Radioactive Dating — Overview & Applications

The earth is 4.5 billion years old. At first glance, that seems pretty old. But, the universe is about 13.8 billion years old. Our planet has only existed for about a third of that total! Attempts to calculate these ages go back hundreds of years. It started in 1862! Lord Kelvin calculated that the earth was between 20 and 40 million years old. That's the same Lord Kevin who developed the absolute temperature scale. He was only off by a couple of orders of magnitude! His calculations were actually quite sophisticated. He calculated the time needed for a molten earth to first solidify. Then he calculated the time required to cool to the current temperatures. What was he missing? The heat created via radioactivity and convection within the mantle.

Do you remember the discovery of radioactivity? It goes back to Earnst Rutherford. Remember, alpha decay emits a helium nucleus. Rutherford assumed that the alpha particles got trapped within the rocks as helium gas. So, let's say we know the rate of alpha decay and the amount of helium. We should be able to calculate the age of the rock. It turns out those are poor assumptions. But it was the first attempt at radioactive dating!

What is Radioactive Dating

Radioactive dating is a technique for measuring a sample's age. We have methods for both organic and inorganic specimens. How does it work? On earth, there are many naturally occurring radioactive isotopes. We've discovered the natural abundance of these isotopes. We also know the ratio of the radioactive isotope to the stable isotope. Radioactive elements decay at a constant rate. That's the element's half-life.

So, let's say we have a sample we want to date. What would we do? First, we can measure the amount of stable and radioactive isotopes. That would give us the current ratio. We know that the proportion will be less than what occurs in nature. Why? The radioactive element has been decaying! But, we can determine the original mass of the radioactive isotope. Then, we use the half-life to work backward and calculate the sample's age.

Let's look at a couple of the most common methodologies.

Carbon-14 Decay

To calculate the age of organic samples, we use carbon-14 dating. How is carbon-14 formed? Most carbon-14 comes from the top of the atmosphere. Our atmosphere is hit constantly by cosmic rays. They come from our sun or far-off supernovas. What are cosmic rays? They are protons or neutrons that come from fusion reactions. They travel through space and collide with our atmosphere. They also have unbelievably large energies! Sometimes, a neutron hits atmospheric nitrogen. The collision is so powerful that it dislodges a proton. What's the result? Carbon-14!

But how do we intake carbon-14? We need to remember that isotopes still undergo chemical reactions. So, carbon-14 can oxidize into carbon dioxide. But remember, nuclei are unaffected by chemical bonding. So, the rate of radioactive decay remains constant. Plants intake carbon dioxide as a part of photosynthesis. Animals then eat the plants. If you're higher up the food chain, you might eat the animals that ate the plants. Anyway, plants and animals have a continuous input of carbon-14. So, the proportion of carbon-12 to carbon-14 equals the ratio in the atmosphere. But what happens when it dies? They're no longer consuming it. Carbon-14 begins to decay. So, the ratio changes.

So, how do we measure age? Our sample will have a specific ratio of carbon-12 to carbon-14. We know that the starting ratio equaled the atmospheric proportions. So, we can compare the current balance to the original. That can tell us the number of half-lives. For example, let's say our ratio decreased by half. We know the sample experience one half-life. The half-life of carbon-14 is 5715 years. So, our sample is 5730 years old!

Increasing the Accuracy of Carbon-14 Dating

How accurate is carbon-14 dating? Let's consider our most significant assumption. We have to assume carbon-14's formation rate is generally constant. Generally, it's a reasonable assumption. We do know that cosmic radiation can fluctuate. But, minor fluctuations average out. Yet, there have been times when cosmic rays have spiked. So, how do we correct for these periods? We calibrate the carbon-14 content with tree rings! Tree rings give an incredibly accurate age because trees grow uniformly. Trees grow in predictable patterns during certain times of the year. So, by counting the tree's rings, we can measure its exact age. We can then compare the tree's age to the carbon-14 test result. During carbon-14 spikes, more of it gets incorporated into the ring. So, the carbon-14 test is inaccurate. But, we know the tree's age at that ring. So we have a correction factor!

Uranium-238 Decay

When we studied decay series, we looked at Uranium-238. It decays to lead-206 with a half-life of 4.5 billion years. Uranium-238 works great for dating rocks. We can generally assume that all lead-206 is from Uranium decay. Can lead-206 come from other sources? It can. But then it's accompanied by lead-208. So, we can correct the amount of lead-206 that came from uranium.

Practice Problems

Let's do a few practice problems.

1. Let's say we have a rock we want to date. The ratio of Uranium-238 to Lead-206 is 1.000 mg to 0.128 mg. The half-life is 4.5 billion years. Calculate the age of the rock.

First, we need to calculate the starting mass of Uranium-238. We know it decayed to lead-206. So, we need to work backward from lead-206. It's like a balanced chemical equation with a one-to-one mole ratio. So, we can use dimensional analysis. Just remember that everything is in milligrams. So convert the molar mass to milligrams per mole.

$U_{238\,initial}=1.000mg + (0.128 mg_{\,Pb-206} \times \frac{1 mol_{\,Pb-206}}{205970.0 mg}\times \frac{\,1 mol_{U-238}}{\,1 mol_{Pb-206}}\times \frac{238050.0mg_{\,U-238}}{1 mol_{\,U-238}})$

$=1.000mg + 0.148 mg= 1.148 mg_{\,U-238}$

Now we need to calculate the decay constant.

$\lambda=\frac{0.693}{t_{\frac{1}{2}}}=\frac{0.693}{4.5\times10^{9}}=\frac{1.5\times10^{-10}}{yr}$

Finally, we can use the half-life equation to calculate time. Remember, all radioactive decay follows the first-order differentiated rate law.

$\ln{(\frac{N_{t}}{N_{o}})}=-\lambda \times t \rightarrow t= \frac{-1}{\lambda}\times ln{(\frac{N_{t}}{N_{o}})}=\frac{-1}{\frac{1.5\times10^{-10}}{yr}}\times \ln{(\frac{1.000}{1.148})}=\frac{-1\times -0.138}{\frac{1.5\times10^{-10}}{yr}}= 9.2 \times 10^{12} years$

1. Let's look at an example using carbon-14. Let's say we wanted to date an archeological artifact made of wood. The artifact's activity is 9.5 disintegrations per second. A modern wood sample's activity is 17.5 disintegrations per second. How old is the artifact?

Do you remember the activity equation? It's $A=\lambda \times N$. Let's see what happens if we take the ratio of the activities.

$\frac{A_{t}}{A_{o}}=\frac{\lambda\times N_{t}}{\lambda\times N_{o}}= \frac{N_{t}}{N_{o}}$

Now we need to calculate the decay constant.

$\lambda=\frac{0.693}{t_{1/2}}=\frac{0.693}{5730yrs}=0.00012yrs^{-1}$

Now, we can calculate the age of the artifact.

$t=\frac{-1}{\lambda}\times \ln{(\frac{N_{t}}{N_{0}})}=\frac{-1}{0.00012yrs^{-1}}\times \frac{9.5}{17.5}=5051.2 yrs$

(Video) Radiometric Dating: Carbon-14 and Uranium-238

By Professor Dave Explains

In this quick video, Professor Dave explains radioactive dating. He starts by explaining what it is and how it works. Then, Dave talks about using multiple dating methods to confirm results. For example, he talks about using carbon-14 and uranium-238 dating together. Throughout, he talks about the practical applications of radioactive dating. This is an excellent and concise video for learning about radioactive dating!