# Calculating the Earth's Age

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Desire to develop mastery; Desire for autonomy; Desire for purpose.

— Daniel Pink, Author of "Drive," talking about the primary employee motivations in today's job environment.

## Introduction

Figure 1: Barringer Crater in Arizona. Scientists first used lead-lead dating on fragments from the meteorite that formed this crater. (USGS)

I have been listening to the audio book The Disappearing Spoon, which is an excellent tale about all of the elements of the periodic table. Of particular interest to me was the discussion of how geologists date the age of the Earth using ratios of uranium and lead. The book also discusses determining the ages of meteorites and the Sun. The discussions were interesting enough that I thought I would look up some additional information. As frequently happens, I was amazed at the amount of information on the web about this subject. This technique has been around since 1956, when it was first used to date meteor fragments from a well-known impact site (Figure 1).

## Computing the Age of the Earth

### Basic Chemistry

The basic chemistry used in the dating process is nicely described in this About.com article. I summarize the contents of this article as follows.

• Tiny zirconium crystals (called zircons) are the oldest rocks in the world.
• The zircons form containing uranium with little or no lead
• Any lead in the crystals comes from the decay of U235 and U238 into Pb207 and Pb206, respectively.
• We can use the ratios of Pb207-to-U235 and Pb206-to-U238 to estimate the age of the zircons. Both ratios can be used to estimate the age of the Earth, so we can cross-check our results for consistency. This technique is referred to as lead-lead dating.

### Basic Mathematics

We can use Equation 1 as our model for the variation of the isotope ratio over time.

 Eq. 1 $\text{Ratio}\left( t,{{t}_{\text{Half-Life}}} \right)=\frac{1-{{\left( \frac{1}{2} \right)}^{\frac{t}{{{t}_{\text{Half-Life}}}}}}}{{{\left( \frac{1}{2} \right)}^{\frac{t}{{{t}_{\text{Half-Life}}}}}}}$

where t is time and tHalf-Life is the half-life of the isotope in question. Using time as a parameter, we can plot x(t) = Ratio(t,tHalf Life U235) and y(t) = Ratio(t,tHalf Life U238). I threw the equations into Mathcad and plotted the data (Figure 1) assuming a U235 half life of 704 million years and a U238 half life of 4.47 billion years. As I would expect, my plot here agrees with the same plot shown on the About.com web site.

Figure 1: Concordia Plot Showing Uranium and Lead Ratios Versus Time.

When geologists measure the ratios and look at the chart, they see that the Earth is about 4.4 billions years old.

## Conclusion

I was able to duplicate the results presented on a couple of different web sites on determining the age of the Earth. This was a good exercise and really shows the power of using radioactive decay to determine the age of the Earth.

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