Question

Naturally occurring uranium is composed mostly of \(^{238} \mathrm{U}\) and \(235 \mathrm{U},\) with relative abundances of 99.28\(\%\) and \(0.72 \%,\) respectively. The half-life for \(^{238} \mathrm{U}\) is \(4.5 \times 10^{9}\) years, and the half-life for 235 \(\mathrm{U}\) is \(7.1 \times 10^{8}\) years. Assuming that the earth was formed 4.5 billion years ago, calculate the relative abundances of the 28 \(\mathrm{U}\) and \(^{235} \mathrm{U}\) isotopes when the earth was formed.

Short answer

When the Earth was formed, the relative abundances of Uranium-238 and Uranium-235 isotopes were \(99.28\%\) and \(100\%\) respectively.

Step-by-step solution

Identify the decay formula

The decay formula is given by: \[ N_t = N_0 \times (1/2)^{t/T}\] Where \(N_t\) is the amount of the isotope at time \(t\), \(N_0\) is the initial amount of the isotope, \(t\) is the time elapsed, and \(T\) is the half-life of the isotope.

Set up the decay formulas for both isotopes

Let the initial relative abundance of Uranium-238 be \(X\%\) and that for Uranium-235 be \((100-X)\%\). Now, we can write the decay formulas for both isotopes as: \[N_t(^{238}U) = X \times (1/2)^{t/T_{238}}\] \[N_t(^{235}U) = (100-X) \times (1/2)^{t/T_{235}}\]

Substitute the known values

We know the present relative abundances and half-lives of both isotopes and the age of the Earth. We can substitute these values into the formulas we derived in Step 2: \[99.28 = X \times (1/2)^{4.5 \times 10^9 / 4.5 \times 10^9}\] \[0.72 = (100-X) \times (1/2)^{4.5 \times 10^9 / 7.1 \times 10^8}\]

Solve for the initial relative abundance of Uranium-238

Solving the first equation for \(X\), we find that the initial relative abundance of Uranium-238 is the same as its present relative abundance: \[X = 99.28\%\]

Solve for the initial relative abundance of Uranium-235

Now, we solve the second equation for the initial relative abundance of Uranium-235: \[0.72 = (100-99.28) \times (1/2)^{4.5 \times 10^9 / 7.1 \times 10^8}\] \[(1/2)^{4.5 \times 10^9 / 7.1 \times 10^8} = 0.0072 = \frac{(100-X)}{Y}\] \[Y = (100-X) / 0.0072\] \[Y \approx 100\] The initial relative abundance of Uranium-235 when the Earth was formed is approximately \(100\%\). So, when the Earth was formed, the relative abundances of Uranium-238 and Uranium-235 isotopes were \(99.28\%\) and \(100\%\) respectively.

Key concepts

Uranium Isotopes

Uranium isotopes, particularly Uranium-238 ( ^{238}U) and Uranium-235 ( ^{235}U), are fascinating components of nuclear chemistry. These isotopes differ in their atomic mass, which affects their physical properties and stability. ^{238}U and ^{235}U are naturally occurring isotopes, with ^{238}U being much more abundant at 99.28%, compared to ^{235}U at a mere 0.72%.
These isotopes are vital for different applications. ^{235}U is especially important in nuclear reactors and nuclear weapons due to its ability to sustain a nuclear chain reaction. This is because it can easily undergo fission when struck by a neutron. In contrast, ^{238}U is more stable and doesn't fission easily, serving more of a supportive role in these processes but is crucial for generating ^{239}Pu, another fissionable material.
Understanding the balance of these isotopes in the past, just after the Earth's formation, offers insights into geological processes and the Earth's history. This balance has shifted over billions of years due to radioactive decay, making isotopic analysis an essential tool in geology and archeology.

Half-Life

Half-life is a fundamental concept in understanding radioactive decay. It refers to the time required for half of the atoms in a radioactive material to decay. In nuclear chemistry, knowing the half-life of isotopes allows scientists to predict their behavior over time.
For ^{238}U, the half-life is extremely long, approximately 4.5 billion years. This means it decays very slowly, remaining prevalent in the Earth's crust to this day. In contrast, ^{235}U has a shorter half-life of about 710 million years. This faster decay rate explains its lower abundance as compared to ^{238}U.
Half-life calculations involve the decay formula: \[N_t = N_0 \times \left(\frac{1}{2}\right)^{t/T}\]where N_t is the remaining quantity of the isotope at time t, N_0 is the initial quantity, and T is the half-life. Understanding this formula helps calculate past abundances of isotopes by mapping the decay process backward, a useful tool in many scientific investigations.

Nuclear Chemistry

Nuclear chemistry focuses on reactions and processes at the atomic nucleus level. Radioactive decay, like the process involving ^{238}U and ^{235}U, is central to this field. It investigates changes in an atom's nucleus and how these changes affect the atom's properties.
This branch of chemistry explores phenomena such as nuclear reactions, fission, and fusion. In the context of uranium isotopes, nuclear chemistry helps explain how isotopes transform over time and contribute to larger processes, be it in nature or technology. ^{238}U's long half-life makes it significant for age dating geological samples, providing crucial data about Earth's history.
Knowledge in nuclear chemistry enables advancements in energy production, medical treatments, and understanding of cosmic events. The decay of uranium isotopes, for example, not only provides energy via nuclear power but also helps scientists determine the age of rocks and fossils, offering a timeline for Earth's history.