What is the age of the pole in years?

Question

You take a course in archaeology that includes field work. An ancient wooden totem pole is excavated from your archaeological dig. The beta decay rate is measured at 610 decays/min.

If a sample from the totem pole contains 235 g of carbon-14 and the ratio of carbon-12 to carbon in living trees is 1.35 x 10^-12,  what is the age of the pole in years?

The molar mass of 14C is 18.035 g/mol. The half‑life of 14C is 5730 years.

Answer 1

Should be: the totem pole contains 235 g of carbon, not carbon-14.

The measure 650 decays/min related to radioactive ¹⁴C. The  initial number of nuclei ¹⁴C we can calculate by using the constant ratio of N_14C/ N_12C 1.35∙10^-12, molar mass, and Avogadro number 6.022∙10²³ nuclei per mol;

that is, N_C12= (235 g/12 g/mol)( 6.022∙10²³ nucleus/mol)=1.18∙10²⁵ nuclei;

initial number of N_14C in the totem is then N_14C,_in=1.18∙10²⁵)( 1.35∙10^-12)=1.59∙10¹³ nuclei.

 ¹⁴C decay constant λ=3.83∙10^-12 s^-1; A(t)= λN(t), and then initial activity of the totem was A(0)= λ (N_14C)_in=60.9 decay per second   and activity measured in t  time 650 decay/min=14.89 decay/s;

  A(t)= A(0)e^{- λt};

solving for t gives t= ln[A(0)/A(t)]=37.3∙10¹⁰ s ≈1.19∙10⁴ years