= Solution
The common energetic principle behind <nuclear fission> and <nuclear fusion> is the <nuclear binding energy>. If the products are more tightly bound in total than the reactants, their smaller rest <mass> releases
$$
\boxed{Q=\left(\sum_i M_i-\sum_f M_f\right)c^2
=\sum_f B_f-\sum_i B_i.}
$$
The two binding-energy expressions assume that the same numbers of <protons> and <neutrons> are counted on both sides; emitted free nucleons have zero nuclear binding. The <energy> appears mainly as product <kinetic energy>, with radiation and subsequent decays contributing according to the reaction.
The <binding energy per nucleon> rises rapidly through light nuclei, has a broad maximum in the iron/nickel region, and decreases more slowly for heavier nuclei. Combining suitable light nuclei can therefore increase their total binding, whereas splitting a sufficiently heavy nucleus into moderately heavy fragments can do the same. This is an <energy> comparison, not a rule that every fusion or fission channel is exothermic: the actual nuclear masses fix the sign of $Q$.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2015/paper-1-binding-curve.png]
{title=Schematic binding-energy trend explaining why light-nucleus fusion and heavy-nucleus fission can release energy}
In <nuclear fission>, the <liquid-drop model> illustrates the competition between attractive short-range nuclear cohesion and long-range <proton> repulsive <Coulomb interaction>. Deformation increases surface area and costs surface <energy>, but separating the charge can reduce <nuclear Coulomb energy>. Their competition creates a <fission barrier>. An excited nucleus may pass over the barrier; <spontaneous fission> can occur through <quantum tunnelling>. The resulting fragments are often excited and neutron-rich, so they may emit prompt <neutrons> and <photons> and undergo later <beta decay>. The typical total <energy> released by a heavy-nucleus fission is of order a few hundred MeV, rather than an electronic-scale <energy>. Its size follows from an approximately MeV-per-nucleon binding gain summed over a heavy nucleus. A <neutron> from one fission can induce another; a <nuclear chain reaction> is the repeated physical process, distinct from the energy-release condition for one event.
In <nuclear fusion>, the initial charged nuclei repel one another. Exothermicity does not remove this entrance <Coulomb barrier>. A reaction requires sufficient overlap of their <wavefunctions>, achieved by energetic encounters and <quantum tunnelling>. For nonresonant low-energy charged-particle fusion, a useful separation is
$$
\sigma(E)=\frac{S(E)}E e^{-2\pi\eta_C},\qquad
\eta_C=\frac{Z_1Z_2\alpha c}{v_{\rm rel}}.
$$
The <astrophysical S factor> isolates the nuclear part from much of the rapidly varying Coulomb penetration probability. In a thermal population, the competition between the decreasing high-energy population and increasing penetration produces the <Gamow peak>. This explains how fusion can proceed below the classical barrier without requiring the mean thermal <energy> to equal its height.
Both processes conserve total <energy>, charge, <baryon> number and <angular momentum>. Their very different entrance barriers and <mass> ranges explain why a favorable $Q$ value by itself does not specify a reaction rate. Short-range attraction gives approximately saturated nuclear binding, while surface, Coulomb, symmetry and pairing terms account for the broad trend and its isotope-dependent departures.
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