Calculating Relative Atomic Mass
Relative atomic mass (Ar) is the weighted average mass of an element's atoms, accounting for the natural abundances of its various isotopes. For the ESAT, you must be able to calculate this value quickly and accurately from percentage or mass spectrum data without a calculator.
Part of the ESAT Chemistry syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The relative atomic mass is a weighted mean, not a simple average; the abundance of each isotope is critical.
- Isotope abundances can be presented as percentages (totalling 100) or as relative values from a mass spectrum (which must be summed to find the total).
- The mass of each isotope in these calculations is taken to be its mass number.
- Ar is rarely a whole number because it's an average of different isotope masses.
- The 'relative' in the term signifies that all atomic masses are compared to a standard: 1/12th the mass of a carbon-12 atom.
Formulae
Ar = ( (%1 × mass1) + (%2 × mass2) + ⋯ ) / 100 When the abundances of isotopes are given as percentages.
Ar = ( (abundance1 × mass1) + (abundance2 × mass2) + ⋯ ) / (total abundance) When data is from a mass spectrum or given as relative abundances/ratios. The 'total abundance' is the sum of all the individual abundances.
Definitions
- Relative Atomic Mass (Ar)
- The weighted mean mass of an atom of an element, compared to one-twelfth of the mass of an atom of carbon-12.
- Isotopes
- Atoms with the same number of protons but different numbers of neutrons. They are the same element but have different mass numbers.
Worked example
A mass spectrometer analysis of a sample of element X shows two peaks. One peak at mass-to-charge ratio 85 has a relative abundance of 70. The other peak at mass-to-charge ratio 87 has a relative abundance of 30. Calculate the relative atomic mass of element X.
- 1
Step 1:
Identify the mass and relative abundance of each isotope from the data.
Isotope 1:
mass = 85, abundance = 70Isotope 2:
mass = 87, abundance = 30 - 2
Step 2:
Since the data is in relative abundances, find the total abundance by summing the individual abundances.
Total abundance = 70 + 30 = 100.
- 3
Step 3:
Apply the weighted average formula.
Ar = [ (abundance1 × mass1) + (abundance2 × mass2) ] / total abundance - 4
Step 4:
Substitute the values and calculate.
Ar = [ (70 × 85) + (30 × 87) ] / 100 - 5
Step 5:
Perform the multiplications:
70 × 85 = 595030 × 87 = 2610 - 6
Step 6:
Sum the results and divide by the total abundance:
Ar = (5950 + 2610) / 100 = 8560 / 100 - 7
Step 7:
Final calculation:
Ar = 85.6
Answer: 85.6
Common mistakes
- ×Simple arithmetic errors in multiplication or addition are the most frequent mistake. Always write out your steps clearly and double-check your working.
- ×Using the wrong denominator. If given percentages, divide by 100. If given relative abundances (e.g. from a mass spectrum), you must remember to first sum the abundances to find the total to divide by.
- ×Calculating a simple average of the masses instead of a weighted average, completely ignoring the abundance data.
No-calculator tips
- ✓Before calculating, estimate the answer. If an element is 75% isotope A and 25% isotope B, the final answer will be three-quarters of the way from B to A. This helps you spot an unreasonable result.
- ✓Break down complex multiplications. For example, 35 x 63 can be calculated as (35 x 60) + (35 x 3) = 2100 + 105 = 2205.
- ✓When working with fractions from relative abundances, look for common factors to simplify before you multiply. E.g., for (35*63 + 15*65)/50, you can divide the top and bottom by 5 to get (7*63 + 3*65)/10, which involves smaller numbers.