Question

The half-life of the uranium isotope ${}^{235}U$is $700$million years. The earth is approximately localid="1650486537957" $4.5$billion years old. How much more ${}^{235}U$was there when the earth formed than there is today? Give your answer as the then-to-now ratio.

Short answer

Then-to-now Ratio of uranium isotope is, $\frac{{\left(N_U\right)}_0}{N_U}=86.2$

Step-by-step solution

Given information

We are given that,

Time is, $t=4500my$

And half life is,$t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=700my$

Simplify 

Let, amount of uranium isotope when earth is formed is,${\left(N_U\right)}_0$

And amount of uranium isotope now is,$N_U$

We know that,

The ${}^{235}U$decay is given by,

$N_U={\left(N_U\right)}_0\times {\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$}\right.}$

Now, ratio is,

localid="1650360018817" $$\begin{gathered} \frac{{\left(N_U\right)}_0}{N_U}=\frac{1}{{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$}\right.}}} \\ ={\left(2\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$}\right.} \\ ={\left(2\right)}^{\raisebox{1ex}{$4500my$}\!\left/ \!\raisebox{-1ex}{$700my$}\right.} \\ =86.2 \end{gathered}$$