Problem 36 - Chemistry and Chemical Reactivity

Question

Calculate the binding energy per mole of nucleons for iron-56. Masses needed for this calculation (in g/mol) are \(_{1}^{1} \mathrm{H}=1.00783,_{0}^{1} \mathrm{n}=1.00867,\) and \(\frac{56}{26} \mathrm{Fe}=55.9349 .\) Compare the result of your calculation to the value for iron-56 in the graph in Figure 25.4

Step-by-Step Solution

Answer

The binding energy per nucleon for iron-56 is approximately \(8.48 \times 10^{11}\) J/nucleon.

Step 1: Define the Objective

The goal is to calculate the binding energy per mole of nucleons for iron-56 \((^{56}_{26} \text{Fe})\). Then, compare this calculated value to the graph in Figure 25.4 (not provided here).

Step 2: Calculate Total Mass of Individual Components

Calculate the total mass of individual nucleons if they were separate, using proton and neutron masses in grams per mole. Iron-56 has 26 protons and 30 neutrons.Proton total mass = \(26 \times 1.00783 \) g/mol = 26.20358 g/mol.Neutron total mass = \(30 \times 1.00867 \) g/mol = 30.26010 g/mol.Total mass of separate nucleons = 26.20358 g/mol + 30.26010 g/mol = 56.46368 g/mol.

Step 3: Calculate Mass Defect

The mass defect is the difference between the mass of separate nucleons and the actual mass of the nucleus of iron-56.\[ \Delta m = 56.46368 \, \text{g/mol} - 55.9349 \, \text{g/mol} = 0.52878 \, \text{g/mol} \]

Step 4: Convert Mass Defect to Energy

Use Einstein's equation \(E=mc^2\), where \(c\) is the speed of light \(3 \times 10^8\) m/s. Convert grams to kilograms (1 g = 0.001 kg).\[ E = 0.52878 \, \text{g/mol} \times 0.001 \, \text{kg/g} \times (3 \times 10^8 \, \text{m/s})^2 = 4.75302 \times 10^{13} \, \text{J/mol} \]

Step 5: Calculate Binding Energy per Nucleon

Iron-56 consists of 56 nucleons. Divide the total binding energy by the number of nucleons to find the energy per nucleon.\[ \text{Binding Energy per Nucleon} = \frac{4.75302 \times 10^{13} \, \text{J/mol}}{56} \approx 8.4839 \times 10^{11} \, \text{J/nucleon} \]

Step 6: Compare with Graph

Assuming reference to Figure 25.4, compare this value with the given data point for iron-56. This value, 8.4839 \(\times 10^{11}\) J/nucleon, is historically close to typical values found in nuclear binding energy charts.

Key Concepts

Iron-56

Iron-56 is a stable isotope of iron with one of the highest nuclear binding energies. This means it's very stable and less likely to undergo radioactive decay.
It's composed of 26 protons and 30 neutrons, giving it an atomic mass of 56.
Iron-56 is significant because elements near it have the highest binding energies and are the most stable.

• Understanding iron-56 helps explain stability in other elements.
• It's a benchmark for measuring binding energies in nuclear physics.

Iron-56's stability explains its abundance in the universe, particularly in the cores of massive stars.

Mass Defect

The mass defect is a critical concept in understanding nuclear energy. It refers to the difference between the total mass of individual protons and neutrons and the actual mass of a nucleus.
When nucleons bind together, energy is released, causing a decrease in mass according to Einstein's famous equation.
In iron-56, for example, the mass defect can be calculated by subtracting the actual mass of the nucleus from the sum of the separated nucleons' masses:\[ \Delta m = \text{Mass of separate nucleons} - \text{Mass of the nucleus}\]

• This mass defect is what contributes to the nucleus's binding energy.
• Without the mass defect, there would be no nuclear power or nuclear reactions.

Understanding the mass defect is essential for calculating nuclear energy as we did for iron-56.

Nucleons

Nucleons are the particles that make up an atomic nucleus, specifically protons and neutrons. Each nucleon has specific characteristics:

• Protons: Positively charged and define the atomic number.
• Neutrons: Neutral particles that add to atomic mass and contribute to nucleus stability.

The number of protons and neutrons (nucleons) in iron-56 is crucial to its calculation for binding energy per nucleon. Iron-56 has 26 protons and 30 neutrons, summing up to 56 nucleons. These nucleons work together to create the strong force that holds the nucleus together.
The total binding energy relates directly to the number of nucleons because each nucleon binds with others, releasing energy and thus creating a stable nucleus.

Einstein's equation

Albert Einstein's equation, \(E=mc^2\), is a fundamental principle in physics that connects energy (\(E\)) and mass (\(m\)). The equation highlights how mass can be converted into energy and vice versa.
In nuclear physics, this equation is essential for calculating the energy released during nuclear reactions. The speed of light squared (\(c^2\)) is a large number, which shows the massive energy potential stored in matter.

• The equation explains the energy release from the mass defect in nuclear reactions.
• It shows why even a small mass loss in nuclear reactions can release vast amounts of energy.

In the case of iron-56, Einstein's equation allows for the conversion of the mass defect into binding energy, showing why nuclear forces are so powerful.

Nuclear Binding Energy

Nuclear binding energy is the energy required to disassemble a nucleus into its component protons and neutrons. It signifies the stability of the nucleus. The higher the binding energy, the more stable the nucleus.
Iron-56 has one of the highest nuclear binding energies per nucleon, making it very stable and common in nature.
Calculating the nuclear binding energy involves determining the mass defect and then using Einstein's formula \(E=mc^2\) to find the energy equivalent.

• It represents why nuclei remain intact rather than disintegrating.
• A vital measure for understanding nuclear reactions and processes like fission and fusion.

For iron-56, the binding energy per nucleon helps compare its stability to other isotopes. This knowledge is crucial for applications in nuclear energy and astrophysics.