 # Relative Atomic Mass Calculations Chemistry Tutorial

## Key Concepts

• The relative atomic mass of an element is also known as the relative atomic weight of an element, or, the atomic weight of an element.
• Relative atomic mass is often abbreviated as r.a.m.
• The relative atomic mass of an element (its atomic weight) is given in the Periodic Table.
• The relative atomic mass of an element is the weighted average of the masses of the isotopes in the naturally occurring element relative to the mass of an atom of the carbon-12 isotope which is taken to be exactly 12.
• The atomic mass unit (u) is defined as a mass equivalent to 1/12 of the mass of one atom of carbon-12.

1 u = 1.66 × 10-27 kg

• We can estmate the the relative atomic mass (atomic weight) of an element E with the naturally occurring isotopes aE, bE, cE, etc, and with the respective abundances of A%, B%, C% etc,

 relative atomic mass (r.a.m.) = ( A   100 × a) + ( B   100 × b) + ( C   100 × c) + etc

• Given the relative atomic mass (r.a.m.) of an element and the estimated mass of each of its isotopes, we can then estimate the relative abundance of each isotope:
let x = %abundance of isotope-a
and 100 - x = %abundance of isotope-b
then, let r.a.m = relative atomic mass of the element:

 r.a.m. = ( x   100 × mass isotope-a) + ( 100 - x 100 × mass isotope-b)

and solve for x

• Note that we can measure the mass of each isotope and its abundance using Mass Spectroscopy

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## Abundance of Naturally Occurring Isotopes

Most elements occur in nature as a mixture of different isotopes.

The element carbon, for example, exists in nature as a mixture of different isotopes: stable1 carbon-12 atoms and carbon-13 atoms.

If you were to take a sample of carbon atoms, for example the soot from a chimney or a lump of coal, you would find that most of the carbon atoms are the carbon-12 isotope and only a few would be the carbon-13 isotope.
We call this amount of each isotope found in the naturally occurring element its abundance, or its isotopic abundance to be more precise.
The abundance of the carbon-12 isotope in naturally occurring bulk carbon is 98.90% while the abundance of the carbon-13 isotope in nature is 1.10%
This means that if you take a lump of coal from nature, 98.90% of the carbon in the coal will be atoms of the carbon-12 isotope, while only 1.10% of it will be atoms of the carbon-13 isotope.

The table below gives the isotopic abundances for some elements on Earth:

Element Isotope Abundance (%) Element Isotope Abundance (%)
hydrogen 1H 99.99 sodium 23Na 100.00
2H 0.01 magnesium 24Mg 78.90
helium 3He 0.0001 25Mg 10.00
4He 99.9999 26Mg 11.10
lithium 6Li 7.42 aluminium 27Al 100.00
7Li 92.58 silicon 28Si 92.23
beryllium 9Be 100.00 29Si 4.67
boron 10B 19.80 30Si 3.10
11B 80.20 phosphorus 31P 100
carbon 12C 98.90 sulfur 32S 95.02
13C 1.10 33S 0.75
nitrogen 14N 99.63 34S 4.21
15N 0.37 36S 0.02
oxygen 16O 99.76 chlorine 35Cl 75.77
17O 0.038 37Cl 24.23
18O 0.20 argon 36Ar 0.34
fluorine 19F 100.00 38Ar 0.063
neon 20Ne 90.60 40Ar 99.60
21Ne 0.26
22Ne 9.20

You can find a more complete list of isotopic abundances at the bottom of this page.

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## Estimating Isotopic Mass

The relative atomic mass of a carbon-12 atom is defined as 12.00
The relative atomic mass of an atom of carbon-13 is found to be 1.08333 times the mass of a carbon-12 atom, that is 1.083 × 12 = 13.00

We can estimate the mass of any isotope of an element, its isotopic mass, using its mass number (A).

The mass number (A) of an isotope tells us how many protons and neutrons are in the nucleus of an atom of this isotope.
Nucleon is the term used to describe both protons and neutrons.
So, the mass number (A) tells us the number of nucleons in the nucleus of an atom of the isotope of the element.

For example:

• nucleus of an atom of the carbon-12 isotope contains 12 nucleons
• nucleus of an atom of the carbon-13 isotope contains 13 nucleons

The mass of a proton is almost exactly the same as the mass of a neutron.
The mass of a proton is about 1 u (1 atomic mass unit), so the mass of a neutron is also about 1 u.
The mass of an electron is so small compared to the mass of a proton or neutron that it can be ignored when estimating the mass of an isotope of an element.2

We can estimate the mass of an atom of an isotope of an element by adding together the mass of its nucleons:3

isotopic mass = number of nucleons × mass of nucleon

isotopic mass = number of nucleons × 1 u

For example:

• isotopic mass of carbon-12 atom: 12 nucleons × 1 u/nucleon = 12 u
• isotopic mass of carbon-13 atom: 13 nucleons × 1 u/nucleon = 13 u

Isotopic masses can be measured using mass spectroscopy. You will find a discussion of calculating relative atomic mass (atomic weight) using these measured isotopic masses in the Mass Spectroscopy tutorial.

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## Calculating the Relative Atomic Mass (Atomic Weight) of an Element

The relative atomic mass of an element is the weighted average of the masses of the isotopes in the naturally occurring element relative to the mass of an atom of the carbon-12 isotope which is taken to be exactly 12.

What is a "weighted average" ?

First lets look at what the "average weight" of carbon would be:

mass of carbon-12 isotope is 12 u

mass of carbon-13 isotope is 13 u

So we can calculate the average mass (average weight) of carbon as:

 average weight = 12 + 13 2 = 12.5 u

If we look up the atomic weight of carbon in the Periodic Table we find that it is 12.01 NOT 12.5

This is because most of the atoms found in naturally occurring carbon are atoms of the carbon-12 isotope while very few of the atoms will be of the carbon-13 isotope.
We need to take this into account when we calculate our "average", and when we do this we refer to the result as a "weighted average".

We need to know the abundance of each isotope, that is we need to know how much of the mass (weight) of the naturally occuring bulk carbon is due to each of the isotopes (carbon-12 and carbon-13).
From the table in the section above we find that the isotopic abundance of carbon-12 is 98.90% and the isotopic abundance of carbon-13 is 1.10%

This means that if I had 100 atoms of bulk carbon (like in coal or soot), then:

• 98.90% of the 100 carbon atoms are carbon-12 atoms with a mass of 12 u
That is, 98.90 of the carbon atoms have a mass of 12 u
• 1.10% of the 100 carbon atoms are carbon-13 atoms with a mass of 13 u
That is, 1.10 of the carbon atoms have a mass of 13 u

So, the total mass of 100 naturally occurring carbon atoms is:

 total mass of 100 atoms = mass of all carbon-12 atoms + mass of all carbon-13 atoms = 98.90 × 12 + 1.10 × 13 = 1186.8 + 14.3 = 1201.1

So the mass of 100 naturally occurring carbon atoms is 1201.1 u
Therefore the "weighted average mass" of 1 carbon atom is 1201.1 ÷ 100 = 12.011 u
This value for the atomic weight of carbon agrees with the value in the Periodic Table.

Note that this does NOT mean that the mass of 1 atom of carbon is 12.011 u
It DOES mean that if we take a sample of naturally occurrring carbon we would find that the average mass of a carbon atom would be 12.011 u

Let's review how we calculated the "weighted average" mass of bulk carbon atoms:

• abundance of the first isotope multiplied by the mass of this isotope given by the mass number
• abundance of the second isotope multiplied by the mass of this isotope given by the mass number
• add these two values together then divide by 100

In general, to calculate the "weighted average" mass of an element that occurs naturally as two different isotopes, isotope 1 and isotope 2, then:

 "weighted average" mass of element = (abundance isotope 1 × mass isotope 1) + (abundance isotope 2 × mass isotope 2) 100

If we estimate the mass of each isotope by using its mass number (A), then we can re-write the expression as:

 "weighted average" mass of element = (abundance isotope 1 × A(isotope 1)) + (abundance isotope 2 × A(isotope 2)) 100

Or, put another way:

 "weighted average" mass of element = ( %abundance isotope 1100 × A(isotope 1)) + ( %abundance isotope 2 100 × A(isotope 2))

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## Calculating Isotopic Abundance from Atomic Weight

The Periodic Table gives us the weighted average for the mass of an element, referred to as the element's atomic weight (or relative atomic mass).

If we know the mass of each isotope making up this naturally occurring element (estimated by its mass number), then we can calculate the abundance of each isotope in nature.

We wrote a general mathematical expression (mathematical equation) above for calculating the "weighted average" mass, also known as the relative atomic mass or atomic weight, of an element, which was:

 "weighted average" mass of element = (abundance isotope 1 × A(isotope 1)) + (abundance isotope 2 × A(isotope 2)) 100

Let's say I wanted to find the abundance (%) of each isotope of nitrogen.

Nitrogen has two naturally occurring stable isotopes: nitrogen-14 and nitrogen-15.

So, substituting these into the mathematical equation above I get:

 "weighted average" mass of nitrogen = (abundance nitrogen-14 × A(nitrogen-14)) + (abundance nitrogen-15 × A(nitrogen-15)) 100

I can look up the atomic weight (relative atomic mass, or, "weighted average" mass) of nitrogen in the Periodic Table:

atomic weight of nitrogen is 14.01

I can estimate the mass of an atom of each isotope by using its mass number (A):

The mass of an atom of nitrogen-14 = its mass number = 14 u

The mass of an atom of nitrogen-15 = its mass number = 15 u

I can substitute these values into the mathematical equation above:

 14.01 = (abundance nitrogen-14 × 14) + (abundance nitrogen-15 × 15) 100

I can multiply both sides of the mathematical equation by 100:

 100 × 14.01 = 100 × (abundance nitrogen-14 × 14) + (abundance nitrogen-15 × 15) 100 1401 = (abundance nitrogen-14 × 14) + (abundance nitrogen-15 × 15)

But how can I solve this equation when there are 2 unknowns, the abundance of nitrogen-14 is unknown and the abundance of nitrogen-15 is unknown.

The trick is to remember that we are talking about percentage abundance! Which means that:

%abundance of nitrogen-14 + %abundance of nitrogen-15 = 100

Or, put a different way:

%abundance of nitrogen-14 = 100 - %abundance of nitrogen-15

So, if I let the %abundance of nitrogen-14 be equal to x then:

%abundance of nitrogen-14 = x

%abundance of nitrogen-15 = 100 - x

If I substitute these into the mathematical equation above I will have only 1 unknown value, x:

 1401 = (abundance nitrogen-14 × 14) + (abundance nitrogen-15 × 15) 1401 = (x × 14) + ([100 - x] × 15)

To solve for x I will clear the brackets first:

 1401 = (x × 14) + ([100 - x] × 15) 1401 = 14x + (15 × 100) - 15x 1401 = 14x + 1500 - 15x

Next I collect like terms, starting with x:

 1401 = 14x + 1500 - 15x 1401 = 14x - 15x + 1500 1401 = -x + 1500

Then subtract 1500 from both sides of the equation:

 1401 - 1500 = -x + 1500 - 1500 -99 = -x

Note that the abundance of an isotope must be a positive number (not a negative number), so I divide both sides of the equation by -1 to find the value of x :

 -99 -1 = -x -1 99 = x

Then substitute this value for x back into the expressions we wrote for the abundance of each isotope:

%abundance of nitrogen-14 = x = 99 %

%abundance of nitrogen-15 = 100 - x = 100 - 99 = 1%

In general, if an element occurs in nature in 2 isotopic forms, isotope 1 and isotope 2, then we can estimate the percentage abundance of each isotope using the mass number (A) of each isotope because the %abundance of isotope 2 equals 100 - %abundance of isotope 1:

 100 × atomic weight of element = (abundance isotope 1 × A(isotope 1)) + ([100 - abundance isotope 1] × A(isotope 2))

let abundance isotope 1 = x

 100 × atomic weight of element = (x × A(isotope 1)) + ([100 - x] × A(isotope 2)) 100 × atomic weight of element = x A(isotope 1) + 100A(isotope 2) - A(isotope 2)x (100 × atomic weight of element) - (100 × A(isotope 2)) = x A(isotope 1) - xA(isotope 2) 100 × (atomic weight of element - A(isotope 2)) = x(A(isotope 1) - A(isotope 2)) 100 × (atomic weight of element - A(isotope 2))(A(isotope 1) - A(isotope 2)) = x

So,

• %abundance isotope 1 = x
• %abundance isotope 2 = 100 - x

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## Worked Example 1: Calculating the Atomic Weight of an Element

Question: Naturally occurring silver is 51.84% silver-107 and 48.16% silver-109.

Calculate the atomic weight of silver.

Solution:

(Based on the StoPGoPS approach to problem solving.)

1. What is the question asking you to do?

Calculate the "weighted average" mass (relative atomic mass, or, atomic weight) of silver

2. What data (information) have you been given in the question?

Extract the data from the question:

isotope name mass number (A) isotopic abundance (%)
silver-107 107 51.84
silver-109 109 48.16
3. What is the relationship between what you know and what you need to find out?
(a) mass of an isotope can be estimated using its mass number (A) expressed in atomic mass units, u

(b)

 "weighted average" mass of silver = ( %abundance silver-107100 × A(silver-107)) + ( %abundance silver-109 100 × A(silver-109))

4. Substitute in the values and solve:

 "weighted average" mass of silver = ( 51.841 × 107) + ( 48.16 100 × 109) = 55.469 + 52.494 = 107.96

Do a rough calculation:
Since about half (≈50%) of the atoms of silver are silver-107 and the other half (≈50%) are silver-109, the weighted average mass for silver will lie about half-way between the mass of silver-107 (107) and silver-109 (109):
atomic weight ≈ ½ × (107 + 109) = 108
Since our calculated value of 107.96 is close to 108 we are reasonably confident that our answer is correct.
You can also look up the atomic weight of silver in the periodic table and find that it is 107.9.
Since our calculated value of 107.96 is about the same as the value in the periodic table we are reasonably confident that our answer is plausible.
6. State your solution to the problem "atomic weight of silver":

atomic weight of silver is 107.96 u

## Worked Example 2: Calculating an Element's Isotopic Abundance

Question: Copper consists of two isotopes, copper-63 and copper-65.
Its atomic weight (relative atomic mass) is given as 63.62 u

Find the % abundance of each isotope.

Solution:

(Based on the StoPGoPS approach to problem solving.)

1. What is the question asking you to do?

(a) Calculate the % abundance of copper-63

(b) Calculate the % abundance of copper-65

2. What data (information) have you been given in the question?

Extract the data from the question:

isotope name mass number (A) approximate mass of isotope (u)
copper-63 63 63
copper-65 65 65

atomic weight of copper = 63.62 u

3. What is the relationship between what you know and what you need to find out?

(a)

 "weighted average" mass of copper = ( %abundance copper-63100 × A(copper-63)) + ( %abundance copper-65 100 × A(copper-65))

(b) let %abundance copper-63 = x
and %abundance copper-65 = 100 - x

4. Substitute the values into the equation and solve:

(a)

 63.62 = ( x100 × 63) + ( 100 - x 100 × 65) 63.62 = 63x100 + (65 × 100) - 65x 100 63.62 = 63x100 + 6500 - 65x 100 63.62 = 63x + 6500 - 65x 100 100 × 63.62 = 100 × 63x + 6500 - 65x 100 6362 = 63x + 6500 - 65x 6362 = -2x + 6500 6362 - 6500 = -2x -138 = -2x -138 -2 = -2x -2 69 = x

(b) %abundance copper-63 = x = 69 %
%abundance copper-65 = 100 - x = 100 - 69 = 31 %

The atomic weight of copper is given as 63.62.
Because this weight is closer to that of isotopic mass of copper-63 than it is to that of the isotopic mass of copper-65 we know that copper-63 must be more abundant (greater than 50%).
Our calculated value for the abundance of copper-63 is 69% which is greater than 50% so we are reasonably confident that our answer is plausible.
Note that the %abundance of both isotopes must add up to 100%.
69% + 31% = 100%
So we are reasonably confident that our calculated values for the abundance of each isotope is plausible.
6. State your solution to the problem "abundance of each isotope of copper":

abundance of copper-63 = 69%

abundance of copper-65 = 31%

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Footnotes

1. A stable isotope is one that does not undergo radioactive decay (nuclear decay).
An unstable isotope is one that does undergo radioactive decay, and therefore, the abundance of a naturally occurring unstable isotope will decrease over time ....
Isotopic abundances can change over time, if a radioactive isotope decays to produce a stable isotope of a different element then the isotopic abundance of this stable isotope will increase over time.
Man-made nuclear reactions will also change the isotopic abundance.
There is a discussion of the variation in isotopic abundance in the Carbon-14 Dating tutorial.

2. Mass of a proton = 1.673 × 10-27 kg = 1.01 u
Mass of a neutron = 1.675 × 10-27 kg = 1.01 u
Mass of an electron = 9.109 × 10-31 kg = 0.000549 u

3. In effect we are estimating the mass of the nucleus rather than the atom (since we are ignoring the mass of the electrons which contribute very little to the mass of an atom).
When we measure the mass of a nucleus we find the observed mass is less than the sum of the masses of all the nucleons. This is called the mass defect.

Isotopic Abundance Data

Element Isotope Abundance (%)
hydrogen 1H 99.99
2H 0.01
helium 3He 0.0001
4He 99.9999
lithium 6Li 7.42
7Li 92.58
beryllium 9Be 100.00
boron 10B 19.80
11B 80.20
carbon 12C 98.90
13C 1.10
nitrogen 14N 99.63
15N 0.37
oxygen 16O 99.76
17O 0.038
18O 0.20
fluorine 19F 100.00
neon 20Ne 90.60
21Ne 0.26
22Ne 9.20
sodium 23Na 100
magnesium 24Mg 78.90
25Mg 10.00
26Mg 11.10
aluminium 27Al 100
silicon 28Si 92.23
29Si 4.67
30Si 3.10
phosphorus 31P 100
sulfur 32S 95.02
33S 0.75
34S 4.21
36S 0.02
chlorine 35Cl 75.77
37Cl 24.23
argon 36Ar 0.34
38Ar 0.063
40Ar 99.60
potassium 39K 93.20
40K 0.012
41K 6.73
calcium 40Ca 96.95
42Ca 0.65
43Ca 0.14
44Ca 2.086
46Ca 0.004
48Ca 0.19
scandium 45Sc 100
titanium 46Ti 8.00
47Ti 7.30
48Ti 73.80
49Ti 5.50
50Ti 5.40
51V 99.75
chromium 50Cr 4.35
52Cr 83.79
53Cr 9.50
54Cr 2.36
manganese 55Mn 100
iron 54Fe 5.80
56Fe 91.72
57Fe 2.20
58Fe 0.28
cobalt 59Co 100
nickel 58Ni 68.27
60Ni 26.10
61Ni 1.13
62Ni 3.59
64Ni 0.91
copper 63Cu 69.17
65Cu 30.83
zinc 64Zn 48.60
66Zn 27.90
67Zn 4.10
68Zn 18.80
70Zn 0.60
gallium 69Ga 60.10
71Ga 39.90
germanium 70Ge 20.50
72Ge 27.40
73Ge 7.80
74Ge 36.50
76Ge 7.80
arsenic 75As 100
selenium 74Se 0.90
76Se 9.00
77Se 7.60
78Se 23.50
80Se 49.60
82Se 9.40
bromine 79Br 50.69
81Br 49.31
krypton 78Kr 0.35
80Kr 2.25
82Kr 11.60
83Kr 11.50
84Kr 57.00
86Kr 17.30
rubidium 85Rb 72.17
87Rb 27.84
strontium 84Sr 0.56
86Sr 9.86
87Sr 7.00
88Sr 82.58
yttrium 89Y 100
zirconium 90Zr 51.45
91Zr 11.27
92Zr 17.17
94Zr 17.33
96Zr 2.78
niobium 93Nb 100
molybdenum 92Mo 14.84
94Mo 9.25
95Mo 15.92
96Mo 16.68
97Mo 9.55
98Mo 24.13
100Mo 9.63
ruthenium 96Ru 5.52
98Ru 1.88
99Ru 12.70
100Ru 12.60
101Ru 17.00
102Ru 31.60
104Ru 18.70
rhodium 103Rh 100
104Pd 11.14
105Pd 22.33
106Pd 27.33
108Pd 24.46
110Pd 11.72
silver 107Ag 51.84
109Ag 48.16
108Cd 0.89
110Cd 12.49
111Cd 12.80
112Cd 24.13
113Cd 12.22
114Cd 28.73
116Cd 7.49
indium 113In 4.30
115In 95.70
tin 112Sn 0.97
114Sn 0.65
115Sn 0.36
116Sn 14.70
117Sn 7.70
118Sn 24.30
119Sn 8.60
120Sn 32.40
122Sn 4.60
124Sn 5.60
antimony 121Sb 57.30
123Sb 42.70
tellurium 120Te 0.096
122Te 2.60
123Te 0.91
124Te 4.82
125Te 7.14
126Te 18.95
128Te 31.69
130Te 33.80
iodine 127I 100
xenon 124Xe 0.10
126Xe 0.09
128Xe 1.91
129Xe 26.40
130Xe 4.10
131Xe 21.20
132Xe 26.90
134Xe 10.40
136Xe 8.90
cesium 133Cs 100
barium 130Ba 0.11
132Ba 0.10
134Ba 2.42
135Ba 6.59
136Ba 7.85
137Ba 11.23
138Ba 71.70
lanthanum 138La 0.09
139La 99.91
cerium 136Ce 0.19
138Ce 0.25
140Ce 88.48
142Ce 11.08
praseodymium 141Pr 100
neodymium 142Nd 27.13
143Nd 12.18
144Nd 23.80
145Nd 8.30
146Nd 17.19
148Nd 5.76
150Nd 5.64
samarium 144Sm 3.10
147Sm 15.00
148Sm 11.30
149Sm 13.80
150Sm 7.40
152Sm 26.70
154Sm 22.70
europium 151Eu 47.80
153Eu 52.20
154Gd 2.18
155Gd 14.80
156Gd 20.47
157Gd 15.65
158Gd 24.84
160Gd 21.86
terbium 159Tb 100
dysprosium 156Dy 0.06
158Dy 0.10
160Dy 2.34
161Dy 18.90
162Dy 25.50
163Dy 24.90
164Dy 28.20
holmium 165Ho 100
erbium 162Er 0.14
164Er 1.61
166Er 33.60
167Er 22.95
168Er 26.80
170Er 14.90
thulium 169Tm 100
ytterbium 168Yb 0.13
170Yb 3.05
171Yb 14.30
172Yb 21.90
173Yb 16.12
174Yb 31.80
176Yb 12.70
lutetium 175Lu 97.40
176Lu 2.60
hafnium 174Hf 0.16
176Hf 5.20
177Hf 18.60
178Hf 27.10
179Hf 13.74
180Hf 35.20
tantalum 180Ta 0.012
181Ta 99.99
tungsten 180W 0.013
182W 26.30
183W 14.30
184W 30.67
186W 28.60
rhenium 185Re 37.40
187Re 62.60
osmium 184Os 0.02
186Os 1.58
187Os 1.60
188Os 13.30
189Os 16.10
190Os 26.40
192Os 41.00
iridium 191Ir 37.30
193Ir 62.70
platinum 190Pt 0.01
192Pt 0.79
194Pt 32.90
195Pt 33.80
196Pt 25.30
198Pt 7.20
gold 197Au 100
mercury 196Hg 0.15
198Hg 10.10
199Hg 17.00
200Hg 23.10
201Hg 13.20
202Hg 29.65
204Hg 6.80
thallium 203Tl 29.52
205Tl 70.48