K323 / 2027
Radioactivity overview

Lesson 3 of 7

Nuclear decay equations

  • I can represent changes in nuclear composition with equations using nuclide notation.Syllabus

    Syllabus K323, 20(f). Represent changes in nuclear composition with equations using nuclide notation.

A nuclear equation records the daughter nucleus and emitted radiation. Check both the total nucleon number and the total charge number.

In nuclide notation, A counts nucleons and Z counts protons. An alpha particle contains two protons and two neutrons; beta-minus is an electron; gamma carries away nuclear energy.

Balance the upper and lower numbers separately

  1. Upper numbers: the total number of nucleons is the same before and after the emission.
  2. Lower numbers: the total charge number is the same. For a nucleus, this is its proton number; for an emitted electron, it is -1.
  3. Identify the daughter: its new proton number determines the element. Use the supplied element information when needed.

The lower -1 on a beta-minus electron does not mean that it contains negative protons. Its upper 0 means zero nucleons, not zero electron mass.

Balance nucleon and charge numbers

For a nucleus, the lower number is its proton number. For the emitted electron, -1 is its charge number; its upper 0 means no nucleons, not zero mass. The nucleus boxes give counts, not a drawing of every nucleon.

Alpha: four nucleons leave together

Alpha: four nucleons leave togetherUranium-238, proton number 92, becomes thorium-234, proton number 90, and a helium-4 nucleus with proton number 2. The parent has 92 protons and 146 neutrons; the daughter has 90 protons and 144 neutrons. The emitted alpha particle contains the remaining two protons and two neutrons. Nucleon totals balance as 238 equals 234 plus 4; lower charge numbers balance as 92 equals 90 plus 2.23892U23490Th42He+Before: nucleusAfter: nucleus92 protons146 neutrons90 protons144 neutronsEmitted alpha particlep+nnp+2 protons + 2 neutronsCharge +2238 = 234 + 492 = 90 + 2

Both nucleon number and total charge balance. Two protons and two neutrons leave the parent nucleus together.

Beta-minus: a neutron changes into a proton

Beta-minus: a neutron changes into a protonCarbon-14, proton number 6, becomes nitrogen-14, proton number 7, with a newly emitted electron. The nucleus changes from 6 protons and 8 neutrons to 7 protons and 7 neutrons. The inset shows one neutron changing into a proton with an electron produced during the change; no electron is drawn waiting in the original nucleus. This school bookkeeping omits the antineutrino, which is also emitted. The upper totals balance as 14 equals 14 plus 0; the lower charge totals balance as 6 equals 7 plus negative 1.146C147N0-1e+Before: nucleusAfter: nucleus6 protons8 neutrons7 protons7 neutronsOne neutron changesnp++e-NeutronProtonElectronThe electron is produced during the change.14 = 14 + 06 = 7 + (-1)

The beta electron is produced during the nuclear change; it is not ejected from an existing electron shell or stored inside the nucleus. An antineutrino is also emitted; the displayed school-level bookkeeping omits it. Antineutrino recall is not needed here.

Gamma: the same nuclide at lower nuclear energy

Gamma: the same nuclide at lower nuclear energyAn excited nucleus X, with nucleon number A and proton number Z, emits a gamma photon and reaches a lower nuclear energy. A star on the parent symbol marks its higher-energy nuclear state. Both nuclei have Z protons and A minus Z neutrons. Neither count changes. The photon carries energy away; its upper and lower bookkeeping numbers are both zero. The starred and unstarred symbols describe the same nuclide, not different isotopes.AZX*AZX00γ+Before: nucleusAfter: nucleusZ protonsA - Z neutronsZ protonsA - Z neutronsHigher energyLower energy* = higher-energy nucleusPhoton carries energy awaySame A and Z; lower nuclear energyA = A + 0Z = Z + 0

The star marks extra nuclear energy. Removing it changes the energy state, while the proton and neutron counts remain the same.

Alpha removes two protons and two neutrons. Beta-minus changes one neutron into a proton while emitting an electron. Gamma lowers nuclear energy without changing either nucleon count. The upper and lower bookkeeping rows must both balance.
Alpha: A falls by 4; Z falls by 2
The daughter loses the four nucleons carried away in the helium nucleus.
Beta-minus: A is unchanged; Z rises by 1
One neutron becomes a proton, so the total nucleon count stays the same. The emitted negative charge balances the daughter's increased positive charge.
Gamma: A and Z are unchanged
The daughter has the same composition but lower nuclear energy. A star on the initial symbol marks a higher-energy nuclear state; it does not mean an extra proton or neutron.

Two emissions in order

Start with A = 214 and Z = 84

The nucleus emits an alpha particle, then a beta-minus particle.

  1. After alpha: A = 214 - 4 = 210; Z = 84 - 2 = 82.
  2. After beta-minus: A stays 210; Z = 82 + 1 = 83.
  3. Count the final neutrons: A - Z = 210 - 83 = 127.

The second step acts on the daughter from the first step. Do not apply both emissions separately to the original nucleus and discard one of the changes.

Try it step by step

Work it out in steps: Nucleon number after the decay, then Proton number after the decay.

  1. Problem 1. A radium nucleus with nucleon number 226 and proton number 88 emits an alpha particle (nucleon number 4, proton number 2). Find the nucleon and proton numbers of the new nucleus.
  2. Problem 2. A carbon-14 nucleus (nucleon number 14, proton number 6) emits a beta-minus particle (an electron, nucleon number 0, charge -1). Find the nucleon and proton numbers of the new nucleus.
  3. Problem 3. A uranium nucleus with nucleon number 238 and proton number 92 emits an alpha particle. Find the nucleon and proton numbers of the new nucleus.
Show the answer

Problem 1.

Nucleon number after the decay: A = 226 - 4 = 222

Proton number after the decay: Z = 88 - 2 = 86 (radon)

Problem 2.

Nucleon number after the decay: A = 14 - 0 = 14

Proton number after the decay: Z = 6 - (-1) = 7 (nitrogen)

Problem 3.

Nucleon number after the decay: A = 238 - 4 = 234

Proton number after the decay: Z = 92 - 2 = 90 (thorium)

A nucleus losing energy need not lose nucleons: gamma emission is the example. Conversely, the beta electron's negative charge does not mean the daughter's proton number decreases.

Check your understanding

A nucleus with A = 214 and Z = 84 emits an alpha particle, then a beta-minus particle. What are its final nucleon, proton and neutron numbers?

  1. A = 214, Z = 85, with 129 neutrons.
  2. A = 210, Z = 83, with 127 neutrons.
  3. A = 210, Z = 81, with 129 neutrons.
Show the answer

A = 210, Z = 83, with 127 neutrons.

Alpha emission subtracts 4 from A and 2 from Z, giving 210 and 82. Beta-minus emission changes one neutron into a proton, leaving A at 210 and increasing Z to 83. Neutrons = 210 - 83 = 127.