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23.1 Mass defect and nuclear binding energy

Syllabus
9702–2028–2029
Topic
23.1
Level
A2

Learning objectives

23.1.1The equivalence between energy and mass as represented by E = mc 2 and• understand the equivalence between energy and mass as represented by E = mc 2 and recall and use this equation23.1.2Simple nuclear reactions by nuclear equations of the form NH eO H7 14 2 4 8• represent simple nuclear reactions by nuclear equations of the form NH eO H7 14 2 4 8 17 1 1"++23.1.3The terms mass defect and binding energy• define and use the terms mass defect and binding energy23.1.4The variation of binding energy per nucleon with nucleon number• sketch the variation of binding energy per nucleon with nucleon number23.1.5What is meant by nuclear fusion and nuclear fission• explain what is meant by nuclear fusion and nuclear fission23.1.6The relevance of binding energy per nucleon to nuclear reactions• explain the relevance of binding energy per nucleon to nuclear reactions, including nuclear fusion and nuclear fission23.1.7The energy released in nuclear reactions using E = c 2∆m• calculate the energy released in nuclear reactions using E = c 2∆m

Mass and energy are equivalent: use E=mc² with consistent units

Mass is a form of stored energy. A system of mass m has rest energy E=mc²; when its mass changes by Δm, the corresponding energy change is ΔE=Δmc².

E=mc2,m=E/c2,c=3.00×108ms−1UseminkgforEinJ.E=mc², m=E/c², c=3.00×10⁸ m s⁻¹ Use m in kg for E in J.

For 1.00 u=1.66×10⁻²⁷ kg, E=(1.66×10⁻²⁷)(3.00×10⁸)²=1.49×10⁻¹⁰ J=931 MeV.

E=mc² is rest-energy equivalence, not the classical kinetic-energy formula. Convert g or u to kg before using SI c, unless using the established 1 u≈931 MeV/c² bridge.

Balance nuclear equations by conserving nucleon number A and proton number Z

In nuclide notation ᴬ_ZX, A is nucleon number and Z is proton number. Across a nuclear reaction, the total A and total Z are each conserved.

Write two balances: ΣA(left)=ΣA(right) and ΣZ(left)=ΣZ(right). Solve both before identifying the missing nuclide or particle from its A and Z.

¹74N+24He→817O+11HA:14+4=17+1;Z:7+2=8+1¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁H A: 14+4=17+1; Z: 7+2=8+1

Useful entries: neutron ¹₀n, proton ¹₁p, beta-minus ⁰₋₁e, beta-plus ⁰₊₁e and gamma ⁰₀γ.

Do not conserve the number of written symbols or electrons as in chemistry. A beta-minus particle contributes A=0 and Z=−1 to the equation.

Mass defect becomes the binding energy that holds a nucleus together

Mass defect Δm is the mass of the constituent nucleons when separated to infinity minus the mass of the bound nucleus.

Binding energy is the minimum energy required to separate all nucleons of the nucleus to infinity. The same energy is released when the nucleus forms from those separated nucleons.

Δm=Zmp+(A−Z)mn−mnucleusEbinding=Δmc2;bindingenergypernucleon=Ebinding/AΔm=Zmp+(A−Z)mn−mnucleus E_binding=Δmc²; binding energy per nucleon=E_binding/A

A positive mass defect means the bound state has lower mass-energy. More binding energy must be supplied to dismantle that nucleus into the same free nucleons.

Mass is not lost from conservation laws: the lower rest mass corresponds to energy released. State the infinitely-separated reference state in definitions.

Sketch binding energy per nucleon: steep rise, iron peak, slow decline

Horizontal axis: nucleon number A, from about 1 to 250. Vertical axis: binding energy per nucleon, usually in MeV.

Starting near A=1, draw a steep rise to one broad maximum near A≈56 (iron/nickel region), then a much shallower continuous decrease toward heavy nuclei.

The right-hand branch stays well above zero at A≈250. Light fusion candidates lie left of the peak; heavy fission or alpha-decay candidates lie right of it.

Greater binding energy per nucleon means nucleons are, on average, more tightly bound and more energy per nucleon is required to separate the nucleus.

Do not draw a symmetric bell curve or return the heavy-nucleus end to zero. The graph is binding energy per nucleon, not total binding energy.

Fusion joins light nuclei; fission splits a heavy nucleus

Nuclear fusion is the joining of two light nuclei to form one heavier nucleus, usually requiring very high temperature so nuclei can approach despite electrostatic repulsion.

Nuclear fission is the splitting of one heavy nucleus into two or more lighter nuclei, commonly after the heavy nucleus absorbs a neutron; further neutrons may be released.

Fusion: light + light → heavier. Fission: heavy → lighter + lighter (and often neutrons). Both conserve total nucleon number and proton number.

Energy release is a consequence, not the definition. Do not call alpha decay fission merely because a heavy nucleus emits a small particle.

Fission and fusion release energy by moving nuclei toward greater binding per nucleon

A reaction releases energy when its products have greater total binding energy than its reactants. The products then have lower total rest mass; the difference appears as released energy.

Fusion starts with light nuclei left of the A≈56 peak. Joining them moves the product toward the peak, increasing binding energy per nucleon and total binding energy.

Fission starts with a very heavy nucleus right of the peak. Splitting it into medium-mass products also moves nucleons toward the peak and increases average binding.

releasedenergy=totalbindingenergy(products)−totalbindingenergy(reactants)wheretotalbindingenergy=A×(bindingenergypernucleon)foreachnucleusreleased energy = total binding energy(products) − total binding energy(reactants) where total binding energy = A×(binding energy per nucleon) for each nucleus

Compare totals when nuclei have different A. Energy is released because binding energy increases and mass-energy decreases—not because binding energy is consumed.

Calculate nuclear energy release from the reaction mass decrease

Δm=Σmreactants−ΣmproductsEreleased=Δmc2Δm=Σm_reactants−Σm_products E_released=Δmc²

For ²₁H+²₁H→⁴₂He with masses 2.013553 u and 4.001505 u: Δm=2(2.013553)−4.001505=0.025601 u.

Convert Δm=(0.025601)(1.66×10⁻²⁷)=4.25×10⁻²⁹ kg, then E=(4.25×10⁻²⁹)(3.00×10⁸)²=3.82×10⁻¹² J per helium nucleus.

For 1.00 mol of helium nuclei, multiply by 6.02×10²³: E=2.30×10¹² J. For reaction rate R, power P=RE_reaction.

If binding energies are given, calculate Ereleased=ΣBEproducts−ΣBEreactants. If binding energy per nucleon is given, multiply each value by that nucleus's A before summing.

Include every coefficient in the mass sum. A positive reactant-minus-product Δm means release; keep per-reaction, per-nucleus, per-mole and per-second quantities distinct.