Balancing equations

Reading the numbers in a nuclear equation

A nucleus is represented as

$^{A}_{Z}\mathrm{X}$

Here, $\mathrm{X}$ is the element symbol, $A$ is the mass number and $Z$ is the atomic number. The mass number counts protons plus neutrons, while the atomic number counts protons. Because the number of protons determines the element, changing $Z$ changes the element.

A radioactive decay equation must balance in two independent ways:

• the upper numbers, representing mass numbers, must have the same total on both sides;
• the lower numbers, representing atomic numbers for nuclei and charges for emitted particles, must also have the same total on both sides.

Think of these as two columns of bookkeeping. Never add an upper number to a lower number.

Alpha emission

An alpha particle contains two protons and two neutrons, so it may be written as $^{4}_{2}\alpha$ or $^{4}_{2}\mathrm{He}$. When a nucleus emits one, its mass number falls by 4 and its atomic number falls by 2:

$^{A}_{Z}\mathrm{X}\rightarrow{}^{A-4}_{Z-2}\mathrm{Y}+{}^{4}_{2}\alpha$

For example,

$^{210}_{84}\mathrm{Po}\rightarrow{}^{206}_{82}\mathrm{Pb}+{}^{4}_{2}\alpha$

The upper numbers balance because $210=206+4$. The lower numbers balance because $84=82+2$. The atomic number changes from 84 to 82, so the daughter nucleus is a different element.

Beta-minus emission

In beta-minus decay, a neutron in the nucleus changes into a proton and an electron. The electron is emitted as a beta particle, written $^{0}_{-1}\beta$ or $^{0}_{-1}\mathrm{e}$.

The total number of protons and neutrons is unchanged, so the mass number stays the same. There is now one more proton, so the daughter nucleus has an atomic number one greater:

$^{A}_{Z}\mathrm{X}\rightarrow{}^{A}_{Z+1}\mathrm{Y}+{}^{0}_{-1}\beta$

For example,

$^{14}_{6}\mathrm{C}\rightarrow{}^{14}_{7}\mathrm{N}+{}^{0}_{-1}\beta$

The upper numbers give $14=14+0$. For the lower numbers, $6=7+(-1)$. Although a particle leaves, the daughter atomic number rises because the emitted beta particle carries negative charge and a new proton remains in the nucleus.

Gamma emission

Gamma radiation is electromagnetic radiation rather than matter. It carries neither mass number nor charge, so it is represented as $^{0}_{0}\gamma$. The nucleus loses energy, but its mass number and atomic number do not change:

$^{A}_{Z}\mathrm{X}^{*}\rightarrow{}^{A}_{Z}\mathrm{X}+{}^{0}_{0}\gamma$

The star indicates an excited, higher-energy nucleus. For instance,

$^{99}_{43}\mathrm{Tc}^{*}\rightarrow{}^{99}_{43}\mathrm{Tc}+{}^{0}_{0}\gamma$

Both number totals balance, and the element and isotope remain the same.

Finding a missing nucleus or emission

Start with the upper numbers. Subtract the known mass numbers on one side from the total on the other to find the missing mass number. Repeat with the lower numbers, taking care with the negative charge of a beta particle. If the missing object is a nucleus, use its resulting atomic number to identify its element.

Suppose uranium-238 emits an alpha particle:

$^{238}_{92}\mathrm{U}\rightarrow{}^{?}_{?}\mathrm{Th}+{}^{4}_{2}\alpha$

The missing mass number is $238-4=234$, and the missing atomic number is $92-2=90$. The completed daughter nucleus is therefore $^{234}_{90}\mathrm{Th}$.

The same method can identify an unknown emission. A missing object with upper and lower numbers 4 and 2 is an alpha particle; one with 0 and $-1$ is a beta-minus particle; one with 0 and 0 may be a gamma ray when the question describes gamma emission.

A reliable balance check

After completing an equation, perform two explicit checks:

1. add every upper number on the left and compare the result with the total on the right;
2. add every lower number on the left and compare the result with the total on the right.

An equation is not balanced unless both comparisons agree. This represents conservation of nucleon number and conservation of electric charge during the nuclear change.

The balancing rule

For $^{A}_{Z}\mathrm{X}$, $A$ is the mass number and $Z$ is the atomic number. In every nuclear equation:

• total upper numbers balance;
• total lower numbers balance.

Changes caused by emissions

Alpha and beta-minus emission form a different element because $Z$ changes. Gamma emission changes only the nucleus's energy.

Completing an equation

1. Balance the upper numbers to find the missing mass number.
2. Balance the lower numbers, including any negative sign.
3. Use the resulting atomic number to identify a missing element.
4. Recheck both totals independently.

Key patterns:

$^{A}_{Z}\mathrm{X}\rightarrow{}^{A-4}_{Z-2}\mathrm{Y}+{}^{4}_{2}\alpha$

$^{A}_{Z}\mathrm{X}\rightarrow{}^{A}_{Z+1}\mathrm{Y}+{}^{0}_{-1}\beta$

$^{A}_{Z}\mathrm{X}^{*}\rightarrow{}^{A}_{Z}\mathrm{X}+{}^{0}_{0}\gamma$