Question

The binding energy of an element is \(64 \mathrm{meV}\). If \(\mathrm{BE} /\) Nucleon is \(6.4\), the number of nucleons are (a) 10 (b) 64 (c) 16 (d) \(6.4\)

Short answer

The number of nucleons is 10.

Step-by-step solution

Understanding Binding Energy per Nucleon

The "Binding Energy per Nucleon" is given as 6.4. This value represents the binding energy when it is equally distributed among all the nucleons of the element in question. It is calculated by dividing the total binding energy by the number of nucleons.

Identify Provided Values

We are told that the total binding energy of the element is 64 meV and the binding energy per nucleon is given as 6.4. We need to use these values to find the number of nucleons.

Set Up the Equation

Given the relationship between total binding energy and binding energy per nucleon, the equation is:\[\text{BE per Nucleon} = \frac{\text{Total Binding Energy}}{\text{Number of Nucleons}}\]Substituting the given values in:\[6.4 = \frac{64}{\text{Number of Nucleons}}\]

Solve for Number of Nucleons

Rearrange the equation to solve for the number of nucleons:\[\text{Number of Nucleons} = \frac{64}{6.4}\]Calculating this gives:\[\text{Number of Nucleons} = 10\]

Verification of Answer

Verify that this solution makes sense: If you multiply 10 nucleons by the binding energy per nucleon (6.4), you should get the total binding energy (64 meV). Indeed, \(10 \times 6.4 = 64\). Therefore, the number of nucleons is verified as correct.

Key concepts

Nucleon

Nucleons are the constituents of the atomic nucleus. They include both protons and neutrons. In nuclear physics, understanding nucleons is fundamental as they determine the characteristics and stability of an atom. The number of nucleons is synonymous with the atomic mass number of an element. Both protons and neutrons contribute almost equally to the mass but not to the charge, since neutrons are electrically neutral.

• Protons are positively charged particles.
• Neutrons carry no charge at all.

The behavior and interaction of nucleons are key to the physical structure of the nucleus. It's important to know that changes in the number of nucleons can significantly alter the element or isotope, as well as its binding energy characteristics.

Nuclear Physics

Nuclear physics is the scientific study of the forces and components inside atomic nuclei. It covers phenomena such as radioactivity, nuclear fission, and fusion. Understanding these phenomena helps explain the distribution of energy within a nucleus, which is pivotal to topics like binding energy.
Nuclear reactions and interactions are governed by complex forces. The nuclear force, also known as the strong force, keeps nucleons bound tightly in the nucleus. Despite the repulsive electric force between protons, the strong force ensures the stability of the nucleus. This force is a fundamental point in nuclear physics.
Further exploration of nuclear physics helps clarify why energy distribution per nucleon is such a crucial measure in understanding nuclear stability and reactions.

Mathematical Calculation

Calculations in nuclear physics often involve understanding ratios and distributions of energy among nucleons. Mathematical equations unveil how binding energy per nucleon relates to the total binding energy and the number of nucleons.
For example, to find the number of nucleons given the total binding energy and the binding energy per nucleon, you use the equation:
\[\text{BE per Nucleon} = \frac{\text{Total Binding Energy}}{\text{Number of Nucleons}}\]This equation was used in the original problem to determine that there are 10 nucleons. Rearranging this formula gives:
\[\text{Number of Nucleons} = \frac{64}{6.4} = 10\]Such calculations are not only useful to solve problems but also to understand how energy is partitioned within atomic structures.

Energy Distribution in Nucleus

Understanding how energy is distributed within a nucleus is central to nuclear physics. Energy distribution is often expressed in terms of binding energy per nucleon. This value is an average measure of the stability of a nucleus, indicating how much energy would be required to remove a nucleon.
A high binding energy per nucleon suggests a stable nucleus as more energy is needed to disassemble it. Conversely, a lower binding energy per nucleon implies a less stable nucleus.

• Binding energy per nucleon = Total Binding Energy / Number of Nucleons.
• This concept helps predict nuclear reactions and their energy outcomes.

The distribution of energy within the nucleus is key to predicting nuclear behaviors. For example, the original problem showed a binding energy of 6.4 meV per nucleon, giving insight into the nucleus's stability.