Nuclear reactions and conservation laws

A nuclear reaction is a process in which the composition of one or more nuclei changes: the reactants and products are different nuclides, written as $^{A}_{Z}\mathrm{X}$, where $A$ is the nucleon number and $Z$ the charge (proton) number. Three bookkeeping rules always hold: the total charge number $Z$ is conserved, the total nucleon number $A$ is conserved, and the total relativistic energy (rest mass plus kinetic energy) is conserved. Two more rules complete the picture: linear momentum is conserved in every collision and decay, and lepton number is conserved in beta processes. Because $Z$ and $A$ must balance, they let you find an unknown product: count $A$ and $Z$ on each side and solve for the missing entry. If the total rest mass of the products is smaller than that of the reactants, the difference $\Delta m$ appears as kinetic energy of the products through $E = \Delta m c^2$, so the reaction releases energy.

Balance the fission reaction $^{235}_{92}\mathrm{U} + ^{1}_{0}\mathrm{n} \to ^{141}_{56}\mathrm{Ba} + ^{92}_{36}\mathrm{Kr} + x\,^{1}_{0}\mathrm{n}$: find $x$ and check both conservation laws.

Find the unknown product in the reaction $^{27}_{13}\mathrm{Al} + ^{4}_{2}\mathrm{He} \to ^{30}_{15}\mathrm{P} + {}^{A}_{Z}\mathrm{X}$ and identify the particle.

For the fusion reaction $^{2}_{1}\mathrm{H} + ^{3}_{1}\mathrm{H} \to ^{4}_{2}\mathrm{He} + ^{1}_{0}\mathrm{n}$, check the conservation laws and estimate the released energy. Rest masses: $m(^{2}_{1}\mathrm{H}) = 2.014102\ \mathrm{u}$, $m(^{3}_{1}\mathrm{H}) = 3.016049\ \mathrm{u}$, $m(^{4}_{2}\mathrm{He}) = 4.002603\ \mathrm{u}$, $m(^{1}_{0}\mathrm{n}) = 1.008665\ \mathrm{u}$; $1\ \mathrm{u} \approx 931.5\ \mathrm{MeV}/c^{2}$.

Goal: test the conservation laws on real reaction data without any radiation hazard.

1. Open a nuclear reaction or fission simulation (for example the PhET Nuclear Fission simulation, or any decay-chain simulator available offline). All observation happens on screen; never handle radioactive sources in this lesson.
2. For each event the simulation shows, record the initial nuclide(s) and the final nuclide(s) in a two-column table with their $Z$ and $A$.
3. Check that $\sum Z$ and $\sum A$ are equal before and after the event. If they are not, the event is impossible — state which law is violated.
4. Repeat for five different events, including at least one beta decay, one alpha decay, and one neutron-induced fission.
5. For each event, write one sentence saying which conservation check was the most useful.

To make the $Z$ and $A$ bookkeeping visible and to show where the mass defect goes in an energy-releasing reaction.

A labeled sketch of a fission event: $^{235}\mathrm{U}$ absorbs a neutron and splits into two fragments and three neutrons, with each nuclide tagged by its $A$ and $Z$; below the sketch a balance table lists $\sum A$ and $\sum Z$ for reactants and products, and a second row shows the total rest mass shrinking by $\Delta m$, with an arrow labeled $E = \Delta m c^{2}$ pointing to the kinetic energy of the fragments.

Above the stove in the kitchen there is an ionization smoke detector. Its chamber holds about $33\ \text{kBq}$ (roughly $0.9\ \mu\text{Ci}$) of americium-241. The $^{241}_{95}\text{Am}$ nucleus emits an $\alpha$ particle $^{4}_{2}\text{He}$ with an energy of $5.49\ \text{MeV}$ and turns into neptunium-237; the half-life of americium-241 is $432$ years.

The reaction $^{241}_{95}\text{Am} \to {}^{237}_{93}\text{Np} + {}^{4}_{2}\text{He}$ shows both conservation rules of this concept. The nucleon number is the same before and after: $241 = 237 + 4$. The electric charge is also conserved: $95 = 93 + 2$. The emitted $\alpha$ particles ionize the air between the electrodes and keep a small current flowing; smoke absorbs them, the current drops and the alarm sounds.

One fission channel in a power reactor is written as $^{1}_{0}\text{n} + {}^{235}_{92}\text{U} \to {}^{141}_{56}\text{Ba} + {}^{92}_{36}\text{Kr} + 3\,{}^{1}_{0}\text{n}$. A unit of such a power plant feeds $1000\ \text{MW}$ of electric power into the grid, which is enough for about a million flats. Every such fission releases about $200\ \text{MeV} \approx 3.2\times 10^{-11}\ \text{J}$.

Both conservation laws are at work here. Charge: on the left $92 + 0$, on the right $56 + 36 + 0$, and $92 = 56 + 36$. Nucleon number: $1 + 235 = 141 + 92 + 3 \cdot 1 = 236$. From the energy of one fission it follows that for $3000\ \text{MW}$ of thermal power the reactor must sustain about $3000\times 10^{6} / 3.2\times 10^{-11} \approx 10^{20}$ fissions per second — the balanced equation is what gives this estimate.

Every second the Sun turns about $600$ million tonnes of hydrogen into helium and radiates $3.8\times 10^{26}\ \text{W}$. The net result of the proton-proton chain is $4\,{}^{1}_{1}\text{H} \to {}^{4}_{2}\text{He} + 2\,e^{+} + 2\,\nu_e$, and $26.7\ \text{MeV}$ is released per helium nucleus. So about $8.9\times 10^{37}$ such chains run every second, and this heat reaches us as light and as weather.

Balancing the reaction gives three checks of this concept. Nucleon number: $4 = 4$. Electric charge: $4 = 2 + 2$. Lepton number: $0 = -2 + 2$, because the positrons contribute $-2$ and the neutrinos $+2$. The positrons then annihilate with electrons and add energy, while the neutrinos carry theirs out of the Sun almost without interacting.