Question

On December \(2,1942,\) the first man-made self-sustaining nuclear fission chain reactor was operated by Enrico Fermi and others under the University of Chicago stadium. In June 1972 , natural fission reactors, which operated billions of years ago, were discovered in Oklo, Gabon. At present, natural uranium contains \(0.72 \%{ }_{92}^{235} \mathrm{U}\). How many years ago did natural uranium contain \(3.0 \%\) \({ }_{92}^{235} \mathrm{U},\) sufficient to sustain a natural reactor? \(\left(t_{1 / 2}\right.\) for \({ }_{92}^{235} \mathrm{U}=\) \(7.04 \times 10^{8}\) years.

Short answer

1.82 billion years ago.

Step-by-step solution

Understand the problem

We need to calculate how many years ago the concentration of \( \text{U}^{235} \) in natural uranium was 3.0%, given that the current concentration is 0.72% and the half-life \( t_{1/2} \) of \( \text{U}^{235} \) is \( 7.04 \times 10^8 \) years.

Use the decay formula

The decay of \( \text{U}^{235} \) can be described by the formula: \[ N(t) = N_0 \exp\left(-\frac{t}{\tau}\right) \]where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time, and \( \tau \) is the mean lifetime (related to half-life by \( \tau = \frac{t_{1/2}}{\ln(2)} \)).

Calculate the mean lifetime (\( \tau \))

Calculate \( \tau \) using the formula: \[ \tau = \frac{7.04 \times 10^8}{\ln(2)} \approx 1.015 \times 10^9 \text{ years} \].

Set up the equation for concentration change

We need to find the time \( t \) when \( \frac{N(t)}{N_0} = \frac{0.72}{3.0} \). Using the decay formula:\[ \frac{0.72}{3.0} = \exp\left(-\frac{t}{1.015 \times 10^9}\right) \].

Solve for time \( t \)

Taking the natural logarithm on both sides, we have:\[ \ln\left(\frac{0.72}{3.0}\right) = -\frac{t}{1.015 \times 10^9} \].Solving for \( t \), we find: \[ t = -\ln\left(\frac{0.72}{3.0}\right) \times 1.015 \times 10^9 \].Calculating it leads to: \[ t \approx 1.82 \times 10^9 \text{ years ago} \].

Key concepts

Nuclear Fission

Nuclear fission is a crucial process in nuclear chemistry where the nucleus of an atom splits into two or more smaller nuclei, along with the release of a significant amount of energy. Put simply, fission is like popping a big atomic bubble into smaller ones, releasing energy in the process. This energy is immense and is utilized in nuclear power plants and atomic bombs.

Here are some key points about nuclear fission:

• It typically involves heavy elements like uranium or plutonium.
• The process can start spontaneously but is often initiated by a neutron hitting the nucleus.
• Fission releases more neutrons, which can then cause further fission reactions, creating a chain reaction.
• Controlled fission is used in nuclear reactors to produce electricity, while uncontrolled fission is used in nuclear weapons.

The amazing fact about the first self-sustaining nuclear chain reaction created by Enrico Fermi and his team is that it marked the dawn of the nuclear age, proving that fission could be sustained and controlled for practical uses.

Radioactive Decay

Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. One common type of decay is the decay of Uranium-235 ( U^{235} ), a process essential to understanding nuclear reactors and radiometric dating.

Key features of radioactive decay include:

• It is a random process at the level of single atoms, but predictable in larger quantities.
• Each isotope has a characteristic half-life, which is the time taken for half the quantity of the isotope to decay.
• The decay process transforms the original atom into a different element or isotope.

In our problem, we used the decay formula to calculate how many years ago the concentration of U^{235} in nature was able to sustain a natural fission reactor. The half-life of U^{235} is approximately 704 million years, which allows us to determine age using exponential decay equations. By knowing these principles, scientists can date ancient artifacts or understand ancient natural reactors, such as those found in Oklo, Gabon.

Uranium Isotopes

Uranium isotopes are different forms of the element uranium, defined by the number of neutrons in their nuclei. The primary isotopes of uranium found in nature are Uranium-238 and Uranium-235, with U^{238} being the most abundant. However, it's U^{235} that is crucial for nuclear reactions.

Here's what makes Uranium-235 special:

• It is a fissile material, meaning it can sustain a nuclear chain reaction.
• It is relatively rare compared to U^{238} , comprising about 0.72% of natural uranium.
• In its enriched form, U^{235} is used in nuclear reactors and weapons.

Understanding the percentage of U^{235} in natural samples helps us gauge the age of natural nuclear reactors, like the ancient ones in Oklo. At one point, natural uranium contained enough U^{235} to sustain such reactors, which modern science can calculate using known decay rates and the decay formula applied in our example.