Question

Which of the following nuclei is likely to have the largest mass defect per nucleon: (a) \({ }^{59} \mathrm{Co},(\mathbf{b}){ }^{11} \mathrm{~B},(\mathrm{c})^{118} \mathrm{Sn}\) (d) \({ }^{243} \mathrm{Cm} ?\) Explain your answer.

Short answer

The nucleus most likely to have the largest mass defect per nucleon is (a) \({ }^{59} \mathrm{Co}\). This is because it is a medium-sized nucleus, which generally has the largest binding energy per nucleon, indicating a more stable nucleus with a greater mass defect per nucleon.

Step-by-step solution

Understand Mass Defect and Binding Energy

Mass defect is the difference between the mass of a nucleus and the sum of masses of its individual nucleons (protons and neutrons). It is due to the conversion of some mass into energy (known as binding energy) when nucleons come together to form a nucleus. The binding energy is the energy required to separate the nucleus into its individual nucleons. A greater mass defect per nucleon indicates stronger binding, which results in a more stable nucleus.

Consider the Stability of each Nucleus

To compare the stability of the given nuclei and determine which one has the largest mass defect per nucleon, we can examine the binding energy per nucleon. Here is a general trend: - Lighter nuclei (such as Hydrogen and Helium) have a relatively small binding energy per nucleon. - Medium-sized nuclei (especially those with a mass number around 60) usually have the largest binding energy per nucleon. This is due to the balance between the attractive nuclear force and the repulsive electromagnetic force between protons. - Heavier nuclei (with a mass number greater than 200) have a smaller binding energy per nucleon compared to medium-sized nuclei, which makes them less stable and more likely to undergo decay processes.

Determine the Largest Mass Defect per Nucleon

Based on the general trend and given options, we can conclude the following: - \({ }^{11}\mathrm{B}\) is a lighter nucleus with a relatively lower mass defect per nucleon. - \({ }^{59}\mathrm{Co}\) is a medium-sized nucleus and likely to have the largest mass defect per nucleon. - \({ }^{118}\mathrm{Sn}\) is also a medium-sized nucleus but has a larger number of protons, so it is less likely to have a greater mass defect per nucleon than \({ }^{59}\mathrm{Co}\). - \({ }^{243}\mathrm{Cm}\) is a heavy nucleus with a smaller mass defect per nucleon compared to medium-sized nuclei. Hence, the nucleus most likely to have the largest mass defect per nucleon is (a) \({ }^{59} \mathrm{Co}\).

Key concepts

Binding Energy

Binding energy is a fundamental concept in nuclear physics, referring to the energy required to disassemble a nucleus into its individual protons and neutrons. It represents the strength of the nuclear bonds holding the nucleus together. At the atomic level, this energy arises because when protons and neutrons bind together, the total energy of the system decreases and, according to Einstein's famous equation, E=mc^2, this loss of energy implies a loss in mass, known as the mass defect.

Understanding the relation between the mass defect and binding energy is crucial for students because it explains why certain isotopes are more stable than others. A larger binding energy means a greater mass defect per nucleon, which translates to more stability within the nucleus. In the context of the exercise, learning to associate a large mass defect per nucleon with high binding energy allows students to predict nuclear stability.

Nuclear Stability

Nuclear stability is the likelihood that a nucleus will remain unchanged over time. It is determined by several factors, including the binding energy per nucleon as previously discussed. In general, a nucleus with high binding energy per nucleon is more stable because the energy required to break it apart is greater.

Understanding nuclear stability helps in explaining natural phenomena such as radioactivity, where unstable nuclei spontaneously break apart. For the purposes of the exercise, recognizing the pattern that medium-sized nuclei (with mass number around 60) are often more stable than very light or heavy nuclei allows students to assess stability and predict which isotopes are more likely to exist in nature. This understanding can be particularly useful when examining the periodic table and nuances within isotopes of chemical elements.

Nucleus Mass Number

The mass number of a nucleus is the combined total number of protons and neutrons it contains, often represented as A in the nuclear symbol \( ^A_ZX \), where X is the chemical element, A is the mass number, and Z is the atomic number (number of protons). Understanding the mass number is vital because it allows us to determine the isotope of an element and gain insights into its nuclear properties.

For the exercise solution, knowledge of the mass number is directly linked to the anticipated mass defect per nucleon. Generally, students must be aware that nuclei with a mass number around 60 have the highest binding energy per nucleon, thus having a larger mass defect per nucleon. This piece of information was key to determining that \( ^{59}Co \) will likely have the largest mass defect per nucleon out of the given options.