Question

Calculate the binding energy for the following two uranium isotopes: a) \({ }_{92}^{238} \mathrm{U},\) which consists of 92 protons, 92 electrons, and 146 neutrons, with a total mass of \(238.0507826 \mathrm{u}\). b) \({ }^{235} \mathrm{U},\) which consists of 92 protons, 92 electrons, and 143 neutrons, with a total mass of \(235.0439299 \mathrm{u} .\) The atomic mass unit \(\mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg} .\) Which isotope is more stable (or less unstable)?

Short answer

Answer: \({}^{235}\mathrm{U}\) is more stable (less unstable) than \({}_{92}^{238}\mathrm{U}\) based on their binding energies.

Step-by-step solution

Mass of Protons and Neutrons

Since 1u is approximately equal to the mass of a proton or neutron, we will consider the mass of both protons and neutrons as 1u. Step 2: Calculate mass defect for \({}_{92}^{238}\mathrm{U}\)

Mass Defect for \({}_{92}^{238}\mathrm{U}\)

First, find the sum of individual masses of 92 protons and 146 neutrons: \(92 \times 1 + 146 \times 1 = 238\,u\). Then, subtract the total mass of \({}_{92}^{238}\mathrm{U}\) from this sum to get the mass defect: \(238 - 238.0507826 = -0.0507826\,u\). Step 3: Calculate binding energy for \({}_{92}^{238}\mathrm{U}\)

Binding Energy for \({}_{92}^{238}\mathrm{U}\)

Convert the mass defect to kilograms by multiplying it with the value of the atomic mass unit, and then use the mass-energy equation to find the binding energy: E = mc^2 E = \((-0.0507826 \times 1.66 \times 10^{-27}) \times (3 \times 10^8)^2\) E = \(-7.846 \times 10^{-11}\,\mathrm{J}\) Step 4: Calculate mass defect for \({}^{235}\mathrm{U}\)

Mass Defect for \({}^{235}\mathrm{U}\)

First, find the sum of individual masses of 92 protons and 143 neutrons: \(92 \times 1 + 143 \times 1 = 235\,u\). Then, subtract the total mass of \({}^{235}\mathrm{U}\) from this sum to get the mass defect: \(235 - 235.0439299 = -0.0439299\,u\). Step 5: Calculate binding energy for \({}^{235}\mathrm{U}\)

Binding Energy for \({}^{235}\mathrm{U}\)

Convert the mass defect to kilograms by multiplying it with the value of the atomic mass unit, and then use the mass-energy equation to find the binding energy: E = mc^2 E = \((-0.0439299 \times 1.66 \times 10^{-27}) \times (3 \times 10^8)^2\) E = \(-6.746 \times 10^{-11}\,\mathrm{J}\) Step 6: Compare binding energies to determine stability

Stability Comparison

Both binding energies are negative, which means the isotopes are unstable. However, the isotope with a higher (less negative) binding energy is more stable, or less unstable. In this case, \({}^{235}\mathrm{U}\) has a binding energy of \(-6.746 \times 10^{-11}\,\mathrm{J}\) while \({}_{92}^{238}\mathrm{U}\) has a binding energy of \(-7.846 \times 10^{-11}\,\mathrm{J}\). Therefore, \({}^{235}\mathrm{U}\) is more stable (less unstable) than \({}_{92}^{238}\mathrm{U}\).

Key concepts

Nuclear Physics

Nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. At the core of this study is the understanding of nuclear stability and the forces that bind particles within the atomic nucleus—a region made of protons and neutrons, collectively known as nucleons.

The stability of a nucleus depends on a delicate balance between the strong nuclear force, which is an attractive force that acts between all nucleons, and the electrostatic repulsion between protons. More stable nuclei have a higher binding energy, meaning a greater amount of energy is required to separate the nucleons. This concept is crucial when assessing isotopes like uranium, an element that can exhibit differing nuclear stability across its isotopes.

Mass-Energy Equivalence

The mass-energy equivalence principle, immortalized by Einstein's equation, E=mc^2, reveals that mass can be converted into energy and vice versa. The 'c' in the equation stands for the speed of light in a vacuum, which is approximately 3 x 10^8 meters per second—a constant that signifies the incredible amount of energy equivalent to a small amount of mass.

In nuclear physics, this principle helps us understand how the mass defect of a nucleus (the difference between the predicted mass based on constituent nucleons and the actual mass of the nucleus) translates to the binding energy. This binding energy is the energy required to break the nucleus apart into its individual protons and neutrons.

Uranium Isotopes Stability

Uranium isotopes, such as U-238 and U-235, differ in the number of neutrons they contain. The stability of a uranium isotope is dependent on its neutron-to-proton ratio, with certain ratios leading to greater nuclear stability. Uranium-238, having three more neutrons than Uranium-235, will exhibit a different level of stability.

In terms of binding energy per nucleon, a good indicator of nuclear stability, U-235 has a slightly higher value, making it more stable relative to U-238. This property is also why U-235 is the preferred isotope for nuclear reactors and weapons—it is more likely to undergo fission when bombarded with neutrons.

Mass Defect

The mass defect is the difference between the sum of the masses of an atom's separate nucleons (protons and neutrons) and the actual mass of the atom's nucleus. It is a manifestation of the mass-energy equivalence principle, as the 'missing' mass has been converted into binding energy to hold the nucleus together.

To calculate the mass defect, you sum the masses of all the individual protons and neutrons that would make up the nucleus if they were separate and then subtract the mass of the nucleus. This mass defect, when converted to energy, reveals how much energy is holding the nucleus together. A larger mass defect signifies a nucleus with higher binding energy, reflecting a more stable atomic structure.