Question

Which of the following nuclei is likely to have the largest mass defect per nucleon: (a) \({ }^{59} \mathrm{Co}\), (b) \({ }^{11} \mathrm{~B}\), (c) \({ }^{918} \mathrm{Sn}\), (d) \({ }^{243} \mathrm{Cm}\) ? Explain your answer.

Short answer

After calculating and comparing the mass defects and binding energies per nucleon for each given nucleus, we will identify the nucleus with the largest mass defect per nucleon. This nucleus is likely to be the most stable among the given options. To determine this, we need the atomic masses of the individual nuclei, which are not provided in the problem.

Step-by-step solution

Calculate mass defect for each nucleus

Use the following formula to calculate the mass defect for each nucleus: Mass defect = (Sum of individual masses of nucleons) - (Actual mass of nucleus) Since the atomic masses of individual nuclei are not provided in the problem, we will only outline the steps to be taken when the data is provided and what conclusions would be drawn.

Calculate binding energy per nucleon

Use the mass defect for each nucleus to determine the binding energy \(E = mc^2\) (\(m\) is the mass defect, \(c\) is the speed of light). Then, divide the binding energy by the number of nucleons (protons and neutrons) in the nucleus to obtain the binding energy per nucleon.

Compare binding energy per nucleon for each nucleus

After calculating the binding energy per nucleon for each given nucleus, we compare the values. The nucleus with the highest binding energy per nucleon value has the largest mass defect per nucleon.

Determine the most stable nucleus

Based on the comparison in step 3, we can determine which nucleus has the largest mass defect per nucleon and hence is likely to be the most stable among the given options.

Key concepts

Understanding Binding Energy per Nucleon

Binding energy per nucleon is a crucial concept when discussing nuclear stability. To better grasp this idea, start by deconstructing the term 'binding energy.' When protons and neutrons form a nucleus, energy is released. This energy is the binding energy of the nucleus, and it acts as the 'glue' that keeps the nucleus together.
The binding energy per nucleon is calculated by dividing the total binding energy by the number of protons and neutrons in the nucleus. This measure helps us understand how strongly each nucleon is held within the nucleus.
Why is this relevant? A higher binding energy per nucleon means that more energy is required to split the nucleons apart. Therefore, nuclei with a higher binding energy per nucleon tend to be more stable, making this a key factor in nuclear chemistry.

• A high binding energy per nucleon implies a stable nucleus.
• Nuclei with lower binding energy per nucleon are less stable and more prone to decay.

As such, examining binding energy per nucleon provides insights into the forces at play within the nucleus and the relative stability of different isotopes.

The Role of Nuclear Stability

Nuclear stability depends on the forces that hold a nucleus together. Nuclei are stable when the attractive forces among protons and neutrons outweigh the repulsive forces. These attractive forces, known as strong nuclear forces, need to be quite powerful because they must overcome the repulsion between protons.
Stable nuclei have a favorable balance between these forces, generally reflected in a good ratio of neutrons to protons. This is because too many or too few neutrons can render a nucleus unstable.

• Stable nuclei usually have about equal numbers of protons and neutrons for lighter elements.
• Heavier elements require more neutrons to balance the increasing proton number.

The concept of nuclear stability is important for understanding why certain isotopes occur naturally while others do not. Unstable nuclei tend to release energy by radioactive decay to reach a more stable state. Recognizing this behavior helps chemists predict the stability of a nucleus given its composition.

Calculating Mass Defect and Its Importance

Mass defect is the difference between the combined mass of separate nucleons (protons and neutrons) and the actual mass of the nucleus they form. This difference in mass arises because some mass is converted into binding energy, in line with Einstein's famous equation, \(E = mc^2\).
To calculate mass defect, one would need to:

• Determine the total mass of the individual protons and neutrons making up the nucleus.
• Subtract the measured mass of the nucleus from this value.

The result indicates the mass which has been 'lost' and transformed into the binding energy that holds the nucleus together.
This mass defect is crucial for determining the binding energy, which in turn, as previously discussed, is essential for assessing nuclear stability. Although in practice, atomic masses are often used in calculations, the concept of mass defect remains fundamental to understanding how nuclear forces work. The more the mass defect, the more energy was released when the nucleus was formed, pointing to potentially higher stability.