# A sample of a fossil was taken to a lab and it contained the element Thorium-230, an unstable radioactive isotope, and radium-226. Thorium's half life is 75,400 years. (Meaning in 75,400 years, half of the sample of Thorium will turn into the stable elemental isotope, Radium-226.) This sample contains 5 grams of Thorium and 15 grams of Radium. Calculate how old the rock is.

When Thorium-240 decays to radium-226 it emits an alpha particle and loses 4 mass units.

15 grams of radium-226 resulted from the decay of this amount of thorium-230:

(15g)(230/226) = 15.26 g

Since 5 g of thorium-230 remains, the original mass of the thorium was (15.26 + 5)g = 20.26g. The inprecision in the mass measurements allows us to round this to 20. grams.

The fraction of the original sample of thorium-230 that remains is 5/20. = 0.25.

The fraction that remains = (1/2)^n where n equals the number of half-lives that have elapsed. Since (1/2)^n=0.25, n=2, making the age of the rock equal to 2 half-lives.

An simpler way to look at this is that it takes two half-lives to go from 20 grams to 10 grams to 5 grams.

The age of the rock is 2 x 74,500 years = 149,000 years

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