Question

All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes 235 U and 238 U should have been created in roughly equal numbers. Today, 99.28% of uranium is 238 U and only 0.72% is 235 U. How long ago did the supernova occur?

Short answer

The supernova occurred approximately$6$billion years ago.

Step-by-step solution

Step 1. Given Information

$$\begin{gathered} {}^{235}Uand{}^{238}UareequalGivenby \\ \left(N235\right)=\left(N238\right) \end{gathered}$$

Given Expression:

$$\begin{gathered} N=No{\left(\frac{1}{2}\right)}^{\frac{-t}{t1/2}} \\ For{}^{235}Utheabovecanbewritten \\ N238=\left(N235\right){\left(\frac{1}{2}\right)}^{\frac{-t}{t1/2}} \end{gathered}$$

Given Expression:

$$\begin{gathered} \frac{N_{238}}{N_{235}}=\frac{{{\left(N_{238}\right)}_{0\left({\displaystyle \frac{1}{2}}\right)}}^{{\displaystyle \frac{-t}{t{\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}_{235}}}}}{{{\left(N_{238}\right)}_{0\left(\frac{1}{2}\right)}}^{\frac{-t}{t\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)8}}} \\ 137.9=\left(\frac{1}{2}\right) \end{gathered}$$

Given Expression:

$$\begin{gathered} t=\frac{7.13}{\left({\displaystyle \frac{1}{\left(4.47x{10}^{9}yr\right)}}-\frac{1}{\left(7.04x{10}^{9}yr\right)}\right)} \\ =5.9x{10}^{9}yr \end{gathered}$$