Problem 8 - Chemistry

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?

Step-by-Step Solution

Answer

To date very old rocks, scientists use the radioactive decay of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) and their half-life of 37 billion years. They first measure the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in the rock sample and then utilize the following formula to estimate the age of the rock: \( t = \frac{1}{\lambda} \ln \left(1 + \frac{N_\mathrm{Hf}}{N_\mathrm{Lu}}\right) \), where \(t\) is the age, \(\lambda\) is the decay constant, and \(N_\mathrm{Hf}\) and \(N_\mathrm{Lu}\) are the amounts of \(^{176}\mathrm{Hf}\) and \(^{176}\mathrm{Lu}\) in the sample, respectively. By analyzing these ratios, scientists have estimated that the Earth's crust was formed 4.3 billion years ago.

Step 1: Understand Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In this case, \(^{176}\mathrm{Lu}\) is decaying to \(^{176}\mathrm{Hf}\). The time it takes for half of the original radioactive substance to decay is called the half-life. For \(^{176}\mathrm{Lu}\), the half-life is 37 billion years.

Step 2: Calculate the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\)

To date a rock using this method, scientists first determine the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in the rock. This requires measuring the amounts of both isotopes present in the rock sample. The ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) will give an indication of how long the decay process has been occurring since the rock's formation.

Step 3: Use half-life to find the age of the rock

With the half-life of \(^{176}\mathrm{Lu}\) and the ratio of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in the rock, the age of the rock can be estimated. The formula to calculate the age is: \[ t = \frac{1}{\lambda} \ln \left(1 + \frac{N_\mathrm{Hf}}{N_\mathrm{Lu}}\right) \] where \(t\) is the age of the rock, \(\lambda\) is the decay constant (which is related to the half-life by the formula \(\lambda = \frac{\ln 2}{T_{1/2}}\), where \(T_{1/2}\) is the half-life), \(N_\mathrm{Hf}\) is the amount of \(^{176}\mathrm{Hf}\) in the sample, and \(N_\mathrm{Lu}\) is the amount of \(^{176}\mathrm{Lu}\) in the sample.

Step 4: Estimate the age of the Earth's crust

In this exercise, scientists have used this method to estimate that the Earth's crust was formed 4.3 billion years ago. By analyzing the ratios of \(^{176}\mathrm{Lu}\) to \(^{176}\mathrm{Hf}\) in very old rocks, they were able to determine the age of formation through radioactive decay processes. This method is critical in understanding the geological history and development of our planet.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei transform into more stable ones by emitting energy. This transformation can release particles like alpha, beta, or gamma rays. For the element \(^{176} ext{Lu}\), radioactive decay results in its conversion to \(^{176} ext{Hf}\). It's a slow process, requiring billions of years.

• This decay helps estimate the age of geological formations.
• It's like a natural clock ticking away over millions of years.

Understanding radioactive decay is key to determining the lifetime of materials on Earth.

Half-life

The concept of half-life is crucial in radiometric dating. It's the time it takes for half of a radioactive substance to decay. For \(^{176} ext{Lu}\), this period is a whopping 37 billion years.

• Half-life provides a measure of stability for an isotope.
• Long half-lives mean the substance changes very slowly.

By knowing the half-life, scientists can use the remaining radioactive material to calculate how old a rock or fossil is. This straightforward concept is a cornerstone of geochronology.

Isotopic Ratios

Isotopic ratios involve comparing the amounts of different isotopes within a sample. For dating rocks, the ratio of \(^{176} ext{Lu}\) to \(^{176} ext{Hf}\) is used.

• This ratio indicates how much decaying has occurred.
• A greater amount of \(^{176} ext{Hf}\) means more time has passed since the rock formed.

By calculating these ratios, scientists can determine the age of rocks, which is vital for understanding Earth's history. It's like having a snapshot of the past stored in stone.

Geochronology

Geochronology is the science of determining the age of rocks and fossils. It's like piecing together a massive chronological puzzle of Earth's history.

• Techniques like radiometric dating provide accurate time estimates.
• This field helps us explore the timeline of Earth's development.

Studying isotopic ratios and half-lives contributes directly to geochronology. It allows researchers to map out significant events like the formation of Earth's crust, offering insights into the dynamics of our planet over billions of years.