The current disintegration rate for carbon-14 is 14.0 Bq. A sample of wood discovered in an archaeological excavation is found to have a carbon-14 decay rate of 0.875 Bq. If the half-life of carbon-14 is 5,700 y, approximately how old is the wood sample?

The wood sample has an age of approximately 22800 years.

Explanation:

The Becquerel () is a SI unit which describes radioactive activity related to decay of radioactive isotopes, which is equivalent to . The decay of radioactive isotope is described by the following ordinary differential equation:

(Eq. 1)

Where:

- Disintegration rate, measured in .

- Amount of remaining radioactive nuclei, dimensionless.

- Time constant, measured in seconds.

By integration the solution of this differential equation is obtained:

(Eq. 2)

Let and different disintegration rates for Carbon-14 samples, so that:

(Eq. 3)

(Eq. 4)

If we divide (Eq. 4) by (Eq. 3), then:

(Eq. 5)

If , we proceed to clear that variable:

(Eq. 6)

Time constant is also a function of half-life (), measured in seconds:

If , and ,  the age of the wood sample is:

The wood sample has an age of approximately 22800 years.

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