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21.3 Radioactive decay  (Page 7/21)

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Radioactive dating using nuclides other than carbon-14

Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events. For example, uranium-238 (which decays in a series of steps into lead-206) can be used for establishing the age of rocks (and the approximate age of the oldest rocks on earth). Since U-238 has a half-life of 4.5 billion years, it takes that amount of time for half of the original U-238 to decay into Pb-206. In a sample of rock that does not contain appreciable amounts of Pb-208, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed. Therefore, by measuring and analyzing the ratio of U-238:Pb-206, we can determine the age of the rock. This assumes that all of the lead-206 present came from the decay of uranium-238. If there is additional lead-206 present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method. K-40 decays by positron emission and electron capture to form Ar-40 with a half-life of 1.25 billion years. If a rock sample is crushed and the amount of Ar-40 gas that escapes is measured, determination of the Ar-40:K-40 ratio yields the age of the rock. Other methods, such as rubidium-strontium dating (Rb-87 decays into Sr-87 with a half-life of 48.8 billion years), operate on the same principle. To estimate the lower limit for the earth’s age, scientists determine the age of various rocks and minerals, making the assumption that the earth is older than the oldest rocks and minerals in its crust. As of 2014, the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4.4 billion years old.

Radioactive dating of rocks

An igneous rock contains 9.58 × 10 –5 g of U-238 and 2.51 × 10 –5 g of Pb-206, and much, much smaller amounts of Pb-208. Determine the approximate time at which the rock formed.

Solution

The sample of rock contains very little Pb-208, the most common isotope of lead, so we can safely assume that all the Pb-206 in the rock was produced by the radioactive decay of U-238. When the rock formed, it contained all of the U-238 currently in it, plus some U-238 that has since undergone radioactive decay.

The amount of U-238 currently in the rock is:

9.58 × 10 −5 g U × ( 1 mol U 238 g U ) = 4.03 × 10 −7 mol U

Because when one mole of U-238 decays, it produces one mole of Pb-206, the amount of U-238 that has undergone radioactive decay since the rock was formed is:

2.51 × 10 −5 g Pb × ( 1 mol Pb 206 g Pb ) × ( 1 mol U 1 mol Pb ) = 1.22 × 10 −7 mol U

The total amount of U-238 originally present in the rock is therefore:

4.03 × 10 −7 mol + 1.22 × 10 −7 mol = 5.25 × 10 −7 mol U

The amount of time that has passed since the formation of the rock is given by:

t = − 1 λ ln ( N t N 0 )

with N 0 representing the original amount of U-238 and N t representing the present amount of U-238.

U-238 decays into Pb-206 with a half-life of 4.5 × 10 9 y, so the decay constant λ is:

λ = ln 2 t 1 / 2 = 0.693 4.5 × 10 9 y = 1.54 × 10 −10 y −1

Substituting and solving, we have:

t = − 1 1.54 × 10 −10 y −1 ln ( 4.03 × 10 −7 mol U 5.25 × 10 −7 mol U ) = 1.7 × 10 9 y

Therefore, the rock is approximately 1.7 billion years old.

Check your learning

A sample of rock contains 6.14 × 10 –4 g of Rb-87 and 3.51 × 10 –5 g of Sr-87. Calculate the age of the rock. (The half-life of the β decay of Rb-87 is 4.7 × 10 10 y.)

Answer:

3.7 × 10 9 y

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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