What a nuclear equation records

Definition

Nuclear equation

A nuclear equation is a way of writing a nuclear change that shows the mass number and the charge of every nucleus and particle taking part.

Definition

Mass number

The total number of protons and neutrons in the nucleus of an atom.

Definition

Charge number

The charge number is the number written at the bottom left of a symbol in a nuclear equation, giving the charge of that nucleus or particle relative to the charge of one proton.

Every nucleus and particle in a nuclear equation is written as $ZAX_{Z}^{A}\text{X}$ , with the mass number $A$ on top and the charge number $Z$ below.
For a nucleus the charge number is simply the proton number, because each proton carries one unit of positive charge and the neutrons carry none.
For an emitted particle the charge number is that particle's own charge, so a beta particle takes $Z = - 1$ and a gamma ray takes $Z = 0$ .
Two quantities are conserved across the arrow, which is what makes the equation solvable.
1. The total mass number on the left equals the total mass number on the right, because nucleons are neither created nor destroyed.
2. The total charge on the left equals the total charge on the right, because charge cannot be created or destroyed.
Balancing a nuclear equation therefore means finding the missing numbers that make both of those totals agree.
The two totals are kept completely separate: the top numbers are added only to other top numbers, and the bottom numbers only to other bottom numbers.
This is not the same task as balancing a chemical equation, where whole atoms are counted and no atom ever changes into another element.

The symbols you need

Definition

Atomic number

The number of protons in the nucleus of an atom.

An alpha particle is $24α_{2}^{4}\alpha$ , sometimes written $24He_{2}^{4}\text{He}$ , because it is two protons and two neutrons.
A beta-minus particle is $−10e_{-1}^{0}\text{e}$ , sometimes written $−10β_{-1}^{0}\beta$ , because it is an electron with no nucleons and one unit of negative charge.
A positron is $+10e_{+1}^{0}\text{e}$ , sometimes written $+10β_{+1}^{0}\beta$ , because it has no nucleons and one unit of positive charge.
A gamma ray is $00γ_{0}^{0}\gamma$ , because it carries energy but neither nucleons nor charge.
A neutron is $01n_{0}^{1}\text{n}$ , because it is one nucleon with no charge.
A proton is $11p_{1}^{1}\text{p}$ , because it is one nucleon carrying one unit of positive charge.
Once these six symbols are known, every balancing task reduces to two subtractions.

The method for balancing

Write the equation out with the unknown labelled, using $A$ for the missing top number and $Z$ for the missing bottom number.
Add the top numbers on the left, then add the top numbers on the right.
Set those two totals equal to each other and solve for the missing mass number.
Add the bottom numbers on the left, then add the bottom numbers on the right, remembering that a negative charge number reduces the total.
Set those two totals equal to each other and solve for the missing charge number.
Use the data supplied with the question, such as a section of the periodic table, to turn the charge number into an element symbol if a symbol is wanted.
Substitute both answers back in and check that each total still agrees, since this catches a sign slip in seconds.

Example

Balancing an alpha decay

The equation to balance is $92238U→ZATh+24α_{92}^{238}\text{U} \rightarrow _{Z}^{A}\text{Th} + _{2}^{4}\alpha$ .
Top numbers give $238 = A + 4$ .
Rearranging gives $A = 238 - 4 = 234$ .
Bottom numbers give $92 = Z + 2$ .
Rearranging gives $Z = 92 - 2 = 90$ .
The balanced equation is $92238U→90234Th+24α_{92}^{238}\text{U} \rightarrow _{90}^{234}\text{Th} + _{2}^{4}\alpha$ .
Checking gives $238 = 234 + 4$ and $92 = 90 + 2$ , so both totals agree.

Example

Finding the daughter after beta-minus emission

The equation to balance is $1940K→ZAX+−10e_{19}^{40}\text{K} \rightarrow _{Z}^{A}\text{X} + _{-1}^{0}\text{e}$ .
Top numbers give $40 = A + 0$ , so $A = 40$ .
Bottom numbers give $19 = Z + (- 1)$ , which rearranges to $Z = 19 + 1 = 20$ .
The negative charge number on the electron is what makes the daughter's charge number larger than the parent's.
An element with $20$ protons is calcium, so the daughter is $2040Ca_{20}^{40}\text{Ca}$ .
The balanced equation is $1940K→2040Ca+−10e_{19}^{40}\text{K} \rightarrow _{20}^{40}\text{Ca} + _{-1}^{0}\text{e}$ .

Example

Identifying an unknown emitted particle

The equation to balance is $1327Al→1227Mg+ZAX_{13}^{27}\text{Al} \rightarrow _{12}^{27}\text{Mg} + _{Z}^{A}\text{X}$ .
Top numbers give $27 = 27 + A$ , so $A = 0$ and the emitted particle contains no nucleons.
Bottom numbers give $13 = 12 + Z$ , so $Z = 1$ and the emitted particle carries one unit of positive charge.
A particle with mass number $0$ and charge $+ 1$ is a positron, written $+10e_{+1}^{0}\text{e}$ .
The balanced equation is $1327Al→1227Mg++10e_{13}^{27}\text{Al} \rightarrow _{12}^{27}\text{Mg} + _{+1}^{0}\text{e}$ .
Reading off the two numbers first, then naming the particle, is what keeps this kind of question quick.

Example

Balancing a decay chain of two steps

The chain to complete is $90232Th→Z1A1Ra+24α_{90}^{232}\text{Th} \rightarrow _{Z_1}^{A_1}\text{Ra} + _{2}^{4}\alpha$ followed by $Z1A1Ra→Z2A2Ac+−10e_{Z_1}^{A_1}\text{Ra} \rightarrow _{Z_2}^{A_2}\text{Ac} + _{-1}^{0}\text{e}$ .
For the first step, top numbers give $A_1 = 232 - 4 = 228$ and bottom numbers give $Z_1 = 90 - 2 = 88$ .
The first daughter is therefore $88228Ra_{88}^{228}\text{Ra}$ .
For the second step, top numbers give $A_2 = 228 - 0 = 228$ .
Bottom numbers give $Z_2 = 88 + 1 = 89$ , because the electron carries away $- 1$ .
The second daughter is $89228Ac_{89}^{228}\text{Ac}$ , so the chain runs $90232Th→88228Ra→89228Ac_{90}^{232}\text{Th} \rightarrow _{88}^{228}\text{Ra} \rightarrow _{89}^{228}\text{Ac}$ .
Each step is balanced on its own, which is always safer than trying to balance a whole chain in one line.

Exam technique

Presenting a balanced equation

Write both totals as a short line of working, such as $238 = A + 4$ , because the marks are usually split between the mass number and the charge number.
Fill in every box a question gives you, including any number that is unchanged, since a blank box scores nothing.
Keep the minus sign on a beta particle's charge number, because dropping it sends the daughter's charge number the wrong way by two.
Give the element symbol only when the question or its data allow you to identify it, and give the numbers regardless.
Check both totals before moving on, because a single arithmetic slip loses both marks in a two-mark part.
If a question supplies a periodic table extract, use it rather than recalling the symbol, since the extract is there to be read.
For an unknown particle, state the mass number and charge first and then name it, because that order is what the working needs to show.

Common Mistake

Do not add a top number to a bottom number, because mass number and charge are conserved separately.
Do not subtract $1$ from the parent's charge number for a beta-minus particle, because a charge of $- 1$ on the right means the daughter must be $+ 1$ higher.
Do not give a gamma ray a mass number or a charge, since $00γ_{0}^{0}\gamma$ leaves both totals untouched.
Do not change an element symbol without changing its charge number to match, because the two must always agree.
Do not treat this like a chemical equation by putting a large number in front of a symbol, because nuclear equations are balanced through the top and bottom numbers only.

Self review

State the two quantities that must balance on both sides of a nuclear equation.
Give the mass number and charge number of an alpha particle, a beta-minus particle, a positron and a gamma ray.
Find $A$ and $Z$ in $88226Ra→ZARn+24α_{88}^{226}\text{Ra} \rightarrow _{Z}^{A}\text{Rn} + _{2}^{4}\alpha$ .
Find $A$ and $Z$ in $614C→ZAN+−10e_{6}^{14}\text{C} \rightarrow _{Z}^{A}\text{N} + _{-1}^{0}\text{e}$ .
A nucleus loses a particle of mass number $0$ and charge $- 1$ . Which particle is it?
Why does the charge number of the daughter rise when an electron is emitted?

What a nuclear equation records

Definition

Nuclear equation

A nuclear equation is a way of writing a nuclear change that shows the mass number and the charge of every nucleus and particle taking part.

Definition

Mass number

The total number of protons and neutrons in the nucleus of an atom.

Definition

Charge number

The charge number is the number written at the bottom left of a symbol in a nuclear equation, giving the charge of that nucleus or particle relative to the charge of one proton.

Every nucleus and particle in a nuclear equation is written as $ZAX_{Z}^{A}\text{X}$ , with the mass number $A$ on top and the charge number $Z$ below.
For a nucleus the charge number is simply the proton number, because each proton carries one unit of positive charge and the neutrons carry none.
For an emitted particle the charge number is that particle's own charge, so a beta particle takes $Z = - 1$ and a gamma ray takes $Z = 0$ .
Two quantities are conserved across the arrow, which is what makes the equation solvable.
1. The total mass number on the left equals the total mass number on the right, because nucleons are neither created nor destroyed.
2. The total charge on the left equals the total charge on the right, because charge cannot be created or destroyed.
Balancing a nuclear equation therefore means finding the missing numbers that make both of those totals agree.
The two totals are kept completely separate: the top numbers are added only to other top numbers, and the bottom numbers only to other bottom numbers.
This is not the same task as balancing a chemical equation, where whole atoms are counted and no atom ever changes into another element.

The symbols you need

Definition

Atomic number

The number of protons in the nucleus of an atom.

An alpha particle is $24α_{2}^{4}\alpha$ , sometimes written $24He_{2}^{4}\text{He}$ , because it is two protons and two neutrons.
A beta-minus particle is $−10e_{-1}^{0}\text{e}$ , sometimes written $−10β_{-1}^{0}\beta$ , because it is an electron with no nucleons and one unit of negative charge.
A positron is $+10e_{+1}^{0}\text{e}$ , sometimes written $+10β_{+1}^{0}\beta$ , because it has no nucleons and one unit of positive charge.
A gamma ray is $00γ_{0}^{0}\gamma$ , because it carries energy but neither nucleons nor charge.
A neutron is $01n_{0}^{1}\text{n}$ , because it is one nucleon with no charge.
A proton is $11p_{1}^{1}\text{p}$ , because it is one nucleon carrying one unit of positive charge.
Once these six symbols are known, every balancing task reduces to two subtractions.

The method for balancing

Write the equation out with the unknown labelled, using $A$ for the missing top number and $Z$ for the missing bottom number.
Add the top numbers on the left, then add the top numbers on the right.
Set those two totals equal to each other and solve for the missing mass number.
Add the bottom numbers on the left, then add the bottom numbers on the right, remembering that a negative charge number reduces the total.
Set those two totals equal to each other and solve for the missing charge number.
Use the data supplied with the question, such as a section of the periodic table, to turn the charge number into an element symbol if a symbol is wanted.
Substitute both answers back in and check that each total still agrees, since this catches a sign slip in seconds.

Example

Balancing an alpha decay

The equation to balance is $92238U→ZATh+24α_{92}^{238}\text{U} \rightarrow _{Z}^{A}\text{Th} + _{2}^{4}\alpha$ .
Top numbers give $238 = A + 4$ .
Rearranging gives $A = 238 - 4 = 234$ .
Bottom numbers give $92 = Z + 2$ .
Rearranging gives $Z = 92 - 2 = 90$ .
The balanced equation is $92238U→90234Th+24α_{92}^{238}\text{U} \rightarrow _{90}^{234}\text{Th} + _{2}^{4}\alpha$ .
Checking gives $238 = 234 + 4$ and $92 = 90 + 2$ , so both totals agree.

Example

Finding the daughter after beta-minus emission

The equation to balance is $1940K→ZAX+−10e_{19}^{40}\text{K} \rightarrow _{Z}^{A}\text{X} + _{-1}^{0}\text{e}$ .
Top numbers give $40 = A + 0$ , so $A = 40$ .
Bottom numbers give $19 = Z + (- 1)$ , which rearranges to $Z = 19 + 1 = 20$ .
The negative charge number on the electron is what makes the daughter's charge number larger than the parent's.
An element with $20$ protons is calcium, so the daughter is $2040Ca_{20}^{40}\text{Ca}$ .
The balanced equation is $1940K→2040Ca+−10e_{19}^{40}\text{K} \rightarrow _{20}^{40}\text{Ca} + _{-1}^{0}\text{e}$ .

Example

Identifying an unknown emitted particle

The equation to balance is $1327Al→1227Mg+ZAX_{13}^{27}\text{Al} \rightarrow _{12}^{27}\text{Mg} + _{Z}^{A}\text{X}$ .
Top numbers give $27 = 27 + A$ , so $A = 0$ and the emitted particle contains no nucleons.
Bottom numbers give $13 = 12 + Z$ , so $Z = 1$ and the emitted particle carries one unit of positive charge.
A particle with mass number $0$ and charge $+ 1$ is a positron, written $+10e_{+1}^{0}\text{e}$ .
The balanced equation is $1327Al→1227Mg++10e_{13}^{27}\text{Al} \rightarrow _{12}^{27}\text{Mg} + _{+1}^{0}\text{e}$ .
Reading off the two numbers first, then naming the particle, is what keeps this kind of question quick.

Example

Balancing a decay chain of two steps

The chain to complete is $90232Th→Z1A1Ra+24α_{90}^{232}\text{Th} \rightarrow _{Z_1}^{A_1}\text{Ra} + _{2}^{4}\alpha$ followed by $Z1A1Ra→Z2A2Ac+−10e_{Z_1}^{A_1}\text{Ra} \rightarrow _{Z_2}^{A_2}\text{Ac} + _{-1}^{0}\text{e}$ .
For the first step, top numbers give $A_1 = 232 - 4 = 228$ and bottom numbers give $Z_1 = 90 - 2 = 88$ .
The first daughter is therefore $88228Ra_{88}^{228}\text{Ra}$ .
For the second step, top numbers give $A_2 = 228 - 0 = 228$ .
Bottom numbers give $Z_2 = 88 + 1 = 89$ , because the electron carries away $- 1$ .
The second daughter is $89228Ac_{89}^{228}\text{Ac}$ , so the chain runs $90232Th→88228Ra→89228Ac_{90}^{232}\text{Th} \rightarrow _{88}^{228}\text{Ra} \rightarrow _{89}^{228}\text{Ac}$ .
Each step is balanced on its own, which is always safer than trying to balance a whole chain in one line.

Exam technique

Presenting a balanced equation

Write both totals as a short line of working, such as $238 = A + 4$ , because the marks are usually split between the mass number and the charge number.
Fill in every box a question gives you, including any number that is unchanged, since a blank box scores nothing.
Keep the minus sign on a beta particle's charge number, because dropping it sends the daughter's charge number the wrong way by two.
Give the element symbol only when the question or its data allow you to identify it, and give the numbers regardless.
Check both totals before moving on, because a single arithmetic slip loses both marks in a two-mark part.
If a question supplies a periodic table extract, use it rather than recalling the symbol, since the extract is there to be read.
For an unknown particle, state the mass number and charge first and then name it, because that order is what the working needs to show.

Common Mistake

Do not add a top number to a bottom number, because mass number and charge are conserved separately.
Do not subtract $1$ from the parent's charge number for a beta-minus particle, because a charge of $- 1$ on the right means the daughter must be $+ 1$ higher.
Do not give a gamma ray a mass number or a charge, since $00γ_{0}^{0}\gamma$ leaves both totals untouched.
Do not change an element symbol without changing its charge number to match, because the two must always agree.
Do not treat this like a chemical equation by putting a large number in front of a symbol, because nuclear equations are balanced through the top and bottom numbers only.

Self review

State the two quantities that must balance on both sides of a nuclear equation.
Give the mass number and charge number of an alpha particle, a beta-minus particle, a positron and a gamma ray.
Find $A$ and $Z$ in $88226Ra→ZARn+24α_{88}^{226}\text{Ra} \rightarrow _{Z}^{A}\text{Rn} + _{2}^{4}\alpha$ .
Find $A$ and $Z$ in $614C→ZAN+−10e_{6}^{14}\text{C} \rightarrow _{Z}^{A}\text{N} + _{-1}^{0}\text{e}$ .
A nucleus loses a particle of mass number $0$ and charge $- 1$ . Which particle is it?
Why does the charge number of the daughter rise when an electron is emitted?

6.3.3 Balancing nuclear equations

What a nuclear equation records

The symbols you need

The method for balancing

Balancing an alpha decay

Finding the daughter after beta-minus emission

Identifying an unknown emitted particle

Balancing a decay chain of two steps

Presenting a balanced equation

Review 6.3.3 Balancing nuclear equations by teaching Genie

6.3.3 Balancing nuclear equations

What a nuclear equation records

The symbols you need

The method for balancing

Balancing an alpha decay

Finding the daughter after beta-minus emission

Identifying an unknown emitted particle

Balancing a decay chain of two steps

Presenting a balanced equation

Review 6.3.3 Balancing nuclear equations by teaching Genie

6.3.3 Balancing nuclear equations

What a nuclear equation records

The symbols you need

The method for balancing

Balancing an alpha decay

Finding the daughter after beta-minus emission

Identifying an unknown emitted particle

Balancing a decay chain of two steps

Presenting a balanced equation

Review 6.3.3 Balancing nuclear equations by teaching Genie

6.3.3 Balancing nuclear equations Revision Guide

6.3.3 Balancing nuclear equations

What a nuclear equation records

The symbols you need

The method for balancing

Balancing an alpha decay

Finding the daughter after beta-minus emission

Identifying an unknown emitted particle

Balancing a decay chain of two steps

Presenting a balanced equation

Review 6.3.3 Balancing nuclear equations by teaching Genie

6.3.3 Balancing nuclear equations Revision Guide