Problem 6 - College Physics

Question

(a) Calculate the total binding energy (in MeV) of the nuclei of \(^{56} \mathrm{Fe}\) (of atomic mass 55.934937 \(\mathrm{u} )\) and of \(^{207} \mathrm{Pb}\) (of atomic mass 206.975897 u). (b) Calculate the binding energy per nucleon for each of these atoms.(c) How much energy would be needed to totally take apart each of these nuclei? (d) For which of these atoms are the individual nucleons more tightly bound? Explain your reasoning.

Step-by-Step Solution

Answer

Iron-56 nucleons are more tightly bound, needing 492.3 MeV to disassemble vs. Lead-207's 1620 MeV with lower binding per nucleon rate.

Step 1: Understand the Problem

We need to find the total binding energy and binding energy per nucleon of two nuclei, which are Iron-56 and Lead-207. Then, we'll compare their binding energies.

Step 2: Gather Needed Formulas and Constants

The binding energy (BE) can be calculated using the formula: \( BE = \left( Zm_p + Nm_n - M(a) \right) c^2\), where \( m_p, m_n \) are the masses of proton and neutron, \( Z \) and \( N \) the number of protons and neutrons, \( M(a) \) the atomic mass, and \( c^2 \approx 931.5 \text{ MeV/u} \).

Step 3: Calculate Total Binding Energy for Iron-56

For \( ^{56}Fe \), there are 26 protons and 30 neutrons. Using the formula from Step 2 and substituting the mass of protons and neutrons \( m_p \approx 1.007825 \text{ u} \) and \( m_n \approx 1.008665 \text{ u} \), we calculate:\[ BE_{Fe} = \left( 26 \times 1.007825 + 30 \times 1.008665 - 55.934937 \right) \times 931.5 \approx 492.3 \text{ MeV} \]

Step 4: Calculate Total Binding Energy for Lead-207

For \( ^{207}Pb \), we have 82 protons and 125 neutrons. Using similar calculations:\[ BE_{Pb} = \left( 82 \times 1.007825 + 125 \times 1.008665 - 206.975897 \right) \times 931.5 \approx 1620 \text{ MeV} \]

Step 5: Calculate Binding Energy per Nucleon for Each Atom

For Iron-56:\[ \text{Binding Energy per Nucleon for Iron} = \frac{492.3}{56} \approx 8.79 \text{ MeV/nucleon} \]For Lead-207:\[ \text{Binding Energy per Nucleon for Lead} = \frac{1620}{207} \approx 7.83 \text{ MeV/nucleon} \]

Step 6: Determine Energy Needed to Disassemble Nuclei

The energy needed to totally disassemble each nucleus is equal to its total binding energy. For Iron-56, it is 492.3 MeV; for Lead-207, it is 1620 MeV.

Step 7: Analyze and Compare Binding Energies

To determine which nucleons are more tightly bound, we compare the binding energy per nucleon. Iron-56, with 8.79 MeV/nucleon, has more tightly bound nucleons than Lead-207, which is 7.83 MeV/nucleon.

Key Concepts

Nuclear Physics

Nuclear physics is the field of physics that explores the components and behavior of atomic nuclei. The nucleus is the dense region at the center of an atom, composed of protons, which are positively charged particles, and neutrons, which have no charge. Together, protons and neutrons are known as nucleons.
Nuclear physics studies several phenomena related to the nucleus such as nuclear reactions, radioactive decay, and nuclear binding. Understanding these requires knowledge of how nucleons interact within the nucleus. There are strong nuclear forces at play, which are significantly more potent than the electromagnetic forces that typically govern the behavior of charged particles. This force binds protons and neutrons together, and it is at the heart of studying nuclear stability and disintegration.

• Nuclear reactions include processes like fission, where a nucleus splits into smaller parts, and fusion, where nucleons combine to form a larger nucleus.
• Radioactive decay refers to the spontaneous emission of particles from an unstable nucleus.
• Binding energy involves understanding the energy required to disassemble the nucleus into individual protons and neutrons.

Atomic Mass

Atomic mass is a crucial concept in understanding the properties of an element's nuclei. It is defined as the mass of an individual atom, expressed in atomic mass units (u), where one atomic mass unit is \(1/12\) the mass of a carbon-12 atom. Atomic mass reflects the total number of protons and neutrons in the nucleus.
Every element on the Periodic Table has a unique atomic mass, which is not always a whole number. This is because it represents an average mass of all isotopes of an element, factoring in their relative abundance. Isotopes are variants of elements with the same number of protons but different numbers of neutrons.

• For instance, take Iron-56, which has an atomic mass of 55.934937 u, and Lead-207, which weighs 206.975897 u.
• The tiny difference in atomic mass can have significant effects when calculating nuclear properties like binding energy.

Atomic mass is pivotal in nuclear physics calculations, often leading to insights about the stability and nuclear reactions of the elements.

Protons and Neutrons

Protons and neutrons are subatomic particles found within the nucleus of an atom. They are collectively known as nucleons, owing to their role as the core constituents of the atomic nucleus.
Protons have a positive charge, while neutrons are neutral, meaning they do not have an electrical charge. The number of protons in an atom determines the element, a fundamental property called the atomic number (Z). Neutrons, on the other hand, contribute to the mass of an atom and help to stabilize the nucleus.

• A balanced number of protons to neutrons leads to a stable nucleus, while an imbalance can lead to instability, causing the atom to become radioactive.
• The interplay of protons and neutrons influences the overall binding energy of a nucleus, with their masses being key components in calculating atomic mass.
• For example, Iron-56 has 26 protons and 30 neutrons, while Lead-207 has 82 protons and 125 neutrons.

Understanding protons and neutrons is crucial for insights into both chemical behavior and nuclear stability.

Binding Energy per Nucleon

Binding energy per nucleon is a measure of the stability of a nucleus. It is the energy required to remove a single nucleon from a nucleus, averaged across all nucleons in that nucleus. This measure helps in comparing the stability of different nuclei.
To calculate the binding energy per nucleon, one divides the total binding energy by the number of nucleons within the nucleus. This concept is invaluable in nuclear physics because:

• It provides insight into how tightly nucleons are held together by the nuclear force.
• Nuclei with higher binding energy per nucleon are generally more stable.

For example, Iron-56 has a binding energy per nucleon of approximately 8.79 MeV, while Lead-207 has about 7.83 MeV per nucleon. Hence, nucleons in Iron-56 are more tightly bound compared to those in Lead-207. This information is not only essential for understanding nuclear stability but also for applications in nuclear technology and energy production.