DEFINING &
CALCULATING RELATIVE ATOMIC MASS
Doc
Brown's Chemistry  GCSE/IGCSE/GCE (basic A level)
O Level
Online Chemical Calculations
1.
Explaining and calculating relative atomic mass RAM or A_{r}
and relative isotopic mass
Quantitative Chemistry
calculations online Help for problem solving
in relative atomic mass calculations. Definitions of relative atomic
mass and relative isotopic mass (A level students only) Practice revision questions on
working out relative atomic mass from isotopic composition (% isotopes,
A level students will learn about very accurate mass spectrometer data). What is relative atomic
mass? How do you calculate the relative atomic mass of an element.
What is the standard mass unit? Relative atomic mass is explained below, with reference to the carbon12
atomic mass scale and the relevance of isotopes and 'u' the unified
atomic mass unit is explained. Detailed examples of
the method of how to calculate relative atomic mass from the isotopic composition are fully explained
with reference to the definition of the relative atomic mass of a
compound. For A level students, how to define and use relative isotopic
masses to calculate relative atomic mass. These notes on defining, explaining and calculating relative
atomic mass and defining relative isotopic mass are
designed to meet the highest standards of knowledge and understanding required
for students/pupils doing GCSE chemistry, IGCSE chemistry, O
Level chemistry, KS4 science courses and A Level chemistry courses.
Spotted any careless error?
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GCSE calculation not covered?
Selfassessment Quizzes:
type in answer
H
only or multiple choice
H only

1.
Explaining and how to calculate the relative atomic mass RAM or A_{r} of
an element
How to calculate relative atomic mass
Introduction
 Every atom has its own unique relative atomic
mass (RAM) based on a standard comparison or relative scale
e.g. it has been based on hydrogen H = 1 amu and oxygen O = 16 amu in the past
(amu = relative atomic mass unit).

The relative atomic
mass scale is now based on an isotope of carbon, namely, carbon12,
nuclide symbol
,
which is given the value of 12.0000 amu.
 The unit 'amu' is now being replaced by a
lower case u, where u is the symbol for the unified atomic mass
unit.
 Therefore one atom of carbon, isotopic mass
12, equals 12 u, or,
 1 u = ^{1}/_{12}th the
mass of one atom of the carbon12 isotope.
 Note that for the standard nuclide notation,
, the top
left number is the mass number (12) and the bottom left number is the
atomic/proton number (6).
 In other words the relative atomic mass
of an element is now based on the arbitrary value of the carbon12 isotope
being assigned a mass of 12.0000 by international agreement!
 Examples are shown in the Periodic Table
diagram above.
 Note
 (i) Because of the presence of
neutrons in the nucleus, the relative atomic mass is usually at least double
the atomic/proton number because there are usually more neutrons than
protons in the nucleus (mass proton = 1, neutron = 1). Just scan the
periodic table above and examine the pairs of numbers.
 You should also notice that generally
speaking the numerical difference between the atomic/proton number and the
relative atomic mass tends to increase with increasing atomic number. This
has consequences for nuclear
stability.
 (ii) For many calculation
purposes, relative atomic masses are usually quoted and used at this
academic level to zero or one decimal place eg.
 e.g. hydrogen H = 1.0 or ~1, calcium Ca= 40.0 or
~40, chlorine Cl = 35.5, copper Cu = 63.6 or ~64, silver Ag 107.9 or ~108
etc.
 At A level, values of relative
atomic masses may be quoted to one or two decimal places.
 Many atomic masses are known to an accuracy
of four decimal places, but for some elements, isotopic composition varies
depending on the mineralogical source, so four decimal places isn't
necessarily more accurate!
 In using the symbol A_{r} for
RAM, you should bear in mind that the letter A on its own usually means the mass number of a particular isotope
and amu is the acronym shorthand for atomic mass units.
 However there are complications due to isotopes and
so very accurate atomic masses
are never whole integer numbers.
 Isotopes are atoms of the same element with different
masses due to different numbers of neutrons. The very accurate relative atomic mass scale
is based on a specific isotope of carbon, carbon12, ^{12}C = 12.0000
units exactly, for most purposes C = 12 is used for simplicity.
 For
example
hydrogen1,
hydrogen2, and
hydrogen3, are
the nuclide notation for the three isotopes of hydrogen, though the vast majority of hydrogen atoms have
a mass of 1. When their accurate isotopic masses, and their % abundance are
taken into account the average accurate relative mass for hydrogen =
1.008, but for most purposes H = 1 is good enough!
 The strict definition of relative
atomic mass (A_{r}) is that it equals the average mass of all the
isotopic atoms present in the element compared to ^{1}/_{12}th
the mass of a carbon12 atom (relative isotopic mass of 12.0000).
 So, in calculating relative atomic mass you
must take into account the
different isotopic masses of the same elements, but also their %
abundance in the element.
 Therefore you need to know the
percentage (%) of each isotope of an element in order to accurately
calculate the element's relative atomic mass.
 For approximate calculations of relative
atomic mass you can just use the mass numbers of the isotopes, which are
obviously all integers ('whole numbers'!) e.g. in the two calculations
below.
 To the nearest whole number, isotopic
mass = mass number for a specific isotope.
Examples of relative atomic mass calculations
for GCSE/IGCSE/AS level students
How do I calculate relative atomic mass?
 For accurate chemical calculations relative atomic
mass must be used and not an individual mass number.
 Therefore relative atomic mass takes into account all
the different 'stable' isotopes of an element which are naturally present.
 The relative atomic mass is the average mass and is
quite easily calculated from the percentage composition (% abundance).
 The presence of isotopes accounts for why some
relative atomic masses are not even close to a whole number.
 Some relative atomic masses are nearly whole numbers
due to coincidence of % isotopes, others because one isotope might dominate
the composition with only tiny amounts of lighter or heavier isotopes.

Example 1.1 Calculating the relative atomic mass of bromine
and
 bromine consists of
two isotopes, 50% ^{79}Br and 50% ^{81}Br, calculate the A_{r} of bromine
from the mass numbers (top left numbers).
 Think of the calculation in terms of '100 atoms'
 A_{r} = [ (50 x 79) + (50
x 81) ] /100 = 80
 So the relative atomic mass of
bromine is 80 or RAM or A_{r}(Br) = 80
 Note the full working shown. Yes, ok, you can do it in your head BUT many students ignore the %'s and
just average all the isotopic masses (mass numbers) given, in this case
bromine79 and bromine81.
 The element bromine is the only case I know where averaging
the isotopic masses actually works! so beware!

Example 1.2 Calculating the relative atomic mass of chlorine
and
 chlorine consists of
two isotopes, 75% chlorine35 and 25% chlorine37, so using
these two mass numbers ...
 ... again think of the data based on 100
atoms, so 75 have a mass of 35 and 25 atoms have a mass of 37.
 The average mass = [ (75 x 35) +
(25 x 37) ] / 100 = 35.5
 So the relative atomic mass of
chlorine is 35.5 or RAM or A_{r}(Cl) = 35.5
 Note: ^{35}Cl and ^{37}Cl are the most common isotopes of chlorine, but, there
are tiny percentages of other chlorine isotopes which are usually
ignored at GCSE/IGCSE and Advanced GCE AS/A2 A level.
 Example 1.3: Calculating the relative atomic
mass of copper from its isotopic composition (isotope abundance)
 Naturally occurring copper consists of 69.2% copper63
(^{63}Cu) and 30.8% copper65 (^{65}Cu)
 Still think in terms of 100 atoms and don't be put
off by decimal fractions, it still works out correctly because 69.2 + 30.8 =
100!
 average mass = relative atomic mass of copper
= {(63 x 69.2) + (65 x 30.8)} / 100 = 63.6
 See below and
mass Spectrometer and isotope analysis
on the GCSEAS(basic) Atomic Structure Notes, with further RAM calculations.
Examples for Advanced Level Chemistry students only
How to calculate relative atomic mass with accurate relative
isotopic masses
Using data from modern very accurate mass spectrometers
(a)
Accurate calculation of relative atomic mass
(need to know and define what relative isotopic mass is)
Relative
isotopic mass
is defined as the accurate mass of a single isotope of
an element compared to ^{1}/_{12}th the mass of a
carbon12 atom e.g. the accurate relative isotopic mass of the cobalt5
is 58.9332
This definition of relative isotopic mass is
a completely different from the definition of relative atomic mass, except
both are based on the same international standard of atomic mass i.e. 1 unit
(1 u)
= 1/12th the mass of a carbon12 isotope (^{12}C).
If we were to redo the calculation of the
relative atomic mass of chlorine (example
1.1 above), which is quite adequate for GCSE purposes (and maybe A level too),
but more accurately at A
level, we might do ....
chlorine is 75.77% ^{35}Cl of
isotopic mass 34.9689 and 24.23% ^{37}Cl of isotopic mass 36.9658
so A_{r}(Cl) = [(75.77 x
34.9689) + (24.23 x 36.9658)] / 100
= 35.4527 (but 35.5 is usually ok in calculations preuniversity!)
See also
Mass Spectrometer and isotope analysis
on the GCSEAS(basic) Atomic Structure Notes, with further RAM calculations.
(b)
Calculations of % composition of isotopes
It is possible to do the reverse
of a relative atomic mass calculation if you know the A_{r} and
which isotopes are present.
It involves a little bit of
arithmetical algebra.
The A_{r} of boron is
10.81 and consists of only two isotopes, boron10 and boron11
The relative atomic mass of
boron was obtained accurately in the past from chemical analysis of reacting
masses but now
mass spectrometers can sort
out all of the isotopes present and their relative abundance.
If you let X = % of boron
10, then 100X is equal to % of boron11
Therefore A_{r}(B) = (X
x 10) + [(100X) x 11)] / 100 = 10.81
so, 10X 11X +1100
=100 x 10.81
X + 1100 = 1081, 1100 
1081 = X (change sides change sign!)
therefore X = 19
so naturally occurring boron
consists of 19% ^{10}B and 81% ^{11}B
(the
data books actually quote 18.7 and 81.3, but we didn't use the very accurate
relative isotopic masses mentioned above!)
On other pages
Atomic structure and Relative Formula
Mass
Selfassessment Quizzes
[ram]
type in answer
H
only or multiple choice
H only
OTHER CALCULATION PAGES

What is relative atomic mass?,
relative isotopic mass & calculating relative atomic mass
(this page)

Calculating relative
formula/molecular mass of a compound or element molecule

Law of Conservation of Mass and simple reacting mass calculations

Composition by percentage mass of elements
in a compound

Empirical formula and formula mass of a compound from reacting masses
(easy start, not using moles)

Reacting mass ratio calculations of reactants and products
from equations
(NOT using
moles) and brief mention of actual percent % yield and theoretical yield,
atom economy
and formula mass determination

Introducing moles: The connection between moles, mass and formula mass  the basis of reacting mole ratio calculations
(relating reacting masses and formula
mass)

Using
moles to calculate empirical formula and deduce molecular formula of a compound/molecule
(starting with reacting masses or % composition)

Moles and the molar volume of a gas, Avogadro's Law

Reacting gas volume
ratios, Avogadro's Law
and GayLussac's Law (ratio of gaseous
reactantsproducts)

Molarity, volumes and solution
concentrations (and diagrams of apparatus)

How to do acidalkali
titration calculations, diagrams of apparatus, details of procedures

Electrolysis products calculations (negative cathode and positive anode products)

Other calculations
e.g. % purity, % percentage & theoretical yield, dilution of solutions
(and diagrams of apparatus), water of crystallisation, quantity of reactants
required, atom economy

Energy transfers in physical/chemical changes,
exothermic/endothermic reactions

Gas calculations involving PVT relationships,
Boyle's and Charles Laws

Radioactivity & halflife calculations including
dating materials
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