Question

A fossil you are radiometrically dating contains 3 micrograms of Uranium 235 and 1 microgram of Lead 207. Uranium 235 decays to Lead 207; the half-life of Uranium 235 is 710 million years. How old is the fossil?

Answer 1

Radioactive decay follows first order kinetics

U_t = U_oe^-λt

Where U_o is the initial undecayed U 235 atoms and Ut is the undecayed U 235 after time t. λ is the decay constant.

The decay of one U 235 atom will result in one atom of Pb 207

Therefore,

U_o = U_t + Pb_t

Here, Pb_t is the number of Pb 207 formed

Decay equation can be rewritten as,

U_t = (U_t + Pb_t)e^-λt

On rearranging,

1+Pb_t/U_t = e^λt

Pb_t/ U_t = e^λt-1   -------(1)

The half – life of U 235 = 710 x 10⁶ years.

Therefore, Decay constant

λ = 0.693/ t_1/2 = 0.693 / (710 x 10⁶ years) = 9.7605634 x 10^-10 year^-1

Moles of Pb²⁰⁷ in 1 micro gram = (1 x 10^-6 g)/ 206.975 = 4.8315 x 10^-9 moles

Moles of U²³⁵ in 3 micro gram = (3 x 10^-6 g)/ 235.0439 = 1.2763 x 10^-8 moles

Applying in equation 1,

(4.8315 x 10^-9 moles)/( 1.2763 x 10^-8 moles) = e^λt-1  

0.37853 = e^λt-1  

eλt = 1.37853

λt = ln(1.37853) = 0.3210

Therefore,

t = 0.32102 /(9.7605634 x 10^-10 year^-1) = 355 x 10⁶ years = 355 million years