How to calculate relative atomic mass

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, carbon-12, 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 = 1/12th 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).
  • In other words the relative atomic mass of an element is now based on the arbitrary value of the carbon-12 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 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! 

Video Tutorial:
  • 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.
How do I calculate relative atomic mass?
  • 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).
    • 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 ...
  • ... 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 ignore      



  • 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!)

    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