Problem 71 - General Chemistry: Principles and Modern Applications

Question

Assume that when Earth formed, uranium-238 and uranium-235 were equally abundant. Their current percent natural abundances are \(99.28 \%\) uranium- 238 and \(0.72 \%\) uranium- \(235 .\) Given half-lives of \(4.5 \times 10^{9}\) years for uranium-238 and \(7.1 \times 10^{8}\) years for uranium-235, determine the age of Earth corresponding to this assumption.

Step-by-Step Solution

Answer

The approximated age of Earth in this scenario is \(4.48 \times 10^{9}\) years.

Step 1: Determine the number of half-lives that has passed

First, find the ratio of the remaining amounts of uranium-238 and uranium-235. This is calculated by dividing the current amounts, which are \(99.28 \%\) for uranium-238 and \(0.72 \%\) for uranium-235.

Step 2: Calculate the difference in half-lives

The ratio calculated in the previous step is equal to \(2^n\), where n is the difference in the number of half-lives that has passed for both uranium isotopes. To find out the difference, solve the equation \(99.28/0.72 = 2^n\) for n using logarithms or approximation.

Step 3: Calculate the age of Earth

Now that the difference in the number of half-lives has been determined, the age of earth can be approximated. Multiply the number of additional half-lives that uranium-238 underwent compared to uranium-235, which is the value for n, by the half-life of uranium-235, which is \(7.1 \times 10^{8}\) years. The result will give an estimation for the age of the Earth.

Key Concepts

Uranium Isotopes

Uranium isotopes are different forms of the element uranium, distinguished by the number of neutrons in their nuclei. Two key isotopes used in radioactive dating are uranium-238 and uranium-235. These isotopes are crucial in estimating geological ages due to their long half-lives. When Earth formed, it is believed these isotopes were present in equal amounts. Over billions of years, the isotopes decay at different rates, allowing scientists to use their remaining abundances to date Earth's formation.

• Uranium-238: Has 146 neutrons and a current natural abundance of 99.28%.
• Uranium-235: Contains 143 neutrons and is found at a much lower abundance of 0.72% today.

These differences in abundance are due to their distinct decay rates, making uranium isotopes essential tools in the study of Earth’s age.

Half-Life Calculation

Understanding half-life calculations is crucial for radioactive dating. The half-life is the time it takes for half of a radioactive substance to decay. For our isotopes:

• Uranium-238: Has a half-life of \(4.5 \times 10^{9}\) years.
• Uranium-235: Has a half-life of \(7.1 \times 10^{8}\) years.

To find how many half-lives have passed, we calculate the ratio of the current abundances of these isotopes. The ratio is given by \[\frac{99.28}{0.72} = 2^n\]where \(n\) represents the number of half-lives. Solving for \(n\) involves using logarithms or approximations, as it shows how many more cycles of decay uranium-238 has undergone compared to uranium-235. This calculation is a key step in determining Earth's age.

Age of the Earth

Calculating the age of Earth involves using the difference in half-lives of uranium isotopes. Once \(n\), the difference in half-lives, is determined, it represents the additional cycles uranium-238 has gone through compared to uranium-235. We multiply \(n\) by the half-life of uranium-235 to estimate Earth's age:
\[\text{Age of Earth} = n \times 7.1 \times 10^8 \text{ years}\]This approach yields an approximation of Earth's age, aligning with scientific estimates of billions of years. Such calculations provide valuable insights into not only how old Earth is, but also how radioactive decay processes have shaped its geological history. Radioactive dating techniques like this underscore the link between natural phenomena and the passage of time.