Problem 41 - Astronomy: At Play in the Cosmos

Question

A rock sample is determined to have a concentration of uranium- 238 (U-238) that is consistent with 0.5 half-life having passed. What percentage of \(\mathrm{U}-238\) remains, and how old is the rock?

Step-by-Step Solution

Answer

70.7% of U-238 remains, and the rock is 2.25 billion years old.

Step 1: Understanding the Half-Life Concept

A half-life is the time required for half of the radioactive nuclei in a sample to decay. After one half-life, 50% of the original substance remains.

Step 2: Applying the Concept to U-238

Given that the rock has undergone 0.5 half-lives, this means that only 0.707 (or 70.7%) of the original U-238 remains. This is calculated using the formula \(R = (0.5)^{t/T}\), where \(t/T = 0.5\). So \(R = (0.5)^{0.5} = \sqrt{0.5} \approx 0.707\). This means 70.7% remains.

Step 3: Calculating the Age of the Rock

The half-life of U-238 is about 4.5 billion years. Since 0.5 half-life has passed, the age of the rock is \(0.5 \times 4.5\) billion years, which equals 2.25 billion years.

Key Concepts

Uranium-238 Decay

Uranium-238 (U-238) is a radioactive isotope of uranium. This means it is unstable and will break down, or decay, into a more stable form over time. In this process, U-238 undergoes a series of transformations. It eventually becomes lead-206. This decay process is known as a decay chain.
One interesting aspect is that U-238 does not decay into lead-206 in one step. It happens through a sequence of up to 14 different radioactive substances. These intermediary substances are themselves radioactive and lead to the final stable isotope.
U-238 is plentiful in nature and is commonly used in radiometric dating. This is because its decay provides a reliable clock for dating geological formations. It has a very long half-life of 4.5 billion years, making it particularly useful for dating rocks and minerals that are very old.

Half-Life Calculation

In radiometric dating, the concept of half-life is crucial. It is the time required for half of the radioactive nuclei in a sample to decay. For uranium-238, this period is about 4.5 billion years.
When calculating the remaining amount of a substance like U-238, we use the half-life formula:

• Let \( t \) be the elapsed time.
• \( T \) is the half-life of the substance.
• \( R \) is the fraction remaining.

The formula is: \( R = \left(0.5\right)^{t/T} \).
In our example, if 0.5 half-lives have passed, \( t/T = 0.5 \). Plugging into the formula gives us \( R = \left(0.5\right)^{0.5} = \sqrt{0.5} \approx 0.707 \). This tells us about 70.7% of U-238 remains.

Age of Rocks Determination

Determining the age of rocks is a fundamental aspect of geology, and radiometric dating serves as a key tool. By analyzing the ratios of parent versus daughter isotopes, geologists can estimate how long certain processes have been going on.
For uranium-238, this involves its decay into lead-206. By measuring the ratio of U-238 to lead-206, scientists can calculate when the rock formed. In our example, since the rock has undergone 0.5 half-lives, we know it is quite old.
With U-238's half-life of 4.5 billion years, if 0.5 half-lives have passed, we calculate the age of the rock as:

• Age = \( 0.5 \times 4.5 \) billion years.
• This equals 2.25 billion years.

This method provides an accurate and scientific means to understand geological timelines.