Problem 67 - University Physics with Modern Physics

Question

Calculate the binding energy per nucleon of a) \({ }_{2}^{4} \mathrm{He}(4.002603 \mathrm{u})\). c) \({ }_{1}^{3} \mathrm{H}(3.016050 \mathrm{u})\) b) \({ }_{2}^{3} \mathrm{He}(3.016030 \mathrm{u}) .\) d) \({ }_{1}^{2} \mathrm{H}(2.014102 \mathrm{u})\).

Step-by-Step Solution

Answer

Question: Calculate the binding energy per nucleon for each of the following isotopes: \({ }_{2}^{4} \mathrm{He}\), \({ }_{1}^{3} \mathrm{H}\), \({ }_{2}^{3} \mathrm{He}\), and \({ }_{1}^{2} \mathrm{H}\). Answer: The binding energy per nucleon for each isotope is: a) For \({ }_{2}^{4} \mathrm{He}\): 7.15025 MeV/nucleon b) For \({ }_{1}^{3} \mathrm{H}\): 2.657333 MeV/nucleon c) For \({ }_{2}^{3} \mathrm{He}\): 2.007 MeV/nucleon d) For \({ }_{1}^{2} \mathrm{H}\): 0.8565 MeV/nucleon

Step 1: Calculate the mass defect

To calculate the mass defect of each isotope, you need to subtract the total mass of the individual protons and neutrons from the actual mass of the isotope. The mass of a proton is 1.007276 u and the mass of a neutron is 1.008665 u. Step 2: Calculate the binding energy

Step 2: Calculate the binding energy

To calculate the binding energy for each isotope, you will need to multiply the mass defect by the atomic mass unit conversion factor: 931.5 MeV/c\(^2\). Step 3: Calculate the binding energy per nucleon

Step 3: Calculate the binding energy per nucleon

To calculate the binding energy per nucleon, simply divide the binding energy by the total number of nucleons (A) in each isotope. a) For \({ }_{2}^{4} \mathrm{He}(4.002603 \mathrm{u})\): Mass defect = (2 x 1.007276 u + 2 x 1.008665 u) - 4.002603 u = 0.030682 u Binding energy = 0.030682 u × 931.5 MeV/c\(^2\) = 28.601 MeV Binding energy per nucleon = 28.601 MeV / 4 = 7.15025 MeV/nucleon b) For \({ }_{1}^{3} \mathrm{H}(3.016050 \mathrm{u})\): Mass defect = (1 x 1.007276 u + 2 x 1.008665 u) - 3.016050 u = 0.008556 u Binding energy = 0.008556 u × 931.5 MeV/c\(^2\) = 7.972 MeV Binding energy per nucleon = 7.972 MeV / 3 = 2.657333 MeV/nucleon c) For \({ }_{2}^{3} \mathrm{He}(3.016030 \mathrm{u})\): Mass defect = (2 x 1.007276 u + 1 x 1.008665 u) - 3.016030 u = 0.006463 u Binding energy = 0.006463 u × 931.5 MeV/c\(^2\) = 6.021 MeV Binding energy per nucleon = 6.021 MeV / 3 = 2.007 MeV/nucleon d) For \({ }_{1}^{2} \mathrm{H}(2.014102 \mathrm{u})\): Mass defect = (1 x 1.007276 u + 1 x 1.008665 u) - 2.014102 u = 0.001839 u Binding energy = 0.001839 u × 931.5 MeV/c\(^2\) = 1.713 MeV Binding energy per nucleon = 1.713 MeV / 2 = 0.8565 MeV/nucleon

Key Concepts

Mass Defect Calculation

Mass defect is a fascinating concept that arises when we delve into the atomic nucleus of an atom.
It involves determining the difference between the combined mass of the protons and neutrons that make up the nucleus of an isotope, and the actual mass of the isotope itself.
This difference might seem small, but it plays a crucial role in the stability of the nucleus. Here's a simple step-by-step guide:

• Identify the number of protons and neutrons in the isotope. You can find this information from the element's atomic number and the mass number.
• Calculate the mass of the protons by multiplying the number of protons by the mass of a single proton, which is approximately 1.007276 u.
• Similarly, calculate the mass of the neutrons using the neutron's mass, about 1.008665 u.
• Then, sum up these masses to get the "expected" mass of the nucleus.
• Finally, subtract the actual mass of the isotope from this expected mass. This gives you the mass defect.

This small "missing mass" reveals itself as energy in the form of nuclear binding energy, demonstrating that masses in atomic nuclei are intimately tied to energy.

Nuclear Binding Energy

Nuclear binding energy is the energy that holds the nucleus together, keeping it stable despite the electromagnetic repulsion forces between protons.
It can also be thought of as the energy required to break the nucleus apart into individual protons and neutrons.To calculate this energy:

• First, calculate the mass defect as discussed earlier.
• Use the conversion factor of 931.5 MeV/c² to convert the mass defect from atomic mass units (u) to energy in mega-electron-volts (MeV). The formula is:
  \[ \text{Binding Energy (MeV)} = \text{Mass Defect (u)} \times 931.5 \ \text{MeV/c}^2 \]

This energy is key to understanding why nuclei are stable.
The higher the binding energy, the more energy required to dismantle the nucleus, and thus, the more stable the nucleus is.
Nucleons within a stable nucleus are bound by the strong nuclear force, and the nuclear binding energy is a measure of this force's strength.

Atomic Mass Unit Conversion

The atomic mass unit (u or amu) is a standard unit of mass used to express atomic and molecular weights.
It is defined as one-twelfth the mass of a carbon-12 atom, approximately equal to 1.660539 x 10⁻²⁷ kilograms.
This unit provides a convenient way to discuss such tiny masses without dealing with the impracticality of using kilograms directly.Converting these atomic mass units into energy is a concept stemming from Einstein’s famous equation, \( E = mc^2 \).

• This equation tells us that mass can be converted into energy, and vice versa.
• For nuclear physics, the conversion factor between atomic mass units and energy is crucial when calculating nuclear binding energy.
• To convert amu to energy, we use the conversion factor 931.5 MeV/c², effectively turning tiny atomic-scale mass differences directly into usable energy measurements.

Understanding this conversion is vital for calculations involving the stability and potential reactions of atomic nuclei, making it an essential component of nuclear chemistry and physics.