SME
University Physics III: Waves, Optics & Modern
Nuclear physics
Use this for “An iron-56 atom, with 26 protons and 30 neutrons, has a mass of 55.934936 u. Find its mass defect, its binding energy and the binding energy per nucleon.”
Diagram
Before: 2.0157 + 2.0173 = 4.033 u
After: 4.0026 u
Δm = 4.033 − 4.0026 = 0.03038 u
B = 0.03038 u × 931.5 MeV/u
B = 28.3 MeV
The axis is broken: the bars start near the masses, so the tiny gap shows.
The lost mass became energy.
This is an example. Type your own values to start a new problem.
Δm (u)
=
Z
(1.007825
u)
+
N
(1.008665
u)
−
M (u)
A
Mass number
worked out (not typed)
Z × mH
Mass of Z hydrogen atoms
worked out (not typed) · u
N × mₙ
Mass of N neutrons
worked out (not typed) · u
B
Binding energy
worked out (not typed) · MeV
B/A
Binding energy per nucleon
worked out (not typed) · MeV
Formulas
A
=
Z
+
N
4 = 2 + 2
Z
×
mH
=
Z
×
1.007825
2.01565 = 2 × 1.007825
N
×
mₙ
=
N
×
1.008665
2.01733 = 2 × 1.008665
Δm
=
Z
×
mH
+
N
×
mₙ
−
M
0.030377 = 2.01565 + 2.01733 − 4.002603
B
=
Δm
×
931.5
28.2962 = 0.030377 × 931.5
B
A
=
B
A
7.074
=
28.2962
4
Assumptions
•
Atomic masses: the atom’s Z electrons are matched by using Z hydrogen atoms (1.007825 u each) rather than bare protons; a neutron is 1.008665 u.
•
The atom weighs less than its parts by the mass defect Δm, the energy that would pull it apart: B = Δm c², 931.5 MeV for each u.
•
B ÷ A is the binding energy per nucleon: the higher it is, the more tightly bound; it peaks near iron, at about 8.8 MeV.
Step-by-step
WE KNOW
Z = 2, N = 2, M = 4.0026 u
FIND
mass number (A), mass of Z hydrogen atoms (Z × mH), mass of N neutrons (N × mₙ), mass defect (Δm), binding energy (B), binding energy per nucleon (B/A)
Step 1 · Find mass number (A)
Use A = Z + N
The mass number counts the nucleons: protons and neutrons.
A
=
2
+
2
A
=
4
Step 2 · Find mass of Z hydrogen atoms (Z × mH)
Use Z × mH = Z × 1.007825
One hydrogen atom (a proton and an electron) for each proton: 1.007825 u each.
Z
×
mH
=
2
×
1.007825
Z
×
mH
=
2.01565
u
Step 3 · Find mass of N neutrons (N × mₙ)
Use N × mₙ = N × 1.008665
Each free neutron is 1.008665 u.
N
×
mₙ
=
2
×
1.008665
N
×
mₙ
=
2.01733
u
Step 4 · Find mass defect (Δm)
Use Δm = Z × mH + N × mₙ − M
The parts apart weigh more than the atom: the difference is the mass defect.
Δm = 2.01565 + 2.01733 − 4.002603
Δm = 0.030377 u
Step 5 · Find binding energy (B)
Use B = Δm × 931.5
E = mc²: each u of mass is 931.5 MeV of energy.
B
=
0.030377
×
931.5
B
=
28.2962
MeV
Step 6 · Find binding energy per nucleon (B/A)
Use B/A = B ÷ A
Share the binding energy among the A nucleons.
B
A
=
28.2962
4
B
A
=
7.074
MeV
Check
Put the numbers back in. Both sides should match.
4 = 2 + 2 ✓
2.01565 = 2 × 1.007825 ✓
2.01733 = 2 × 1.008665 ✓
0.030377 = 2.01565 + 2.01733 − 4.002603 ✓
28.2962 = 0.030377 × 931.5 ✓
7.074
=
28.2962
4
✓
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The decay law: nuclei left and activity
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