Question

The most stable nucleus in terms of binding energy per nucleon is \({ }^{56} \mathrm{Fe}\). If the atomic mass of \({ }^{56} \mathrm{Fe}\) is \(55.9349 \mathrm{amu}\), calculate the binding energy per nucleon for \({ }^{56} \mathrm{Fe}\).

Short answer

The binding energy per nucleon for \({ }^{56}\mathrm{Fe}\) is approximately \(1.36948\times10^{-12}\,\mathrm{J}\).

Step-by-step solution

Calculate the mass defect

To find the mass defect of \({}^{56}\mathrm{Fe}\), we first need to find out the total mass of the individual nucleons. In this case, we have 26 protons and 30 neutrons. For protons, we'll use the mass of \({}^1\mathrm{H}\) because they are essentially the same as the hydrogen atom. The mass of a proton is approximately \(1.007276\,\mathrm{amu}\). For neutrons, we'll approximate their mass to be \(1.008665\,\mathrm{amu}\). Total mass of protons = number of protons × mass of a proton Total mass of protons = 26 × \(1.007276\,\mathrm{amu}\) = \(26.189176\,\mathrm{amu}\) Total mass of neutrons = number of neutrons × mass of a neutron Total mass of neutrons = 30 × \(1.008665\,\mathrm{amu}\) = \(30.25995\,\mathrm{amu}\) Mass defect = (Total mass of protons + Total mass of neutrons) - Atomic mass of \({}^{56}\mathrm{Fe}\) Mass defect = (\(26.189176\,\mathrm{amu}\) + \(30.25995\,\mathrm{amu}\)) - \(55.9349\,\mathrm{amu}\) = \(0.514226\,\mathrm{amu}\)

Calculate the total binding energy

We'll use Einstein's famous mass-energy equivalence formula, \(E=mc^2\), to find the total binding energy. To do this, we need to convert the mass defect (\(0.514226\,\mathrm{amu}\)) into kilograms, and the speed of light, \(c\), is \(2.998\times10^8\,\mathrm{m/s}\). Mass defect in kg = \(0.514226\,\mathrm{amu}\) × \(1.66054\times10^{-27}\,\mathrm{kg/amu}\) = \(8.54254\times10^{-28}\,\mathrm{kg}\) Total binding energy = mass defect × \(c^2\) Total binding energy = \(8.54254\times10^{-28}\,\mathrm{kg}\) × \((2.998\times10^8\,\mathrm{m/s})^2\) = \(7.66513\times10^{-11}\,\mathrm{J}\)

Calculate the binding energy per nucleon

Finally, we'll divide the total binding energy by the number of nucleons (\({ }^{56}\mathrm{Fe}\) has 56 nucleons). Binding energy per nucleon = Total binding energy / Number of nucleons Binding energy per nucleon = \(7.66513\times10^{-11}\,\mathrm{J}\) / 56 = \(1.36948\times10^{-12}\,\mathrm{J}\) Thus, the binding energy per nucleon for \({ }^{56}\mathrm{Fe}\) is approximately \(1.36948\times10^{-12}\,\mathrm{J}\).

Key concepts

Nuclear Stability

The concept of nuclear stability is fundamental to understanding why certain nuclei are more tightly bound than others. A stable nucleus implies that it has a high binding energy per nucleon; this means it would require a significant amount of energy to break the nucleus apart into individual protons and neutrons.

For instance, ^{56}Fe, which is the most stable nucleus in terms of binding energy per nucleon, can be taken as a benchmark for stability. Nuclear stability is governed by a delicate balance between the strong nuclear force, which works to hold the nucleons together, and the electrostatic repulsion between protons. Nuclei are most stable when they have an optimal ratio of neutrons to protons, which for light elements is approximately 1:1, but shifts towards a higher proportion of neutrons in heavier elements. ^{56}Fe's stability comes from having an almost ideal number of neutrons and protons, which minimizes the repulsive forces while maximizing the strong nuclear force.

Mass Defect Calculation

Mass defect refers to the difference in mass between the sum of the separate nucleons (protons and neutrons) and the actual mass of the nucleus. It's a manifestation of the energy released when a nucleus is formed from its constituent particles.

To calculate this, you must sum the masses of the protons and neutrons when outside the nucleus and then subtract the measured atomic mass of the nucleus, just as seen in the solution for ^{56}Fe. Here's the formula simplified:

Mass Defect = (Total mass of protons + Total mass of neutrons) - Atomic mass of the nuclide

This calculation is important because the mass defect directly relates to the binding energy—the energy required to split the nucleus into its individual nucleons. In teaching the notion of mass defect, highlighting the accuracy of measured atomic masses is critical, as they can affect the resulting calculation of nuclear stability.

Einstein's Mass-Energy Equivalence

Einstein's famous equation,

E=mc^2

, known as the mass-energy equivalence, is pivotal in nuclear physics. This equation tells us that mass can be converted into energy and vice versa, implying that the energy which binds the nucleus together is actually derived from the mass defect.

When calculating binding energy, like in the provided solution for ^{56}Fe, this equation is used to convert the mass defect from atomic mass units into energy (joules). A high binding energy signifies a strong binding force between nucleons, thereby contributing to a more stable nucleus. It's these interactions at the subatomic level that illustrate the profound link between mass and energy, a cornerstone of modern physics that revolutionized scientific understanding.