Question

Uranium-238 has a half-life of 4.5 billion years. Given that scientists estimate Earth's age to be 4.6 billion years, what is the most likely percentage of parent to daughter isotopes of this element currently existing on Earth?

Answer 1

Answer:

103.3 %

Explanation:

For a radioactive isotope, the number of radioactive nuclei left (parent nuclei) after a time t, N(t), is

$N(t)=N_0 (\frac{1}{2})^{\frac{t}{\tau}}$

where

N0 is the initial number of radioactive nuclei

t is the time

$\tau$ is the half-life of the isotope

Here we have

$t=4.5 \cdot 10^9 y\\\tau = 4.6\cdot 10^9 y$

So we find

$\frac{N(t)}{N_0}=(\frac{1}{2})^{\frac{4.5\cdot 10^9}{4.6\cdot 10^9}}=0.508$

Which means that the fraction of parent nuclei left after this time is 0.508 (50.8% of the initial value). So the fraction of daugther nuclei at this time is

$\frac{N_d}{N_0}=1-0.508=0.492$

So the percentage of parent to daughter isotopes is

$\frac{N(t)}{N(d)}=\frac{0.508}{0.492}=1.033$

Which corresponds to 103.3 %.