Question

A layer of peat beneath the glacial sediments of the last ice age had a carbon-14 content of 25 % of that found in living organisms. How long ago was this ice age?

Express your answer using four significant figures

Answer 1

Let R(t) be the ratio of concentration of C-14 to the concentration of stable C-12 in the sample as a function of time. The concentration of C-12 does not change with time because this element is stable. The concentration of C-14, however, changes at a rate of:

dC-14/dt = - k*C-14

where k is the decay constant for C-14.

When integrated, we have:

C-14(t) = C-14(0)*exp(-k*t). Dividing both sides of this equation by the constant concentration of C-12 gives us:

R(t) = Ro*exp(-k*t)

where Ro is the C-14/C-12 ratio at time t = 0 (the time that the carbon in the sample became isolated from the atmosphere).

Solving for t gives:

ln(R(t)/Ro) = - k*t

t = (1/k)*ln(Ro/R(t))

The decay constant and halflife are related by:

k = ln(2)/halflife

1/k = halflife/ln(2)

so we can also write this equation as:

t = (halflife/ln(2))*ln(Ro/R(t))

The halflife of C-14 is 5730 years, so the age of the sample in this case (assuming that the atmospheric C-14/C-12 ratio when the sample formed is the same as that today, which is generally not correct) would be:


t = (5730 yr/ln(2))*ln(1/0.25)

t = 12668 yr, which makes sense because there is a little less than 25% of the initial C-14 remaining in the sample, so the sample should be a little older than two halflifes of C-14.