Question

A certain shale deposit containing \(0.006 \%\) U by mass is being considered for use as a potential fuel in a breeder reactor. Assuming a density of \(2.5 \mathrm{g} / \mathrm{cm}^{3},\) how much energy could be released from \(1.00 \times 10^{3} \mathrm{cm}^{3}\) of this material? Assume a fission energy of \(3.20 \times 10^{-11} \mathrm{J}\) per fission event (that is, per U atom).

Short answer

The total energy that could be released from the sample of shale deposit is \(1.213 \times 10^{10} J\).

Step-by-step solution

Find the total mass of the sample

Initial mass of the sample can be found by multiplying the volume with the density. Given volume is \(1.00 \times 10^{3} cm^{3}\) and the density is \(2.5 g/cm^{3}\), thus, mass = volume * density = \(1.00 \times 10^{3} cm^{3} * 2.5 g/cm^{3} = 2500 g.

Calculate the mass of Uranium

Given that the shale deposit contains \(0.006\%\) Uranium by mass, the mass of Uranium is \(2500 g * 0.006\% = 0.15 g\).

Calculate the number of Uranium atoms

To find the number of Uranium atoms, we need to use Avogadro's number \(6.022 \times 10^{23} atoms/mol\). Before that, the mass of Uranium needs to be converted into moles, using the molar mass of Uranium which is \(238.03 g/mol\). Thus, number of moles = \(0.15g /238.03 g/mol= 6.3 \times 10^{-4} mol\), and then the number of Uranium atoms = Number of moles * Avogadro's number = \(6.3 \times 10^{-4} mol * 6.022 \times 10^{23} atoms/mol = 3.79 \times 10^{20} atoms\).

Calculate the total energy released

Each uranium atom can release \(3.20 \times 10^{-11} J\) per fission event. Thus, total energy = number of Uranium atoms * Energy per atom = \(3.79 \times 10^{20} atoms * 3.20 \times 10^{-11} J/atom = 1.213 \times 10^{10} J\).

Key concepts

Uranium Fission

Uranium fission is a nuclear reaction where the nucleus of a uranium atom splits into smaller parts, releasing a significant amount of energy in the process. This reaction is fundamental to nuclear power and is the basis of how we harness energy from uranium, the most commonly used fuel in nuclear reactors.

In a fission event, a neutron collides with a uranium nucleus, causing it to become unstable and split. This splitting results in the release of heat and additional neutrons, which can then go on to trigger more fission events in a chain reaction. Because each fission event releases energy, by harvesting this heat, we can generate electricity.

In the exercise concerning a shale deposit with a small percentage of uranium, it's this fission process that we look into to calculate the potential energy output. The question essentially asks us to calculate how many uranium atoms are in the sample and then estimate how much energy could be released if each one underwent fission.

Avogadro's Number

Avogadro's number, named after the scientist Amedeo Avogadro, is a fundamental constant in chemistry that represents the number of atoms or molecules in one mole of a substance. The value of Avogadro's number is approximately \(6.022 \times 10^{23}\) entities per mole.

This constant lets us bridge the gap between the atomic scale and the macroscopic scale, converting between the mass of a sample and the number of atoms or molecules it contains. It's a crucial step in our exercise, enabling us to calculate how many uranium atoms are present in a given mass of uranium.

Using Avogadro's number in conjunction with the molar mass of uranium, we can determine the number of fissionable atoms in the sample, which subsequently allows us to estimate the energy potential from a mass of uranium.

Breeder Reactor

A breeder reactor is a type of nuclear reactor specifically designed to generate more fissile material than it consumes. This is achieved by converting fertile isotopes, such as uranium-238 or thorium-232, into fissile isotopes like plutonium-239 or uranium-233 through a process known as neutron capture and nuclear transmutation.

The significance of breeder reactors lies in their ability to utilize a much larger portion of the available nuclear fuel, effectively extending the fuel supply of nuclear power generation. Additionally, they reduce the need for mining new fuel and can minimize the amount of long-lived nuclear waste.

In the context of the exercise, if the shale deposit is used as fuel in a breeder reactor, not only the initially fissile uranium-235 may be considered but also the conversion of uranium-238 into more fissile material over time, which may enhance the long-term energy yield of the deposit.

Mass-to-Energy Conversion

Mass-to-energy conversion is a principle made famous by Albert Einstein's equation, \(E = mc^2\), which shows that mass can be converted directly into energy at a rate proportional to the square of the speed of light \(c\). In nuclear fission, a tiny amount of mass from the nucleus of each atom is converted into a large amount of energy.

The conversion of mass to energy in a nuclear reactor is not immediately noticeable as mass loss because the change in mass is minuscule; however, due to the immense value of the speed of light squared, even a tiny lost mass results in a considerable amount of energy.

In our case, we use the known energy release per fission event to calculate the total energy that could be released from the uranium in the sample. Here, each uranium atom's fission converts a small amount of mass to a quantifiable amount of energy, illustrating the astonishing efficiency of mass-to-energy conversion in nuclear reactions.