Finding the mass defect and binding energy involves calculating the difference between the mass of a nucleus's individual components (protons and neutrons) and the actual mass of the nucleus itself, then converting that mass difference into energy. Here's a step-by-step guide:
1. Determine the Number of Protons and Neutrons:
- Identify the element and its atomic number (Z), which equals the number of protons.
- Determine the mass number (A) of the specific isotope.
- Calculate the number of neutrons (N) using the formula: N = A - Z.
2. Calculate the Total Mass of Individual Nucleons:
- Find the mass of a single proton (approximately 1.00728 amu) and a single neutron (approximately 1.00866 amu). These values can also be expressed in kg (1 amu = 1.66054 x 10^-27 kg).
- Multiply the number of protons (Z) by the mass of a proton.
- Multiply the number of neutrons (N) by the mass of a neutron.
- Add these two values together to get the total mass of the individual nucleons.
3. Determine the Actual Mass of the Nucleus:
- Find the experimentally determined mass of the nucleus. This value is usually given in atomic mass units (amu) or kg. Note: This is the mass of the nucleus only, not the mass of the entire atom. If given the mass of the atom, you'll need to subtract the mass of the electrons (number of electrons = number of protons) from the atomic mass to obtain the nuclear mass.
4. Calculate the Mass Defect (Δm):
- Subtract the actual mass of the nucleus (step 3) from the total mass of the individual nucleons (step 2).
- Mass Defect (Δm) = (Total mass of individual nucleons) - (Actual mass of the nucleus)
- The mass defect will be a positive value and represents the mass that was converted into binding energy.
5. Calculate the Binding Energy (E):
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Use Einstein's famous equation, E = mc², where:
- E is the binding energy (in Joules if 'm' is in kg, or in MeV if 'm' is in amu).
- m is the mass defect (Δm) in kg or amu. If the mass defect is in amu, you'll need to multiply it by a conversion factor to get MeV directly. 1 amu = 931.5 MeV
- c is the speed of light (approximately 3.0 x 10⁸ m/s).
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If mass defect (m) is in kg: E = (Mass defect in kg) * (3.0 x 10⁸ m/s)² (Result will be in Joules)
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If mass defect (m) is in amu: E = (Mass defect in amu) * (931.5 MeV/amu) (Result will be in MeV)
Example:
Let's consider Helium-4 (⁴₂He):
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Number of Protons and Neutrons: Z = 2, A = 4, N = 4 - 2 = 2.
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Total Mass of Individual Nucleons:
- 2 protons * 1.00728 amu/proton = 2.01456 amu
- 2 neutrons * 1.00866 amu/neutron = 2.01732 amu
- Total = 2.01456 + 2.01732 = 4.03188 amu
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Actual Mass of Helium-4 Nucleus: 4.00150 amu (This is an experimental value).
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Mass Defect: Δm = 4.03188 amu - 4.00150 amu = 0.03038 amu
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Binding Energy: E = 0.03038 amu * 931.5 MeV/amu ≈ 28.3 MeV
Therefore, the mass defect for Helium-4 is 0.03038 amu, and the binding energy is approximately 28.3 MeV.
