# Balancing Nuclear Equations

## To balance a nuclear equation, the mass number and atomic numbers of all particles on either side of the arrow must be equal.

#### Key Points

• A balanced nuclear equation is one wehre the sum of the mass numbers (the top number) and the sum of the atomic numbers balance on either side of an equation.

• Nuclear equation problems will often be given such that one particle is missing. To determine that particle, figure out what the mass number and atomic number needed to balance the equation is, and provide it.

• Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of of the form A(b,c)D, which is equivalent to A + b gives c + D.

#### Terms

• A heavy subatomic particle created by the binding of quarks by gluons; a hadron containing three quarks. Baryons have half-odd integral spin and are thus fermions. This category includes the common proton and neutron of the atomic nucleus.

#### Figures

1. ##### Lithium-6 plus deuterium gives two Helium-4s.

The visual representation of the equation penned above is shown here.

2. ##### Writing Nuclear Equations

Describes how to write the nuclear equations for alpha, beta decay.

Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass (the mass not considering the mass defect) must balance for each side of the equation, and in which transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number (total atomic mass number, number of protons and neutrons). An example of this notation follows:

$^6_3Li+^2_1H\rightarrow ^4_2He+?$

To balance the equation above for mass, charge and mass number, the second nucleus to the right must have atomic number 2 and mass number 4; it is therefore also helium-4. The complete equation therefore reads:

$^6_3Li+^2_1H\rightarrow ^4_2He+^4_2He$

Or, more simply:

$^{ 6 }_{ 3 }Li+^{ 2 }_{ 1 }H\rightarrow 2^{ 4 }_{ 2 }He$

Graphically, this reaction would look like this: (Figure 1)

Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of the form A(b,c)D, which is equivalent to A + b gives c + D. Common light particles are often abbreviated in this shorthand, typically p for proton, n for neutron, d for deuteron, α representing an alpha particle or helium-4, β for beta particle or electron, γ for gamma photon, etc. The reaction above would be written as Li-6(d,α)α.

In balancing a nuclear equation, it is useful to remember that the sum of all the top numbers (mass number) must be equal on both sides of the equation, as well as the sum of all the bottom numbers (the atomic number). In addition, problems will also often be given as word problems, so it is useful to know the various names of radioactively emitted particles.

Here's a sample problem:

$^{ 235 }_{ 92 }U \rightarrow ^{ 231 }_{ 90 }Th + ?$

This could be written out as Uranium-235 gives Thorium-231 plus what? In order to solve, we find the difference between the atomic masses and atomic numbers in the reactant and product. The result is an atomic mass difference of 4 and an atomic number difference of 2. This fits the description of an alpha particle. Thus, we arrive at our answer:

$^{ 235 }_{ 92 }U \rightarrow ^{ 231 }_{ 90 }Th + ^{ 4 }_{ 2 }He$

Here's another sample problem:

$^{ 214 }_{ 84 }Po+2^{ 4 }_{ 2 }He+2^0_{-1}e\rightarrow ?$

This could also be written out as Polonium-214 plus two alpha particles plus two electrons gives what? In order to solve this equation, we simply add the mass numbers, 214 for Polonium plus 8 (two times four) for helium (two alpha particles), plus zero for the electrons gives a mass number of 222. For the atomic number, we take 84 for Polonium, add 4 (two times two) for helium, then subtract two (two times -1 for two electrons, or beta emission) to give 86, the atomic number for Radon, Rn. Therefore, the equation should read:

$^{ 214 }_{ 84 }Po+2^{ 4 }_{ 2 }He+2^0_{-1}e\rightarrow ^{222}_{86}Rn$

Figure 2

#### Key Term Glossary

alpha particle
an object consisting of two protons and two neutrons bound together, identical to a helium nucleus
##### Appears in these related concepts:
atomic mass
the quantity of matter of a particle, sub-atomic particle, or molecule
##### Appears in these related concepts:
atomic number
The number, equal to the number of protons in an atom that determines its chemical properties. Symbol: Z
##### Appears in these related concepts:
baryon
A heavy subatomic particle created by the binding of quarks by gluons; a hadron containing three quarks. Baryons have half-odd integral spin and are thus fermions. This category includes the common proton and neutron of the atomic nucleus.
##### Appears in this related concept:
beta particle
An energetic electron or positron produced as the result of a nuclear reaction or nuclear decay.
##### Appears in this related concept:
charge
An electric charge.
##### Appears in these related concepts:
chemical equation
a symbolic representation of a chemical reaction; reactants are represented on the left, and products on the right
##### Appears in these related concepts:
electron
The subatomic particle having a negative charge and orbiting the nucleus; the flow of electrons in a conductor constitutes electricity.
##### Appears in these related concepts:
emission
That which is emitted or sent out.
##### Appears in these related concepts:
mass
The quantity of matter that a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
mass defect
the difference between the unbound system calculated mass and experimentally measured mass of nucleus
##### Appears in these related concepts:
mass number
The total number of protons and neutrons in an atomic nucleus.
##### Appears in these related concepts:
neutron
A subatomic particle forming part of the nucleus of an atom and having no charge; it is a combination of an up quark and two down quarks
##### Appears in these related concepts:
nucleus
The massive, positively charged central part of an atom, made up of protons and neutrons.
##### Appears in these related concepts:
photon
The quantum of light and other electromagnetic energy, regarded as a discrete particle having zero rest mass, no electric charge, and an indefinitely long lifetime. It is a gauge boson.
##### Appears in these related concepts:
product
a chemical substance formed as a result of a chemical reaction
##### Appears in these related concepts:
proton
A positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element; the nucleus of the most common isotope of hydrogen; composed of two up quarks and a down quark
##### Appears in these related concepts:
reactant
Any of the participants present at the start of a chemical reaction. Also a molecule before it undergoes a chemical change.