Questions: Nuclear Mass, Binding Energy, and the Mass-Energy Relation

Question 1

A star fuses two carbon nuclei (A=12) to form magnesium (A=24). Does this reaction release or absorb energy, and why?

• A. Absorbs energy — combining nuclei always costs energy
• B. Releases energy — both carbon and magnesium are lighter than iron on the binding energy curve, so moving toward iron releases energy
• C. Releases energy — magnesium has a higher total binding energy than two separate carbon nuclei
• D. Absorbs energy — the product has more protons, increasing Coulomb repulsion

Answer: (B)

The key is binding energy *per nucleon*, not total binding energy. Both carbon and magnesium lie on the rising portion of the BE/A curve below the iron-56 peak. Fusing them moves the products closer to iron — up the curve — which releases energy equal to the mass difference times c². Option C tempts students who correctly note that Mg has more total binding energy, but the *reason* this releases energy is the per-nucleon gain, not just the total.

Question 2

A student says 'uranium-238 is more stable than helium-4 because it has a much larger total binding energy.' What is wrong with this reasoning?

• A. Nothing — total binding energy is the correct measure of nuclear stability
• B. Total binding energy grows with mass number, so it cannot distinguish stability; binding energy per nucleon shows that helium-4 is actually more tightly bound per nucleon than uranium-238
• C. Uranium is actually more stable per nucleon because the strong force acts on more nucleons
• D. The comparison is invalid because uranium and helium are in different decay chains

Answer: (B)

Total binding energy always increases with the number of nucleons — a uranium nucleus has 238 nucleons contributing. But stability depends on how tightly bound *each* nucleon is on average. Uranium-238 has a BE/A of about 7.6 MeV/nucleon, while helium-4 has about 7.1 MeV/nucleon and iron-56 has the maximum at 8.8 MeV/nucleon. The binding energy *curve* (BE/A vs. A) is the correct measure, and iron-56 sits at its peak.

Question 3

A nucleus always has less mass than the sum of its free constituent protons and neutrons.

• T. True
• F. False

Answer: True

This is the mass defect: when nucleons bind together, the energy released (the binding energy) comes at the cost of mass, precisely as E = mc² predicts. The bound nucleus is lighter than its separated parts because some mass has been converted to the energy that holds it together. There are no exceptions — every stable or long-lived nucleus has a positive binding energy and therefore a mass defect.

Question 4

Heavier nuclei typically have greater binding energy per nucleon than lighter nuclei.

• T. True
• F. False

Answer: False

The binding energy per nucleon curve peaks at iron-56 (≈8.8 MeV/nucleon) and *decreases* for heavier nuclei. In heavy elements like uranium, the Coulomb repulsion between the many protons grows (∝Z²) while the short-range strong force cannot compensate at large nuclear radii, reducing the per-nucleon binding energy to about 7.6 MeV/nucleon. This is precisely why fission of heavy nuclei releases energy — the fragments are closer to the iron peak and more tightly bound per nucleon.

Question 5

Why can both nuclear fusion (of light nuclei) and nuclear fission (of heavy nuclei) release energy, even though these processes seem like opposites?

Model answer: Both processes move their products closer to the peak of the binding energy per nucleon curve at iron-56. Light nuclei (hydrogen, helium) are below the peak — fusing them increases binding energy per nucleon, releasing the mass difference as energy. Heavy nuclei (uranium, plutonium) are above the peak — splitting them produces fragments closer to iron with higher binding energy per nucleon, again releasing energy. Iron itself cannot release energy by either process because it sits at the top of the curve.

The binding energy curve is not monotonic — it has a single maximum at iron-56. Any nuclear reaction that moves products toward that peak releases energy; any reaction moving away from it requires energy. This single curve explains why stellar fusion terminates at iron (further fusion costs energy rather than releasing it), why fission reactors use heavy elements, and why fusion reactors need light isotopes like deuterium and tritium.