Question

It was determined that 1g of pure Carbon from a dead mammoth contained 1.428x10^9 atoms of the C-14 isotope. Assuming that the abundance of C-14 in living organisms has remained constant, estimate how long ago the mammoth died.

Answer 1

You are given that \[t_{1/2}(_{}^{14}C) = 5730y\] and that 1 mole of Carbon weighs 12g.
The abundance of C-1 is 1 atom for every 10^12 atoms of carbon.

Answer 2

Find how much C-14 would be there without decay. m(0).
Find how much was left after x years, m(x), and then use

m(x)/m(0) = (0.5)^(x/5730)

ln[m(x)/m(0)] = (x/5730) ln (0.5)

e.g., ln(1/4) = (x/5730) ln (0.5)

x= 5730 ln(1/4)/ln(0.5) = 5730 (2) = 11460 years old

Answer 3

Just rewriting to make it easier for myself to read:

Find how much C-14 would be there without decay. m(0).
Find how much was left after x years, m(x), and then use\[\frac{ m_x }{ m_0 } = (0.5)^{(\frac{ x }{ 5730 })}\]\[ \ln[\frac{ m_x }{ m_0 }] = (\frac{x}{5730}) \ln (0.5)\]e.g., \[\ln(\frac{1}{4}) = (\frac{x}{5730}) \ln (0.5) \]\[x= 5730 \frac{\ln(\frac{1}{4})}{\ln(0.5)} = 5730 (2) = 11460 years\]