Question

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?

Answer 1

The wood sample has an age of approximately 22800 years.

Step-by-step explanation:

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

`(dN)/(dt) = -(N)/(\tau)` (Eq. 1)

Where:

`(dN)/(dt)` - Disintegration rate, measured in
`(1)/(s)`.

`N` - Amount of remaining radioactive nuclei, dimensionless.

`\tau` - Time constant, measured in seconds.

By integration the solution of this differential equation is obtained:

`\int {(dN)/(N) } = -(t)/(\tau)\int dt`

`\ln N = -(t)/(\tau) + C`

`N(t) = N_(o)\cdot e^{-(t)/(\tau) }` (Eq. 2)

Let
`N_(1)` and
`N_(2)` different disintegration rates for Carbon-14 samples, so that:

`N_(1) = N_(o) \cdot e^{-(t_(1))/(\tau) }` (Eq. 3)

`N_(2) = N_(o)\cdot e^{-(t_(2))/(\tau) }` (Eq. 4)

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

`(N_(2))/(N_(1)) = \frac{N_(o)\cdot e^{-(t_(2))/(\tau) }}{N_(o)\cdot e^{-(t_(1))/(\tau) }}`

`(N_(2))/(N_(1)) = e^{-(1)/(\tau)\cdot (t_(2)-t_(1)) }` (Eq. 5)

If
`\Delta t = t_(2)-t_(1)`, we proceed to clear that variable:

`\ln (N_(2))/(N_(1)) = -(1)/(\tau)\cdot \Delta t`

`\Delta t = -\tau\cdot \ln (N_(2))/(N_(1))` (Eq. 6)

Time constant is also a function of half-life (
`t_(1/2)`), measured in seconds:

`\tau = (t_(1/2))/(\ln 2)`

If
`t_(1/2) = 1.798* 10^(11)\,s`,
`N_(1) = 14\,(1)/(s)` and
`N_(2) = 0.875\,(1)/(s)`, the age of the wood sample is:

`\tau = (1.798* 10^(11)\,s)/(\ln 2)`

`\tau = 2.594* 10^(11)\,s`

`\Delta t = -(2.594* 10^(11)\,s)\cdot \ln \left((0.875\,(1)/(s) )/(14\,(1)/(s) ) \right)`

`\Delta t \approx 7.192* 10^(11)\,s`

`\Delta t \approx 22805.682\,yr`

The wood sample has an age of approximately 22800 years.