Question

Estimate the age of the Earth using isotopic abundances of 235U and 238U. State all assumptions.

Answer 1

we need some data like decay constants, initial and present day ration of element so we use some mostly useble data.

we use R₀ of 1.05, and 1.35

present day ratio of 7.257x10^‐2 ,

and decay constants of 9.849 x 10^‐10 and 1.551x10^‐10 y^‐1 for 235U and 238U.

²³⁵U decays faster than ²³⁸U due to its shorter half‐life. The ratio of the two is subsequently decreasing with time. We have the two radioactive decay equations, given by

²³⁵U = ²³⁵U₀ e^{-λ₂₃₅ t}   .........(i)

and

²³⁸U = ²³⁸U₀ e^{-λ₂₃₈ t}   ...........(ii)

Dividing the first equation by the second, we now have

R(t) = R₀e ^{-(λ₂₃₅ - λ₂₃₈) t}    .....(iii)

Where R(t) is the ratio of ²³⁵U to ²³⁸U at some time t, and R₀ is the initial ratio.. This just looks like a regular radioactive decay equation (but is not identical). Thus we can take the natural log of both sides to get

ln(R / R₀) = -(λ₂₃₅ - λ₂₃₈) t .............(iv)

now put the value of assuming data

we use R₀ of 1.05, and 1.35

present day ratio of 7.257x10^‐2 ,

and decay constants of 9.849 x 10^‐10 and 1.551x10^‐10 y^‐1 for 235U and 238U.

we obtain ages of  6.00, and 6.30 Ga or

6 and 6.30 billion years (10⁹ years)