Question

What should be the mass ratio \(^{208} \mathrm{Pb} /^{232} \mathrm{Th}\) in a meteorite that is approximately \(2.7 \times 10^{9}\) years old? The half-life of \(^{232} \mathrm{Th}\) is \(1.39 \times 10^{10}\) years. [Hint: One \(^{208} \mathrm{Pb}\) atom is the final decay product of one \(^{232}\) Th atom.

Short answer

The mass ratio \(^{208} \mathrm{Pb} /^{232} \mathrm{Th}\) in a meteorite that is approximately \(2.7 \times 10^{9}\) years old is 0.14.

Step-by-step solution

Find Number of Half-lives

The age of the meteorite is \(2.7 \times 10^{9}\) years and the half-life of \(^{232} \mathrm{Th}\) is \(1.39 \times 10^{10}\) years. The number of half-lives that has passed can be found by dividing the age of the meteorite by the half-life of the isotope. Thus, \(\frac{2.7 \times 10^{9}}{1.39 \times 10^{10}} \) which is approximately 0.19.

Calculate the Decay

Next, for every half-life, half of the initial isotope will have decayed. The remaining amount of the parent isotope can be found by using the equation \(N = N_0 \times (0.5)^n\), where \(N_0\) is the initial quantity, \(n\) is the number of half-lives that have passed, and \(N\) is the remaining quantity. Here \(N_0 = 1\) ( the original ratio ), and \(n = 0.19\). Substituting these values gives \(N = 1 \times (0.5)^{0.19}\) which is approximately 0.86.

Determine the Ratio

The ratio \(^{208} \mathrm{Pb} / ^{232} \mathrm{Th}\) is found by subtracting the remaining \(^{232} \mathrm{Th}\) from the initial amount (which was 1), which gives \(1 - 0.86\), which equals 0.14.

Key concepts

Half-Life of Isotopes

Understanding the half-life of isotopes is crucial when studying the decay process of radioactive materials. Half-life is the time required for half of the radioactive nuclei in a sample to undergo radioactive decay. It is a constant value for each isotope, which makes it a powerful tool to date materials such as meteorites. For example, if an isotope has a half-life of 1,000 years, this means that after 1,000 years, only half of the original amount of the isotope remains; after another 1,000 years (a total of 2,000 years), only a quarter of the original amount would remain, and so on.

This predictable pattern of decay allows scientists to use isotopes as a 'clock' – by measuring the current amount of an isotope and knowing its half-life, we can calculate the age of the material. This ties directly into the next concept concerning how we date an object from space, such as a meteorite.

Meteorite Dating

Meteorite dating is a prime example of nuclear chemistry at work in astrophysical contexts. By applying our understanding of half-life, we can calculate the age of meteorites - remnants of the early solar system that can offer insights into its formation and age.

Since meteorites are not influenced by terrestrial processes, the isotopic composition is preserved since its formation. We use an isotope like Thorium-232 and measure its decay product Lead-208, to determine the time elapsed since the meteorite solidified from molten material. The isotope, ²³²Th, decays very slowly, with a half-life that often surpasses the age of the Earth, making it ideal for dating geological samples. By knowing the initial ratio of isotopes and measuring the current ratio, we are able to calculate the absolute age, as we did in our example problem.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions, processes, and properties of atomic nuclei. It is central to understanding radioactive decay, which includes the transformation of one element into another as a result of changes to the nucleus. When a nucleus undergoes decay, it emits particles or radiation, which leads to a loss of mass and a change in proton count - transforming the original atom (parent isotope) into a different element (daughter isotope).

This transmutation through decay provides the basis for radioactive dating methods used in geology, archaeology, and even environmental science. In our example involving meteorites, we are witnessing nuclear chemistry in action - the parent isotope ²³²Th decays into its daughter isotope ²⁰⁸Pb over billions of years.

Isotopic Mass Ratio

The concept of isotopic mass ratio is fundamental for discerning the age of geological and extraterrestrial samples. It refers to the proportion between the masses or abundances of different isotopes in a sample. In the context of our exercise, this ratio is between ²⁰⁸Pb and ²³²Th. These ratios often shift over time as parent isotopes decay into daughter isotopes. By analyzing the isotopic mass ratio of elements in a meteorite, scientists can use the known half-life of an isotope to find the age of the sample.

When addressing textbook exercises related to this concept, it's important to first understand the decay pathway and then apply the known half-life to calculate how much of the parent isotope remains. This in turn informs us how much of the daughter isotope has formed, enabling us to determine the isotopic mass ratio at any given point in the sample's history.