Answer :
None of the provided answer choices match the calculated nuclear binding energy for Ar (39.962384 amu), which is approximately 14.61 MeV. Option D
To calculate the nuclear binding energy (NBE) for an atom, determine the mass defect, which is the difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons. The NBE can then be calculated using Einstein's mass-energy equivalence equation, E = mc².
The given atomic mass of Ar (39.962384 amu) represents the mass of the atom, which includes electrons as well. To find the nuclear binding energy, subtract the total mass of the electrons from the atomic mass.
The atomic mass of Ar is 39.962384 amu.
Let's assume it has 18 electrons (since Ar has atomic number 18).
The mass of 18 electrons can be calculated as:
(18 electrons) * (1/1836 amu per electron) ≈ 0.009831 amu.
Subtracting the mass of the electrons from the atomic mass gives us the mass of the nucleus:
39.962384 amu - 0.009831 amu ≈ 39.952553 amu.
Next, determine the number of protons and neutrons in the Ar nucleus. Ar has 18 protons (equal to the atomic number) and 22 neutrons (39.952553 amu - 18 protons).
Now, calculate the nuclear binding energy using the mass-energy equivalence equation:
E = Δm * c²,
where Δm is the mass defect (difference between the mass of the nucleus and the sum of the masses of its individual protons and neutrons) and c is the speed of light (approximately 3.00 x 10^8 m/s).
Calculating the mass defect:
Δm = (18 protons * 1.007825 amu/proton) + (22 neutrons * 1.008665 amu/neutron) - 39.952553 amu.
Δm ≈ 18.14614 amu.
Now, we can calculate the nuclear binding energy:
E = Δm * c².
E ≈ (18.14614 amu) * (1.66053906660 x 10^(-27) kg/amu) * (3.00 x 10^8 m/s)²,
Note: The conversion factor from amu to kg is 1.66053906660 x 10^(-27) kg/amu.
E ≈ 2.603 x 10^(-11) kg * (3.00 x 10^8 m/s)²,
E ≈ 2.603 x 10^(-11) kg * 9.00 x 10^16 m²/s²,
E ≈ 2.3427 x 10^6 kg * m²/s²,
E ≈ 2.3427 x 10^6 J,
Converting to MeV (1 MeV = 1.602 x 10^(-13) J):
E ≈ (2.3427 x 10^6 J) / (1.602 x 10^(-13) J/MeV),
E ≈ 14.61 x 10^6 MeV,
E ≈ 14.61 MeV.
Therefore, none of the provided answer choices match the calculated nuclear binding energy for Ar (39.962384 amu), which is approximately 14.61 MeV.
