Fission and fusion · printable packet

Energy from both sides of the binding curve: balancing fission, chain reactions and reactors, fusion Q-values and their sharing, and fuel energy per gram.

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1. What you will learn

By the end of this lesson you will be able to balance fission reactions, compute fusion energies, and estimate how much fuel reactors and fusion shots consume.

2. What you already have

From lesson 21 you know the binding energy per nucleon rises from light nuclei to a peak near iron and falls slowly beyond it, and that mass defects convert to energy at $931.5$ MeV per u. From lesson 22 you know how Q-values are computed and shared by momentum. This lesson uses both to explain the two ways nuclear energy is released.

3. Words for this lesson

4. Both ways up the binding curve

Because binding energy per nucleon peaks near iron, any change that moves nuclei toward the peak releases energy:

In both, the number of nucleons is conserved. The products are more tightly bound, so their total mass is less, and the missing mass appears as kinetic energy: $Q = \Delta m \, c^2$.

Term	What it means
Fission	A heavy nucleus splits into two middle-sized nuclei and a few neutrons.
Fusion	Light nuclei join into a heavier one.
Chain reaction	Neutrons from each fission cause further fissions.
Critical mass	The least fuel that sustains a chain reaction.
Moderator	A material, such as water, that slows neutrons so they fission uranium-235 more readily.
Plasma	A gas of bare nuclei and electrons, hot enough for fusion.

Another way: picture

Picture the binding curve as a hillside with the most stable nuclei at the top, near iron. Energy is released by climbing it: heavy nuclei climb by breaking apart, sliding in from the right, while light nuclei climb by joining together, from the left. A nucleus at the peak has nowhere higher to go, which is why iron is where stars' fusion ends.

Another way: steps

Balance the reaction: mass numbers and charges on both sides.
Compute $Q$ from mass loss ($931.5$ MeV per u) or binding energies (products minus reactants).
Share $Q$ between two products by momentum conservation.
Scale up: nuclei per gram is $N_A/A$; reactions per second is power over energy per reaction.
Check: fission gives about $200$ MeV, fusion reactions a few to twenty MeV.

5. The discovery of fission

In December 1938, Otto Hahn and Fritz Strassmann in Berlin found barium, an element half the mass of uranium, in uranium they had bombarded with neutrons. Lise Meitner, who had fled Nazi Germany, and her nephew Otto Frisch realized over the Christmas holidays that the nucleus had split, and they used the liquid-drop model to estimate that about $200$ MeV would be released, enough to make the fragments fly apart visibly.

Within weeks, physicists across the United States confirmed it. Leo Szilard and Enrico Fermi at Columbia University showed that each fission also released more neutrons, raising the possibility of a chain reaction, and Einstein signed Szilard's letter warning President Roosevelt.

6. Chain reactions and critical mass

Each fission of uranium-235 releases two or three neutrons. If on average at least one of them causes another fission, the reaction sustains itself. In a small piece of uranium, too many neutrons escape through the surface before they are absorbed; as the piece grows, volume increases faster than surface, and at the critical mass the chain just sustains itself.

On December 2, 1942, Fermi's team achieved the first controlled chain reaction in Chicago Pile-1, a stack of graphite and uranium under the stands of the University of Chicago's Stagg Field. It produced half a watt. Cadmium control rods, which absorb neutrons, kept the reaction in check, as they still do in reactors today.

7. Nuclear reactors

A power reactor keeps its chain reaction exactly critical: each fission leads, on average, to exactly one more. Natural uranium is only $0.7$ percent uranium-235, so most American reactors burn fuel enriched to $3$ to $5$ percent. Water serves both as the moderator, slowing neutrons so they fission uranium-235 efficiently, and as the coolant that carries heat to steam turbines.

A reactor producing $3000$ MW of heat fissions about $3$ kg of uranium-235 per day. The United States has about ninety operating reactors, supplying nearly a fifth of its electricity and about half of its carbon-free electricity.

8. Fusion in the Sun

The Sun shines by fusing hydrogen into helium in its core, at $15$ million kelvins. Protons must approach close enough for the strong force to act, but their electrical repulsion forbids it classically; they get through by quantum tunneling, and only rarely, which is why the Sun burns steadily for ten billion years instead of exploding.

Hans Bethe worked out the reaction chains at Cornell University in 1938, earning the 1967 Nobel Prize. Four protons become one helium-4 nucleus, releasing $26.7$ MeV. The Sun converts about four million tons of mass into energy every second.

9. The method, step by step, and how to check it

Checking an answer. Nucleons and charge must balance. Fission gives about $200$ MeV, fusion reactions $3$ to $20$ MeV. A gram of uranium gives about $80{,}000$ MJ, roughly the energy of three tons of coal.

10. Why each step is allowed

Conservation of nucleon number and charge holds in every nuclear reaction; they are the rules the balancing enforces. The energy released equals the mass lost because rest energy is part of total energy, which is conserved: the products' kinetic energy is paid for by the drop in rest energy.

Sharing by momentum conservation assumes the reactants start nearly at rest, which holds when their kinetic energy, a few keV in a fusion plasma, is small next to the MeV released. Energy per fission of $200$ MeV is an average; individual fissions give different fragment pairs with slightly different releases.

11. Fusion on Earth

Controlled fusion needs deuterium and tritium heated above $100$ million kelvins, held long enough and dense enough for reactions to outpace losses. Magnetic confinement uses donut-shaped tokamaks; ITER, under construction in France with the United States as a partner, is designed to produce ten times the heating power it absorbs.

Inertial confinement crushes a tiny fuel capsule with lasers. On December 5, 2022, the National Ignition Facility at Lawrence Livermore in California released $3.15$ MJ from $2.05$ MJ of laser light, the first fusion experiment to produce more energy than was delivered to its fuel. Later shots have exceeded $5$ MJ.

12. Waste, safety and fuel

Fission fragments are neutron-rich and radioactive, some for centuries, like cesium-137 and strontium-90 at about thirty years, and a few for far longer. American spent fuel is stored in water pools and then in steel and concrete casks at reactor sites, awaiting a permanent repository.

Fusion produces no long-lived fission fragments; its product, helium, is inert. Its neutrons do make reactor walls radioactive, but for decades rather than millennia. Deuterium is abundant in seawater; tritium, with a $12.3$-year half-life, must be bred from lithium inside the reactor using the fusion neutrons themselves.

13. Where the energy of fission goes

Of the roughly $200$ MeV released in a uranium fission, about $170$ MeV appears at once as the kinetic energy of the two fragments, which are driven apart by their electrical repulsion. The fragments slow down within a few thousandths of a millimeter of fuel, turning their energy into heat. A few MeV goes to the prompt neutrons and gamma rays, and the rest emerges over the following hours and years as the fragments themselves decay by beta and gamma emission.

That delayed share matters for safety. When a reactor shuts down, the chain reaction stops within seconds, but the radioactive fragments keep producing heat, about seven percent of full power at first and still about one percent after a few hours. The accident at Three Mile Island in Pennsylvania in 1979 and the Fukushima disaster in Japan in 2011 both came from losing the cooling needed to remove this decay heat, not from the chain reaction running away. Modern American reactor designs add passive cooling that works without pumps or power.

14. Weapons and their legacy

The first use of fission was not power but weapons. The Manhattan Project built reactors at Hanford in Washington to make plutonium and enrichment plants at Oak Ridge in Tennessee to separate uranium-235, and in July 1945 tested the first bomb in the New Mexico desert. A fission weapon assembles more than a critical mass in microseconds, so the chain reaction multiplies before the material blows itself apart. Only about a kilogram of material fissioned in the Hiroshima bomb.

Thermonuclear weapons use a fission explosion to compress and heat fusion fuel, releasing hundreds of times more energy. Today the cleanup of Hanford's radioactive waste is one of the largest environmental projects in the country, a reminder that the same physics that lights cities carries long-term costs.

15. In the world: nuclear power in the United States

About ninety reactors at over fifty American plants generate roughly a fifth of the nation's electricity. The Palo Verde plant in Arizona, the country's largest, runs three reactors of about $4000$ MW of heat each, fissioning roughly $4$ kg of uranium-235 per reactor per day and supplying power to about four million people.

Because a gram of uranium-235 releases as much heat as about three tons of coal, a reactor's annual fuel fits on a few trucks, while a coal plant of the same output burns a train of coal every day. Reactors emit no carbon dioxide while running, which is why interest in new designs, including small modular reactors, has grown as states set clean-energy goals.

16. In the world: ignition at Livermore

The National Ignition Facility in Livermore, California, focuses $192$ laser beams onto a gold cylinder smaller than a pencil eraser. The lasers heat the cylinder's walls, which bathe a peppercorn-sized capsule of deuterium and tritium in X-rays, crushing it to a hundred times the density of lead and more than $100$ million kelvins.

In December 2022 a shot released $3.15$ MJ from $2.05$ MJ of laser light, achieving ignition: the fusion reactions heated the fuel faster than it cooled. The fuel actually fused was only about nine micrograms. The lasers themselves still draw far more energy from the grid than the shot releases, so fusion power plants remain a long engineering road away.

17. No nucleons are destroyed

It is natural to think nuclear energy comes from destroying matter, perhaps nucleons turning into energy. In fission and fusion the number of protons and neutrons is unchanged. The energy comes from binding: the products are more tightly bound, so their rest energy, and hence their mass, is slightly smaller than the reactants', by less than one percent.

A related error is to think fusion of any nuclei releases energy. Fusing nuclei heavier than iron absorbs energy, just as splitting nuclei lighter than iron does; only moves toward the peak of the binding curve release it.

18. Balancing a fission

19. Deuterium–tritium fusion

20. Energy in a gram of fuel

21. Your turn: uranium-235 captures a neutron and splits into xenon-140 and strontium-94. How many neutrons are released?

22. Guided practice

A uranium-235 nucleus absorbs a slow neutron and splits into cesium-137 and rubidium-96. How many neutrons are released?

23. Guided practice

Complete the worked solution: uranium-236, formed when uranium-235 captures a neutron, has $7.59$ MeV of binding energy per nucleon. It splits into two fragments holding $233$ nucleons, bound at $8.35$ MeV per nucleon, plus three free neutrons. Find the total binding energy of the parent and of the fragments, and the energy released, all in MeV.

24. Guided practice

25. Practice

For the fusion reaction deuterium + helium-3 → helium-4 + proton, the reactants' atomic masses exceed the products' by $0.019703$ u. With $1$ u $= 931.5$ MeV/$c^2$, fill in the mass lost in u, the Q-value in MeV, and the kinetic energy in MeV of the proton, which takes the fraction $0.8$ of $Q$ when the reactants start nearly at rest.

26. Practice

Each fission of plutonium-239 releases about $200$ MeV. Using $N_A = 6.022 \times 10^{23}$ per mole and $1$ MeV $= 1.602 \times 10^{-13}$ J, write the energy in MJ released by fissioning $m$ grams of it, rounded to whole MJ per gram.

27. Practice

A reactor core produces $2500$ MW of heat. Each fission of uranium-235 releases $200$ MeV ($3.204 \times 10^{-11}$ J). How many kilograms of uranium-235 does it fission per day? Use $N_A = 6.022 \times 10^{23}$ per mole.

28. Somewhere new

At Lawrence Livermore's National Ignition Facility in California, a laser shot fused deuterium and tritium, releasing $5.2$ MJ. Each reaction releases $17.59$ MeV ($1$ MeV $= 1.602 \times 10^{-13}$ J) and consumes $5.030$ u of fuel ($1$ u $= 1.6605 \times 10^{-27}$ kg). How many micrograms of fuel fused?

29. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

30. Test question

Each fission of uranium-235 releases about $200$ MeV. Using $N_A = 6.022 \times 10^{23}$ per mole and $1$ MeV $= 1.602 \times 10^{-13}$ J, write the energy in MJ released by fissioning $m$ grams of it, rounded to whole MJ per gram.

31. What you can do now

You can analyze fission and fusion. Explain to someone why both splitting uranium and joining hydrogen release energy.

Working for the steps left to you

	value
mass lost (u)
Q-value (MeV)
light product's energy (MeV)

21. Your turn: uranium-235 captures a neutron and splits into xenon-140 and strontium-94. How many neutrons are released?, step 3