Question

Scientists have estimated that the earth's crust was formed 4.3 billion years ago. The radioactive nuclide \(^{176} \mathrm{Lu},\) which decays to \(^{176} \mathrm{Hf}\), was used to estimate this age. The half-life of \(^{176} \mathrm{Lu}\) is 37 billion years. How are ratios of \(^{176} \mathrm{Lu}\) to \(^{176} \mathrm{Hf}\) utilized to date very old rocks?

Short answer

In order to date very old rocks, scientists use the radioactive decay of \(^{176}\mathrm{Lu}\) into \(^{176}\mathrm{Hf}\) as a timekeeper. They calculate the number of half-lives that have passed since the earth's crust formation, find the remaining fraction of \(^{176}\mathrm{Lu}\), and then compare the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in rock samples. Smaller ratios indicate older rocks because more of the \(^{176}\mathrm{Lu}\) has decayed into \(^{176}\mathrm{Hf}\). By measuring these ratios, scientists can estimate the age of ancient rocks and the earth's crust.

Step-by-step solution

Radioactive decay and half-life

Radioactive decay is a random process where unstable isotopes, such as \(^{176}\mathrm{Lu}\), lose energy by emitting radiation and transform into their decay products, in this case, \(^{176}\mathrm{Hf}\). Half-life is the time required for half of the unstable isotopes in a sample to decay. In this exercise, the half-life of \(^{176}\mathrm{Lu}\) is given as 37 billion years.

Calculate the number of half-lives

To find out the number of half-lives that the system has gone through since the crust's formation, we need to divide the age of the earth's crust by the half-life of \(^{176}\mathrm{Lu}\). Number of half-lives = Age of the earth's crust / Half-life of \(^{176}\mathrm{Lu}\) Number of half-lives = \( \frac{4.3\,\text{billion years}}{37\,\text{billion years}} \)

Calculate the remaining fraction of \(^{176}\mathrm{Lu}\)

The remaining fraction of the original \(^{176}\mathrm{Lu}\) is given by the equation: Remaining fraction = \( \left( \frac{1}{2} \right) ^\text{number of half-lives} \)

Utilize ratios of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\)

By comparing the remaining fraction of \(^{176}\mathrm{Lu}\) and the amount of the decay product \(^{176}\mathrm{Hf}\) present in the sample, scientists can make estimates about the age of the rock. The smaller the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\), the older the rock, since more of the \(^{176}\mathrm{Lu}\) has decayed into \(^{176}\mathrm{Hf}\). Thus, by measuring the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in rocks, scientists can establish their age and support the age estimate of very old rocks. So, the basic idea is that we use the radioactive decay of \(^{176}\mathrm{Lu}\) into \(^{176}\mathrm{Hf}\) as a timekeeper. By investigating the remaining proportion of the parent isotope (\(^{176}\mathrm{Lu}\)) and the amount of daughter isotope (\(^{176}\mathrm{Hf}\)) in the sample, we can estimate the age of the earthly crust or ancient rocks.

Key concepts

Radioactive Decay

Understanding the fundamentals of radioactive decay is crucial in many scientific fields, including geology and archeology. At its core, radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. When we say an isotope is unstable, we mean that it has an excess of energy or mass, or both, that can be lost by emitting particles or radiation. Thus, the isotope transforms into a different element or a different isotope of the same element. This decay occurs naturally and spontaneously, but at a rate that is characteristic of each isotope. Each different unstable isotope, or 'parent', will decay into 'daughter' isotopes at a rate described by its half-life.

For example, in the exercise given, the decay process transforms the parent isotope\thinspace \text{\(^{176}\mathrm{Lu}\)}\thinspace into its daughter isotope \thinspace \text{\(^{176}\mathrm{Hf}\)}. Understanding this transformation allows scientists to study and date geological materials, such as the rocks comprising Earth's crust, by observing the accumulations of the daughter products relative to the remaining parent isotope.

Half-Life

The half-life of a radioactive isotope is fundamental in determining the age of various objects and materials on Earth. Defined simply, the half-life is the time it takes for half of the radioactive isotope in a sample to decay. This property is specific to each isotope and remains consistent over time, which makes it very useful for dating purposes.

In the context of the exercise, the half-life of\thinspace \text{\(^{176}\mathrm{Lu}\)}\thinspace is 37 billion years. Why is this information important? It allows us to calculate the number of half-lives that have passed since Earth's crust formed. Shorter half-lives are more useful for dating young materials, while longer half-lives, such as that of\thinspace \text{\(^{176}\mathrm{Lu}\)}, enable the dating of the Earth's crust and very old rocks.

Isotope Ratios

Ratio of Parent and Daughter Isotopes

Isotope ratios are pivotal in techniques used to date materials, with parent to daughter isotope ratios being particularly informative. The ratio of an unstable 'parent' isotope to its stable 'daughter' product can reveal how much time has passed since part of a material stopped exchanging with the environment—its 'closure' date.

The ratio directly links to the number of half-lives that have passed: the more half-lives, the smaller the ratio of parent to daughter isotopes, because more of the parent isotope has had time to decay. By determining the current ratio in a sample, and understanding the half-life of the parent isotope, scientists can back-calculate to find out when the rock was formed.

Earth's Crust Age Estimation

Estimating the age of Earth's crust is like forensic work with clues left behind by isotopes. By analyzing the isotope ratios present in rock samples, geologists can piece together the history of our planet. Radioactive dating, specifically, exploits the natural decay process of radioactive isotopes like\thinspace \text{\(^{176}\mathrm{Lu}\)}\thinspace to estimate the age of very old materials.

We use the known half-life of these isotopes and their current ratios in the rocks to calculate backwards to the time of the rock's formation. This time, found by measuring the proportion of isotopes that have decayed, provides a quantifiable date that supports theories and models concerning the age and development of the Earth's crust. Thus, the Earth's crust age estimation is an analytical process that marries elemental chemistry with the passage of geologic time.