You can find the atomic mass of a nuclide if you know its binding energy by using the following formula, which relates binding energy to the difference between the mass of the individual nucleons (protons and neutrons) and the mass of the nucleus:
BE = [Zm(¹H) + Nmn − m(AX)]c²
Where:
- BE is the binding energy of the nucleus.
- Z is the number of protons (atomic number).
- m(¹H) is the mass of a hydrogen atom (approximately the mass of a proton since it contains one proton and one electron; the electron's mass is relatively negligible).
- N is the number of neutrons.
- mn is the mass of a neutron.
- m(AX) is the atomic mass of the nuclide (what we want to find), where A is the mass number (A = Z + N) and X is the element symbol.
- c is the speed of light (approximately 2.998 x 10⁸ m/s).
To find the atomic mass, m(AX), you can rearrange the formula as follows:
m(AX) = Zm(¹H) + Nmn - BE/c²
Here's a step-by-step breakdown of how to use this formula:
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Determine the Binding Energy (BE): This value will be given in a problem or can be calculated from mass defect. Make sure the binding energy is in appropriate units (usually MeV, which you'll need to convert to kg⋅m²/s² or Joules using appropriate conversion factors if m(AX) is needed in kg or atomic mass units (amu or u)).
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Find the Number of Protons (Z) and Neutrons (N): Identify the element and its isotope. The atomic number (Z) is the number of protons. The number of neutrons (N) is the mass number (A) minus the atomic number (Z), i.e., N = A - Z.
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Obtain the Masses of a Hydrogen Atom and a Neutron: You can find these values in tables of physical constants:
- m(¹H) ≈ 1.007825 u (atomic mass units) or 1.67353 x 10⁻²⁷ kg
- mn ≈ 1.008665 u (atomic mass units) or 1.67493 x 10⁻²⁷ kg
- Note: If your binding energy is in MeV, it's common to work in atomic mass units (u) and convert at the end. You can use the conversion factor 1 u = 931.5 MeV/c².
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Calculate the Total Mass of Individual Nucleons: Calculate Zm(¹H) + Nmn. Be sure to keep track of units!
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Calculate BE/c²: Divide the binding energy by the speed of light squared (c²). If you are using atomic mass units (u) and MeV for the binding energy, this step becomes simpler because 1 u = 931.5 MeV/c². Thus, BE/c² (in u) = BE (in MeV) / 931.5 MeV/u.
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Calculate the Atomic Mass: Subtract BE/c² from the total mass of individual nucleons (Zm(¹H) + Nmn) to obtain the atomic mass, m(AX).
Example:
Let's say you want to find the atomic mass of Helium-4 (⁴He), given that its binding energy is 28.3 MeV.
- BE = 28.3 MeV
- Z = 2, N = 4 - 2 = 2
- m(¹H) = 1.007825 u, mn = 1.008665 u
- Zm(¹H) + Nmn = (2 1.007825 u) + (2 1.008665 u) = 2.01565 u + 2.01733 u = 4.03298 u
- BE/c² = 28.3 MeV / 931.5 MeV/u = 0.03038 u
- m(⁴He) = 4.03298 u - 0.03038 u = 4.00260 u
Therefore, the atomic mass of Helium-4 is approximately 4.00260 u.
