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

The ratio of \({ }^{176}\mathrm{Lu}\) to \({ }^{176}\mathrm{Hf}\) is utilized to date very old rocks by examining the changes in the radioactive isotope \(^{176}\mathrm{Lu}\) over time due to radioactive decay. Since \({ }^{176}\mathrm{Hf}\) does not change (it's not radioactive), it serves as a reference point for measuring the changes in the amount of \({ }^{176}\mathrm{Lu}\). The half-life equation, \( N_t = N_0(1/2)^{t/T} \), is used to calculate the remaining amount of \({ }^{176}\mathrm{Lu}\) after a given time period, and by comparing this ratio with the initial ratio, scientists can estimate the time that has passed, thus determining the age of the rock.

Step-by-step solution

Understanding Radioactive Decay and Half-Life

Radioactive decay is a process by which unstable atomic nuclei lose energy by emitting radiation. This decay eventually leads to the formation of stable isotopes. A radioactive substance's half-life is the time it takes for half of the radioactive atoms to decay. In this exercise, we are given the radioactive nuclide \({ }^{176}\mathrm{Lu}\) that decays into \({ }^{176}\mathrm{Hf}\). The half-life of Lu-176 is 37 billion years, meaning that after 37 billion years, half of the initial amount of Lu-176 will decay into Hf-176.

The Initial Ratio of \({ }^{176}\mathrm{Lu}\) and \({ }^{176}\mathrm{Hf}\)

Initially, during the Earth's formation approximately 4.3 billion years ago, we can assume that the rocks had a certain amount of Lu-176 and Hf-176. As time progresses, the amount of Lu-176 decreases (due to radioactive decay), while the amount of Hf-176 increases. Since the amount of Hf-176 does not change (it's not radioactive), it serves as a reference point for measuring the changes in the amount of Lu-176. The ratio between these two isotopes can be used to determine the age of the rock.

Calculating the Remaining Amount of \({ }^{176}\mathrm{Lu}\)

To determine the age of a rock using the isotopes, we need to find the remaining fraction of the radioactive isotope (Lu-176) after a given time period has passed. Using the half-life equation, we can write: \( N_t = N_0(1/2)^{t/T} \) Where: \(N_t\) = remaining amount of Lu-176 after time t, \(N_0\) = initial amount of Lu-176, t = time (in years), and T = half-life of Lu-176 (37 billion years).

Using the Ratio to Determine the Age of the Rock

The ratio of \({ }^{176}\mathrm{Lu}\) to \({ }^{176}\mathrm{Hf}\) can then be used to estimate the age of the rock. As the radioactive decay takes place, the ratio will change, and by comparing this ratio with the initial ratio, we can estimate the time that has passed. R = \(\frac{N_t}{N_{Hf}}\), where R is the ratio of Lu-176 to Hf-176. By measuring the current ratio of these isotopes, scientists can compare it to the initial ratio and use this information to estimate the age of the rock. Using the known half-life, it becomes possible to determine the age of rocks dating back billions of years, including the Earth's crust itself.

Key concepts

Half-Life

The half-life of a radioactive isotope is a fundamental concept that helps us understand how isotopes transform over time. Radioactive decay occurs when an unstable isotope loses energy and transforms into a more stable form, releasing particles in the process. This transformation occurs over a period defined by its half-life.
For instance, in the case of

• extsuperscript{176}Lu decaying to extsuperscript{176}Hf, the half-life is an extremely long 37 billion years, meaning that every 37 billion years, half of the remaining extsuperscript{176}Lu will have decayed into extsuperscript{176}Hf.
• Understanding half-life is crucial, as it provides a timeframe for when these transformations will occur and how many isotopes will remain at any given time.

This concept is foundational in fields such as radiometric dating, where scientists measure the ratios of isotopes to estimate ages of materials.

Isotope Decay

Isotope decay is the process through which an unstable isotope converts into a stable isotope. This occurs through radioisotopes losing small amounts of energy and particles. The decaying isotope, called the parent isotope, will eventually stabilize as a daughter isotope. This transformation is:

• An essential part of radiometric dating techniques, allowing scientists to date rocks and fossils by comparing the abundance of parent and daughter isotopes.
• When applied to extsuperscript{176}Lu and extsuperscript{176}Hf, scientists measure how the ratios change over time to estimate geological ages.

Because isotopes decay at known rates, by understanding these rates, we can backtrack to determine when a rock or piece of ancient Earth material first formed.

Earth's Crust Age

By utilizing isotope decay, scientists can accurately estimate the age of the Earth’s crust. Determining its age involves measuring how much of the original radioactive substance remains versus how much has turned into a stable form.

• For the Earth, scientists have pinned the age of the crust at approximately 4.3 billion years.
• This estimate is based on the ratio of extsuperscript{176}Lu to extsuperscript{176}Hf isotopes in rocks.
• The known half-life of extsuperscript{176}Lu helps to create a timeline of events, providing a record of the Earth’s geological history.

By knowing the current isotope ratio and using the half-life, scientists can compute backward to confirm ancient ages, revealing extensive information about Earth's formation.

Radiometric Dating Technique

Radiometric dating is a powerful technique used to ascertain the age of rocks and geological features. By measuring the ratio of parent to daughter isotopes in a sample, and knowing the decay rate, scientists can deduce ages. Different isotopic systems are useful depending on the timescale needed.

• For example, the extsuperscript{176}Lu/ extsuperscript{176}Hf system is ideal for dating very old geological formations, like the Earth’s crust.
• This technique is precise and allows the ages of rocks to be calculated over billions of years, helping to map out the history of our planet in incredible detail.
• With radiometric dating, scientists can tell when a particular rock solidified from liquid magma, thus giving insights into early Earth and ongoing geological processes.

Through these methods, we gain a deeper appreciation of geological timescales and the dynamic processes that have shaped the Earth.