Atomic masses, isotopic abundance and composition

Separate a count from a measured mass

The mass number A is an integer count of protons and neutrons in one nucleus. Atomic mass is the mass of an atom; a convenient unit is the dalton, Da, also called the unified atomic mass unit, u. One dalton is defined using one twelfth of the mass of a free neutral carbon-12 atom at rest in its ground state. Atomic masses expressed in Da generally do not equal their integer mass numbers.

Relative atomic mass compares an atomic mass with that reference unit, so it is dimensionless. The same numerical value can therefore appear as “35.45 Da” for a mean atomic mass and “35.45” for the corresponding relative atomic mass. State which quantity you mean. A periodic-table atomic-weight value often represents an isotope-weighted mean rather than the mass of one isotope.

Weight each mass by its fraction

Suppose a specified chlorine sample uses the representative isotope fractions 0.7576 for chlorine-35 and 0.2424 for chlorine-37. Those fractions add to one. Using isotope masses about 34.968852682 Da and 36.965902602 Da, its mean is the first mass times 0.7576 plus the second mass times 0.2424. The more abundant isotope contributes more strongly to the average.

The result is about 35.45294 Da, often rounded to 35.45 Da for an introductory calculation. This is a number-weighted average over the atoms, not a mass-weighted average over portions weighed separately. Equal weighting would be appropriate only if the isotope number fractions were equal. Always make the fractions and source explicit.

The average does not create a third isotope

No individual chlorine atom in this two-isotope example has the calculated mean mass of 35.45294 Da. Each has the mass of its isotope, while the sample mean lies between those masses. The average summarizes a population. This is the same mathematical distinction as an average of two different measurements not necessarily matching either individual measurement.

The difference between an isotope mass and the mass number also matters. Chlorine-35 is not exactly 35 Da. Replacing measured isotope masses with 35 and 37 can be useful for a rough illustration, but it produces a different approximation and should be labelled as such. Keep accurate inputs when the numerical question supplies them.

State the sample and the rounding

Natural isotope fractions can vary between materials. CIAAW gives chlorine a standard atomic-weight interval [35.446, 35.457], reflecting this variation. The representative fractions used above describe one source-specific example, not exact universal proportions for every chlorine sample. A table value or interval must be read with its stated purpose.

A formula mass adds the constituent atomic masses with the formula’s atom counts. With supplied values H = 1.008 Da and O = 16.00 Da, the calculated mass for H₂O is 2(1.008) + 16.00 = 18.016 Da before final rounding. The next chapter connects such mass sums to molar mass and laboratory-scale amounts. Do not confuse the small-particle mass unit with grams per mole.

Worked examples

Try it yourself

1. Does mass number A have the unit Da?

   Show the answer and reasoning

   No.

   It is a dimensionless integer count of protons and neutrons.

2. Two isotope masses are 10 Da and 12 Da, each with number fraction 0.5. What is their mean?

   Show the answer and reasoning

   11 Da.

   The weighted mean is 10 × 0.5 + 12 × 0.5. This invented arithmetic example does not assert a particular real element.

3. Must one chlorine atom have exactly the sample’s mean mass?

   Show the answer and reasoning

   No.

   The mean describes the population; each atom has an isotope-specific mass.

4. Using H = 1.008 Da and O = 16.00 Da, give the unrounded H₂O formula-mass sum.

   Show the answer and reasoning

   18.016 Da.

   Two hydrogen contributions plus one oxygen contribution give 2 × 1.008 + 16.00.

What to carry forward

• Mass number is a count; atomic mass is a mass.
• Relative atomic mass is dimensionless.
• Isotope fractions weight a sample mean by number of atoms.
• Retain stated inputs, isotope-composition scope and rounding.