All living things contain carbon. Most of this carbon is stable carbon-12. Howev

Question

All living things contain carbon. Most of this carbon is stable carbon-12. However, a small percentage will be carbon-14, a radioactive isotope of carbon that decays with time. As living things eat and breath, carbon is constantly recycled and therefore the percentage of carbon-14 remains constant. It's not until death that the percentage of carbon-14 will begin to diminish from decay. Because radioactive decay is a first-order process, the integrated rate law for a first-order reaction can be rewritten as

fraction remaining=0.5(t/t1/2)

where t is the time elapsed and t1/2 is the half-life. The half-life of C-14 is 5730 years.  

A: A fossil was analyzed and determined to have a carbon-14 level that is 21.0 % that of living organisms. How old is the fossil?

Explanation / Answer

radioactive decay is a first-order process

Half life

t1/2 = ln 2 / rate constant (k)

Rate constant

k = ln2 / t1/2

= 0.6932/5730

= 1.21*10^-4 yr-1

For first order reaction

ln([A] / [A0]) = -k*t

[A] = concentration of A after time t

[A0] = initial concentration

t = -ln([A] / [A0]) / k

[A] / [A0] = 21% = 0.21

t = -ln(0.21) / 1.21*10^-4

= 1.561 / 1.21*10^-4

= 12897.9 yr

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