Question

A proposed nuclear theory suggests that the relative abundances of the uranium isotopes \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\) were approximately equal at the time they were formed. Today, the observed ratio of these isotopes \({ }^{235} \mathrm{U} /{ }^{238} \mathrm{U}\) is \(7.25 \times 10^{-3}\). Given that the half-lives for radioactive decay are \(7.04 \times 10^{8} \mathrm{y}\) for \({ }^{235} \mathrm{U}\) and \(4.47 \times 10^{9} \mathrm{y}\) for \({ }^{238} \mathrm{U}\), calculate the age of the elements.

Short answer

The elements are approximately 2 billion years old.

Step-by-step solution

Understand the Problem

We need to find out how long ago the isotopes of Uranium, specifically \(^{235}U\) and \(^{238}U\), had nearly equal relative abundances. Given the current ratio is \(7.25 \times 10^{-3}\), and known half-lives for both isotopes, we need to calculate their decay over time and determine their original condition.

Apply Decay Formula

The decay of an isotope is given by the formula \( N(t) = N_0 e^{-\lambda t} \), where \(N(t)\) is the number of atoms at time \(t\), \(N_0\) is the initial number of atoms, and \(\lambda\) is the decay constant. The decay constant \(\lambda\) is related to half-life \(t_{1/2}\) by \(\lambda = \frac{\ln(2)}{t_{1/2}}\).

Calculate Decay Constants

Compute the decay constants (\(\lambda\)) for both isotopes:\[\lambda_{235} = \frac{\ln(2)}{7.04 \times 10^8} \approx 9.85 \times 10^{-10}\,\text{y}^{-1}\]\[\lambda_{238} = \frac{\ln(2)}{4.47 \times 10^9} \approx 1.55 \times 10^{-10}\,\text{y}^{-1}\]

Set Up Decay Ratio Equation

Since the ratios were equal initially, set the initial ratio \( N_{0,235} / N_{0,238} = 1 \). Therefore, using the current ratio and decay formula, the equation becomes:\[\frac{N_{0,235} e^{-\lambda_{235} t}}{N_{0,238} e^{-\lambda_{238} t}} = 7.25 \times 10^{-3}\]

Solve for Time (t)

Substitute the initial conditions and the decay constants to solve for \(t\):\[e^{(\lambda_{238} - \lambda_{235}) t} = 7.25 \times 10^{-3}\]Solve for \(t\):\[t = \frac{\ln(7.25 \times 10^{-3})}{\lambda_{238} - \lambda_{235}}\]Calculate the value:\[t \approx \frac{\ln(7.25 \times 10^{-3})}{1.55 \times 10^{-10} - 9.85 \times 10^{-10}}\]Calculate \(t\) which gives\[t \approx 2 \times 10^{9}\,\text{years}\]

Verify and Conclude

Verify the calculations by considering the values and ensuring all expressions are consistent along the solution. Once verified, state the conclusion:The isotopes \(^{235}U\) and \(^{238}U\) had nearly equal abundances around 2 billion years ago.

Key concepts

Uranium Isotopes

Uranium, a naturally occurring heavy metal, has several isotopes, with its most notable ones being

• \(^{235}\text{U}\)
• \(^{238}\text{U}\)

These isotopes differ in the number of neutrons in their nucleus, resulting in different properties and stability over time.
\(^{235}\text{U}\) is particularly important in nuclear chemistry because it can sustain a chain reaction required for nuclear power and weaponry. This isotope, though comprising less than 1% of natural uranium, is highly significant because it is fissile.
On the other hand, \(^{238}\text{U}\) makes up more than 99% of natural uranium. Although it is not directly usable in reactors, its neutron capture can transform it into plutonium-239, another fissile isotope.
Understanding the differences between these isotopes helps us grasp how they decay and evolve over geologic time scales, shaping the understanding of nuclear chemistry and energy.

Half-lives

The half-life of a radioactive isotope provides essential information about its rate of decay. It is the time required for half the quantity of the isotope to decay into another element or isotope.
For uranium isotopes, the half-lives vary significantly:

• \(^{235}\text{U}\) has a half-life of \(7.04 \times 10^{8}\) years.
• \(^{238}\text{U}\) has a much longer half-life of \(4.47 \times 10^{9}\) years.

This stark difference means that \(^{238}\text{U}\) decays much more slowly than \(^{235}\text{U}\).
The concept of half-life is crucial not only in evaluating the remaining quantity of an isotope but also in calculating their age and understanding the timeframe over which certain isotopic ratios might have changed.

Decay Formula

The decay of radioactive isotopes is modeled by an exponential decay formula: \[ N(t) = N_0 e^{-\lambda t} \]Where:

• \(N(t)\) is the remaining quantity of the isotope at time \(t\).
• \(N_0\) is the original quantity of the isotope.
• \(\lambda\) is the decay constant.

The decay constant \(\lambda\) is related to the half-life \(t_{1/2}\) by the equation \(\lambda = \frac{\ln(2)}{t_{1/2}}\).
In the case of uranium isotopes, precise calculations using this formula allow scientists to determine how much of an isotope remains and infer the isotope's age or the time since it had a certain ratio of isotopes. It is a key tool in both nuclear chemistry and geochronology.

Nuclear Chemistry

Nuclear chemistry focuses on the reactions and properties of atomic nuclei, particularly those involving radioactive elements such as uranium. In this realm, we study:

• Nuclear reactions including fission and fusion.
• Transmutation of elements where one element transforms into another.
• Decaying processes of isotopes and the energy releases involved.

Understanding nuclear chemistry is vital as it explains mechanisms underlying nuclear energy production and the natural radioactive decay processes that have shaped Earth's evolution over billions of years.
This branch of chemistry also looks at how isotopes are used in practical applications, such as dating archaeological finds, medical imaging, and treatment, as well as fundamentally understanding the forces at play within atomic nuclei.