# Nuclear Equations

Nuclear reactions involve changes in the atomic nuclei such as changes in

• mass number
• atomic number

It is important to understand the types of particles involved in a nuclear reaction.

### Particles involved in nuclear reactions

1. Alpha Particle
• $$^{4}_{2}\mathrm{He}$$ or $$^{4}_{2}\alpha$$
• helium nucleus
• has a +2 charge
2. Beta Particle
• $$^{\phantom{-}0}_{-1}e$$ or $$^{\phantom{-}0}_{-1}\beta$$
• electron
• negatively charged
3. Positron
• $$^{\phantom{+}0}_{+1}e$$ or $$^{\phantom{+}0}_{+1}\beta$$
• same mass as electron but with a positive charge
4. Proton
• $$^{1}_{1}\mathrm{H}$$ or $$^{1}_{1}\mathrm{p}$$
• a hydrogen nucleus
• has a positive charge
5. Neutron
• $$^{1}_{0}\mathrm{n}$$
• no charge
• approximately the mass of a proton
6. Gamma Ray
• $$\gamma$$

There are various types of decay processes that radioactive (unstable) nuclei may undergo to increase their stability. As you peruse these examples, notice the mass-balance of the nuclear equations (both mass number and atomic number).

### Alpha (α) Decay

An α particle is emitted.

In the following example, an unstable uranium-238 nucleus undergoes an alpha decay (converting into thallium-234) and an alpha particle is emitted.

$^{238}_{\phantom{0}92}\mathrm{U} \longrightarrow ^{234}_{\phantom{0}90}\mathrm{Th} +^{4}_{2}\mathrm{He}$

$^{238}_{\phantom{0}92}\mathrm{U} \longrightarrow ^{234}_{\phantom{0}90}\mathrm{Th} +^{4}_{2}\mathrm{\alpha}$

### Beta (β–) Decay

A β particle is emitted.

In the following example, an unstable radium-228 nucleus undergoes an beta decay (converting into the heavier actinium-228) and a beta particle is emitted.

$^{228}_{\phantom{0}88}\mathrm{Ra} \longrightarrow ^{228}_{\phantom{0}89}\mathrm{Ac} + ^{\phantom{-}0}_{-1}e$

$^{228}_{\phantom{0}88}\mathrm{Ra} \longrightarrow ^{228}_{\phantom{0}89}\mathrm{Ac} + ^{\phantom{-}0}_{-1}\beta$

Note that the atomic number changed (+1 proton) but the mass number did not (-1 neutron). One can rationalize that a neutron been converted into a proton and an electron such that

$^{1}_{0}\mathrm{n} \longrightarrow ^{1}_{1}\mathrm{p} + ^{\phantom{-}0}_{-1}e$

though this is a bit misleading as an electron antineutrino is also created (and its discussion lies beyond the scope of this course).

### Positron Emission (β+ Decay)

A positron emission (i.e. a β+ decay) emits a positron.

Here, oxygen-15 decays into nitrogen-15.

$^{15}_{\phantom{1}8}\mathrm{O} \longrightarrow ^{15}_{\phantom{1}7}\mathrm{N} + ^{\phantom{+}0}_{+1}e$

$^{15}_{\phantom{1}8}\mathrm{O} \longrightarrow ^{15}_{\phantom{1}7}\mathrm{N} + ^{\phantom{+}0}_{+1}\beta$

Note that the atomic number changed (-1 proton) but the mass number did not change (+1 neutron). One can rationalize that a proton converted into a neutron and a positron (though as discussed above, this is a bit misleading).

$^{1}_{1}\mathrm{p} \longrightarrow ^{1}_{0}\mathrm{n} + ^{\phantom{+}0}_{+1}e$

### Electron Capture

An electron is captured by the nucleus.

Here, potassium-40 captures an electron in its nucleus and becomes argon-40.

$^{40}_{19}\mathrm{K} + ^{\phantom{-}0}_{-1}e \longrightarrow ^{40}_{18}\mathrm{Ar}$

Note that the atomic number decreased (-1 proton) yet the mass number stayed the same (+1 neutron). ONe can rationalize that a neutron was formed from a proton and an electron (though as discussed above, this is a bit misleading).

$^{1}_{1}\mathrm{p}+ ^{\phantom{-}0}_{-1}e \longrightarrow ^{1}_{0}\mathrm{n}$

### Gamma Ray Emission

A gamma ray emission process commonly accompanies radioactive decay processes and can be written explicitly. A gamma ray emission occurs when a nucleus is in an excited state and relaxes down to a lower energy state (giving off energy in the form of a gamma ray.)

$^{238}_{\phantom{0}92}\mathrm{U} \longrightarrow ^{234}_{\phantom{0}90}\mathrm{Th} +^{4}_{2}\mathrm{He} + \gamma$ $^{40}_{19}\mathrm{K} + ^{\phantom{-}0}_{-1}e \longrightarrow ^{40}_{18}\mathrm{Ar} + \gamma$

## Practice

See Example 21.4 in the textbook and Chapter 21 questions.

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