The binding energy of an atom represents the energy required to completely separate its nucleus into its individual protons and neutrons. It is typically expressed in units of electron volts (eV) per atom or nucleon.
The binding energy of sodium, specifically referring to the most common isotope, sodium-23 (Na-23), can be calculated by subtracting the mass of the individual protons and neutrons from the mass of the sodium-23 nucleus and converting the mass difference into energy using Einstein's mass-energy equivalence equation (E=mc²).
The mass of a sodium-23 nucleus is approximately 22.98976928 atomic mass units (AMU), which is equivalent to 3.82323464 x 10^(-26) kilograms. Each atomic mass unit is approximately 931.5 MeV/c².
The mass of 11 protons (since sodium has an atomic number of 11) is approximately 1.00727647 AMU, and the mass of 12 neutrons is approximately 1.00866492 AMU. Thus, the combined mass of the protons and neutrons is approximately 23.08704199 AMU or 3.83080515 x 10^(-26) kilograms.
Subtracting this combined mass from the mass of the sodium-23 nucleus gives a mass difference of approximately 0.09780735 AMU or 1.60705653 x 10^(-28) kilograms.
Using the mass-energy equivalence equation, E=mc², and converting kilograms to electron volts (1 eV = 1.60218 x 10^(-19) joules), the binding energy can be calculated.
E = (1.60705653 x 10^(-28) kg) x (299792458 m/s)^2 = 1.43786849 x 10^(-11) joules = 8.96777 MeV
Therefore, the binding energy of sodium-23 is approximately 8.96777 MeV per atom or nucleon.