons, wood, paper, fur, food material, and anything else made of organic matter. It is quite accurate for ages to about 15,000 years, roughly three half-lives of 14C. Beyond that, the difficulty of measuring such a small ratio and some uncertainties about the cosmic ray flux in the past combine to decrease the accuracy. Even so, items are dated to about 50,000 years with a fair degree of reliability.

Other isotopes with longer half-lives are used to date geological samples. Potassium-argon dating, using 40K with a half-life of 1.25 billion years, is especially useful for dating rocks of volcanic origin.

EXAMPLE 42.4 Carbon dating

Archeologists excavating an ancient hunters’ camp have recovered a 5.0 g piece of charcoal from a fireplace. Measurements on the sample find that the 14C activity is 0.35 Bq. What is the approximate age of the camp?

MODEL Charcoal, from burning wood, is almost pure carbon. The number of 14C atoms in the wood has decayed exponentially since the branch fell off a tree. Because wood rots, it is reasonable to assume that there was no significant delay between when the branch fell off the tree and the hunters burned it.

SOLVE The 14C/12C ratio was 1.3×10−12 when the branch fell from the tree. We first need to determine the present ratio, then use the known 14C half-life t1/2=5730 years to calculate the time needed to reach the present ratio. The number of ordinary 12C nuclei in the sample is

N(12C)=(5.0 g12 g/mol)(6.02×1023 atoms/mol)=2.5×1023 nuclei

The number of 14C nuclei can be found from the activity to be N(14C)=R/r, but we need to determine the 14C decay rate r. After converting the half-life to seconds, t1/2=5730 years =1.807×1011 s, we can compute

r=1τ=1t1/2 /ln 2=3.84×10−12 s−1

Thus

N(14C)=Rr=0.35 Bq3.84×10−12 s−1=9.1×1010 nuclei

and the present 14C/12C ratio is N(14C)/N(12C)=0.36×10−12. Because this ratio has been decaying with a half-life of 5730 years, the time needed to reach the present ratio is found from

0.36×10−12=(1.3×10−12)(12)t/t1/2

To solve for t, we first write this as

0.361.3=0.277=(12)t/t1/2

Now we take the logarithm of both sides:

ln(0.277)=−1.28=tt1/2 ln(0.5)=−0.693tt1/2

Thus the age of the hunters’ camp is

t=1.85t1/2=11,000 years

REVIEW This is a realistic example of how radioactive dating is done.