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 halflife 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 \({ }^{238} \mathrm{U}\) and \({ }^{235} \mathrm{U}\) isotopes when the earth was formed.

Short answer

The relative abundances of the \({ }^{238}\mathrm{U}\) and \({ }^{235}\mathrm{U}\) isotopes when the earth was formed were 99.9822% and 0.0178%, respectively.

Step-by-step solution

Write down the given information

We are given the following information: - Current relative abundance of \({ }^{238}\mathrm{U}\): 99.28\% - Current relative abundance of \({ }^{235}\mathrm{U}\): 0.72\% - Half-life for \({ }^{238}\mathrm{U}\): \(4.5 \times 10^9\) years - Half-life for \({ }^{235}\mathrm{U}\): \(7.1 \times 10^8\) years - Earth's age: 4.5 billion years

Calculate the decay constant for both isotopes

The decay constant, \(\lambda\), is given by the formula: \(\lambda = \frac{ln(2)}{T}\) where T is the half-life of the isotope. Calculate the decay constants for both isotopes: 1. For \({ }^{238}\mathrm{U}\): \(\lambda_{238} = \frac{ln(2)}{4.5 \times 10^9} = 1.54 \times 10^{-10}/year\) 2. For \({ }^{235}\mathrm{U}\): \(\lambda_{235} = \frac{ln(2)}{7.1 \times 10^8} = 9.76 \times 10^{-10}/year\)

Re-arrange the decay equation to find the initial abundance

Re-arrange the decay equation to solve for the initial number of nuclei (N_0): \(N_0 = \frac{N(t)}{e^{-\lambda t}}\) We are given the current relative abundances, so we can write them as the ratio of their initial abundance: \(\frac{N_{0_{238}}}{N_{0_{235}}} = \frac{N_{238}(t)}{N_{235}(t)} \times \frac{e^{-\lambda_{238}t}}{e^{-\lambda_{235}t}}\) where \(N_{238}(t)\) and \(N_{235}(t)\) are the final abundances after time t.

Calculate the initial abundance ratio

Substitute the current relative abundances, the decay constants, and the time (the age of Earth) into the equation: \(\frac{N_{0_{238}}}{N_{0_{235}}} = \frac{99.28}{0.72} \times \frac{e^{-(1.54 \times 10^{-10})(4.5 \times 10^{9})}}{e^{-(9.76 \times 10^{-10})(4.5 \times 10^{9})}}\) \(\frac{N_{0_{238}}}{N_{0_{235}}} = 137.61 \times \frac{e^{-0.693}}{e^{-4.40}}\) \(\frac{N_{0_{238}}}{N_{0_{235}}} = 137.61 \times \frac{0.50}{0.0122}\)

Find the initial relative abundances

Solve the equation to obtain the initial abundance ratio: \(\frac{N_{0_{238}}}{N_{0_{235}}} = 5625.20\) To find the initial relative abundances, we can first set their sum to be 100%: \(N_{0_{238}} + N_{0_{235}} = 100\) Now, use the ratio calculated in the previous step to express the two abundances: \(N_{0_{238}} = 5625.20 \times N_{0_{235}}\) Substitute this equation into the sum equation: \(5625.20 \times N_{0_{235}} + N_{0_{235}} = 100\) Solve for \(N_{0_{235}}\): \(N_{0_{235}} = 0.0178\%\) Now solve for \(N_{0_{238}}\): \(N_{0_{238}} = 100 - N_{0_{235}} = 99.9822\%\)

Final Answer

The relative abundances of the \({ }^{238}\mathrm{U}\) and \({ }^{235}\mathrm{U}\) isotopes when the earth was formed were 99.9822% and 0.0178%, respectively.

Key concepts

Relative Abundance of Isotopes

Understanding the relative abundance of isotopes is fundamental in fields such as geology, archaeology, and physics. Isotopes are variants of a particular chemical element that differ in neutron number, while their proton number remains the same. This results in isotopes of the same element with different mass numbers. The key aspect is that these isotopes typically occur in variable but consistent ratios known as their relative abundances, which can be measured using various spectrometric techniques.

For instance, the naturally occurring uranium isotopes, Uranium-238 and Uranium-235 , have relative abundances of 99.28% and 0.72%, respectively. These figures are crucial for multiple purposes, including dating rocks and minerals using uranium-lead dating methods, understanding nuclear fuel composition, and studying the origins of the solar system. The relative abundance of isotopes provides a snapshot of the isotopic composition, which is then used to calculate the original concentrations or date the age of a specimen using principles of radioactive decay.

Isotope Half-Life

The isotope half-life is the time it takes for half of the atoms in a radioactive isotope sample to decay. This concept is critical to nuclear chemistry and various applications such as radiometric dating. Each isotope has a characteristic half-life that remains constant regardless of the initial amount of substance or its physical state.

Take the exercise's isotopes: Uranium-238 has a half-life of 4.5 billion years, whereas Uranium-235's is about 707 million years. By knowing these half-lives and the current abundances, we can trace back in time to determine the original abundances of these substances when the Earth formed. This backward calculation involves elaborate formulae that use the concept of half-lives to unravel the history hidden within the samples. Through this process, scientists can gain insight into the dynamics of our planet's formation and the nature of the energetic processes that took place eons ago.

Radioactive Decay Constant

The radioactive decay constant, denoted as λ (lambda), is key to understanding the rate at which radioactive isotopes decay over time. It is inherently linked to an isotope's half-life—mathematically expressed as λ = ln(2) / T, where T is the half-life. The decay constant represents the probability per unit time that a specific atom will decay. In the context of our exercise, knowing the decay constant allows us to calculate the initial relative abundances of Uranium-238 and Uranium-235 when Earth was formed.

For any radioactive isotope, the number of atoms, N, present after time, t, can be calculated using the formula N(t) = N_0 e^(-λt), where N_0 is the initial number of atoms and e is the base of the natural logarithms. When dealing with relative abundances, we can rearrange this equation to solve the problem at hand, requiring an understanding of exponential decay processes. This concept enables us to peer back through billions of years of geological history just by analyzing present-day materials.