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 Ar 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 carbon-12 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.

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Self-assessment Quizzes on relative atomic mass:

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study examples carefully1. Explaining and how to calculate the relative atomic mass RAM or Ar of an elementstudy examples carefully

What is the relative atomic mass of an element? What scale is relative atomic mass based on? What is the formula to work out the relative atomic mass 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).
  • top sub-indexThe relative atomic mass scale is now based on an isotope of carbon, namely, carbon-12, nuclide symbol , which is given the arbitrary value of 12.0000 amu by international agreement.
    • 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 = 1/12th the mass of one atom of the carbon-12 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).
  • Since the relative atomic mass of an element is now based on the carbon-12 isotope it can now be defined as ...
    • ... relative atomic mass equals the average mass of all the atoms in an element compared to 1/12th the mass of a carbon-12 atom (carbon-12 isotope).
    • 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 (GCSE/IGCSE/O level) to zero or one decimal place eg.
        • e.g. hydrogen H = 1.008 or ~1; calcium Ca = 40.08 or ~40.0; chlorine Cl = 35.45 ~35.5, copper Cu = 63.55 or ~63.5/64, silver Ag = 107.9 or ~108 etc.
      • At Advanced 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 Ar 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, carbon-12, 12C = 12.0000 units exactly, for most purposes C = 12 is used for simplicity.
    • For example hydrogen-1, hydrogen-2, and hydrogen-3, 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!
    • (c) doc b See also GCSE/IGCSE/AS Atomic Structure Notes
  • The strict definition of relative atomic mass (Ar) is that it equals the average mass of all the isotopic atoms present in the element compared to 1/12th the mass of a carbon-12 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.

Above is typical periodic table used in GCSE science-chemistry specifications

and I've 'usually' used these values in my exemplar calculations to cover most syllabuses

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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% 79Br and 50% 81Br, calculate the Ar of bromine from the mass numbers (top left numbers).
    • Think of the calculation in terms of '100 atoms'
    • Ar = [ (50 x 79) + (50 x 81) ] /100 = 80
    • So the relative atomic mass of bromine is 80 or RAM or Ar(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 bromine-79 and bromine-81.
    • 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% chlorine-35 and 25% chlorine-37, 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 Ar(Cl) = 35.5
    • Note: 35Cl and 37Cl 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% copper-63 (63Cu) and 30.8% copper-65 (65Cu)
    • 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 GCSE-Advanced A Level (basic) Atomic Structure Notes, with further relative atomic mass calculations.

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Examples for Advanced A 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 1/12th the mass of a carbon-12 atom e.g. the accurate relative isotopic mass of the cobalt-5 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 carbon-12 isotope (12C).

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% 35Cl of isotopic mass 34.9689 and 24.23% 37Cl of isotopic mass 36.9658

so Ar(Cl) = [(75.77 x 34.9689) + (24.23 x 36.9658)] / 100

= 35.4527 (but 35.5 is usually ok in calculations pre-university!)

See also (c) doc b Mass Spectrometer and isotope analysis on the GCSE-AS(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 Ar and which isotopes are present.

It involves a little bit of arithmetical algebra.

The Ar of boron is 10.81 and consists of only two isotopes, boron-10 and boron-11

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 100-X is equal to % of boron-11

Therefore Ar(B) = (X x 10) + [(100-X) 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% 10B and 81% 11B

(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

Self-assessment Quizzes on relative atomic mass

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APPENDIX 1. A typical periodic table used in pre-university examinations

Above is typical periodic table used in GCSE science-chemistry specifications in doing chemical calculations, and I've 'usually' used these values in my exemplar calculations to cover most syllabuses


APPENDIX 2. Table of relative atomic masses for elements 1 to 92

Notes: (i) The list of relative atomic mass are in alphabetical order by element name, together with chemical symbol and proton/atomic number.

(ii) The relative atomic masses are quoted to two decimal places, though it is essential to be aware that values in pre-university examinations might be rounded to the nearest integer or one decimal place.

(iii) Trans-uranium elements have been eliminated because their isotopic composition varies depending on the source e.g. cyclotron, nuclear reactor etc. AND all their isotopes are highly radioactive and most are very unstable (so your relative atomic mass changes all the time!)

(iv) * radioactive, mass number of most stable isotope quoted

Chemical Symbol

Element name

Atomic No. Z

Relative atomic mass
Ac Actinium 89 227.03
Al Aluminium/Aluminum 13 26.98
Sb Antimony 51 121.75
Ar Argon 18 39.95
As Arsenic 33 74.92
At Astatine 85 210 *
Ba Barium 56 137.33
Be Beryllium 4 9.01
Bi Bismuth 83 208.98
B Boron 5 10.81
Br Bromine 35 79.90
Cd Cadmium 48 112.41
Cs Caesium 55 132.91
Ca Calcium 20 40.08
C Carbon 6 12.01
Ce Cerium 58 140.12
Cl Chlorine 17 35.45
Cr Chromium 24 52.00
Co Cobalt 27 58.93
Cu Copper 29 63.55
Dy Dysprosium 66 162.50
Er Erbium 68 167.26
Eu Europium 63 151.97
F Fluorine 9 19.00
Fr Francium 87 223 *
Gd Gadolinium 64 157.25
Ga Gallium 31 69.72
Ge Germanium 32 72.60
Au Gold 79 196.97
Hf Hafnium 72 178.49
He Helium 2 4.00
Ho Holmium 67 164.93
H Hydrogen 1 1.01
In Indium 49 114.82
I Iodine 53 126.90
Ir Iridium 77 192.22
Fe Iron 26 55.85
Kr Krypton 36 83.80
La Lanthanum 57 138.91
Pb Lead 82 207.20
Li Lithium 3 6.94
Lu Lutetium 71 174.97
Mg Magnesium 12 24.31
Mn Manganese 25 54.94
Hg Mercury 80 200.59
       

Chemical Symbol

Element name

Atomic No. Z

Relative atomic mass
Mo Molybdenum 42 95.94
Nd Neodymium 60 144.24
Ne Neon 10 20.18
Ni Nickel 28 58.69
Nb Niobium 41 92.91
N Nitrogen 7 14.01
Os Osmium 76 190.20
O Oxygen 8 16.00
Pd Palladium 46 106.42
P Phosphorus 15 30.97
Pt Platinum 78 195.08
Po Polonium 84 209 *
K Potassium 19 39.10
Pr Praseodymium 59 140.91
Pm Promethium 61 145 *
Pa Protactinium 91 231.04
Ra Radium 88 226.03
Rn Radon 86 222 *
Re Rhenium 75 186.21
Rh Rhodium 45 102.91
Rb Rubidium 37 85.47
Ru Ruthenium 44 101.07
Sm Samarium 62 150.36
Sc Scandium 21 44.96
Se Selenium 34 78.96
Si Silicon 14 28.09
Ag Silver 47 107.87
Na Sodium 11 23.00
Sr Strontium 38 87.62
S Sulphur/Sulfur 16 32.07
Ta Tantalum 73 180.95
Tc Technetium 43 98.91
Te Tellurium 52 127.60
Tb Terbium 65 158.93
Tl Thallium 81 204.38
Th Thorium 90 232.04
Tm Thulium 69 168.93
Sn Tin 50 118.71
Ti Titanium 22 47.88
W Tungsten 74 183.85
U Uranium 92 238.03
V Vanadium 23 50.94
Xe Xenon 54 131.29
Yb Ytterbium 70 173.04
Y Yttrium 39 88.91
Zn Zinc 30 65.39
Zr Zirconium 40 91.22

 


OTHER CALCULATION PAGES

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

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

  3. Law of Conservation of Mass and simple reacting mass calculations

  4. Composition by percentage mass of elements in a compound

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

  6. 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

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

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

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

  10. Reacting gas volume ratios, Avogadro's Law and Gay-Lussac's Law (ratio of gaseous reactants-products)

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

  12. How to do acid-alkali titration calculations, diagrams of apparatus, details of procedures

  13. Electrolysis products calculations (negative cathode and positive anode products)

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

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

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

  17. Radioactivity & half-life calculations including dating materials

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of 12 Magnesium Mg relative atomic mass of 13 Aluminium Al relative atomic mass of 14 Silicon Si relative atomic mass of 15 Phosphorus P relative atomic mass of 16 Sulphur S relative atomic mass of 17 Chlorine Cl relative atomic mass of 18 Argon Ar relative atomic mass of 19 Potassium K relative atomic mass of 20 Calcium Ca relative atomic mass of 21 Scandium Sc relative atomic mass of 22 Titanium Ti relative atomic mass of 23 Vanadium V relative atomic mass of 24 Chromium Cr relative atomic mass of 25 Manganese Mn relative atomic mass of 26 Iron Fe relative atomic mass of 27 Cobalt Co relative atomic mass of 28 Nickel Ni relative atomic mass of 29 Copper Cu  relative atomic mass of 30 Zinc Zn relative atomic mass of 31 Gallium Ga relative atomic mass of 32 Germanium Ge relative atomic mass of 33 Arsenic As relative atomic mass of 34 Selenium Se relative atomic mass of 35 Bromine Br relative atomic mass of 36 Krypton Kr relative atomic mass of 37 Rubidium Rb relative atomic mass of 38 Strontium Sr relative atomic mass of 39 Yttrium Y relative atomic mass of 40 Zirconium Zr relative atomic mass of 41 Niobium Nb  relative atomic mass of 42 Molybdenum Mo  relative atomic mass of 43 Technetium Tc relative atomic mass of 44 Ruthenium Ru relative atomic mass of 45 Rhodium Rh relative atomic mass of 46 Palladium Pd relative atomic mass of 47 Silver Ag relative atomic mass of 48 Cadmium Cd relative atomic mass of 49 Indium In relative atomic mass of 50 Tin Sn relative atomic mass of 51 relative atomic mass of Antimony Sb relative atomic mass of 52 Tellurium Te relative atomic mass of 53 Iodine I relative atomic mass of 54 Xenon Xe relative atomic mass of 55 Caesium Cs relative atomic mass of 56 Barium Ba relative atomic mass of 57 Lanthanum La relative atomic mass of 58 Cerium Ce relative atomic mass of 59 Praseodymium Pr relative atomic mass of 60 Neodymium Nd relative atomic mass of 61 Promethium Pm relative atomic mass of 62 Samarium Sm relative atomic mass of 63 Europium Eu relative atomic mass of 64 Gadolinium Gd relative atomic mass of 65 Terbium Tb relative atomic mass of 66 Dysprosium Dy relative atomic mass of 67 Holmium Ho relative atomic mass of 68 Erbium Er relative atomic mass of 69 Thulium Tm relative atomic mass of 70 Ytterbium Yb relative atomic mass of 71 Lutetium Lu relative atomic mass of 72 Hafnium Hf relative atomic mass of 73 Tantalum Ta relative atomic mass of 74 Tungsten W relative atomic mass of 75 Rhenium Re relative atomic mass of 76 Osmium Os relative atomic mass of 77 Iridium Ir relative atomic mass of 78 Platinum Pt relative atomic mass of 79 Gold Au relative atomic mass of 80 Mercury Hg relative atomic mass of 81 Thallium Tl relative atomic mass of 82 Lead Pb relative atomic mass of 83 Bismuth Bi relative atomic mass of 84 Polonium Po 85 Astatine At relative atomic mass of 86 Radon Rn relative atomic mass of 87 Francium Fr relative atomic mass of 88 Radium Ra relative atomic mass of 89 Actinium Ac relative atomic 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