Molar Mass

Molar mass is the mass of one mole of a substance, in grams per mole. Computing one is addition and multiplication with no concepts hidden inside it, which is why it is often the first thing a course teaches after the mole — and why it is so frustrating to keep getting wrong.

Almost every lost mark comes from one of four places: mis-reading a formula, forgetting that seven elements are not single atoms, rounding too early, or copying the atomic number out of the periodic table instead of the atomic weight. This page works through each.

Two numbers that happen to be the same

Carbon's entry on the periodic table reads 12.011. That single number answers two different questions:

• the average mass of one carbon atom, in unified atomic mass units (u, also called daltons, Da);
• the mass of one mole of carbon atoms, in grams.

They agree because the mole was sized to make them agree. The u is defined as one twelfth of the mass of a carbon-12 atom, and the mole was originally defined as the number of atoms in 12 g of carbon-12 — so scaling up by a mole converts u into grams with a factor of essentially exactly one. Since the 2019 revision of the SI the correspondence is no longer exact by definition, but it holds to about one part in a billion, which is nine significant figures further than any school calculation goes.

The practical upshot: you never convert units when computing a molar mass. You read numbers off the table, multiply by counts, add, and write g/mol on the end.

Adding up Ca(NO₃)₂ without losing a subtotal

The arithmetic is easy. Keeping track of which subtotal belongs to which element is the part that fails. Lay it out in a column, always in the order the formula is written.

Calcium nitrate, Ca(NO₃)₂. The subscript 2 outside the bracket multiplies both the nitrogen and the three oxygens, so the atom counts are one Ca, two N, six O.

Now ammonium sulfate, (NH₄)₂SO₄, where the outer bracket catches eight hydrogens and it is very easy to write four:

• N: 2 × 14.007 = 28.014
• H: 8 × 1.008 = 8.064
• S: 1 × 32.06 = 32.06
• O: 4 × 15.999 = 63.996
• Total: 132.134 g/mol

And glucose, C₆H₁₂O₆, which is worth doing once because the answer, 180.156 g/mol, turns up constantly in biology as well as chemistry:

C: 6 × 12.011 =  72.066
H: 12 × 1.008 =  12.096
O: 6 × 15.999 =  95.994
                --------
                180.156

The molar mass calculator prints the same breakdown, which makes it useful for finding where a hand calculation went wrong rather than only whether it did.

The seven elements you must not write as single atoms

Ask for "the molar mass of oxygen" and there are two defensible answers, 15.999 and 31.998, and only one of them is what the question meant. Oxygen as a substance is O₂. Its molar mass is about 32 g/mol, not 16.

Seven elements exist as diatomic molecules in their standard state: hydrogen, nitrogen, oxygen, fluorine, chlorine, bromine and iodine. In any equation, any gas-law calculation and any stoichiometry problem, those elements are H₂, N₂, O₂, F₂, Cl₂, Br₂, I₂ — never lone atoms.

Two more are worth knowing because they turn up in equations at the same level: white phosphorus is P₄ and elemental sulfur is S₈, so a mole of sulfur molecules has a mass of about 256 g while a mole of sulfur atoms has a mass of about 32 g.

Metals are the opposite case and cause no trouble: a mole of iron means a mole of Fe atoms, because a metal is a continuous lattice rather than a collection of molecules.

Where 12.011 comes from, and why it is not 12

Carbon-12 has a mass of exactly 12 u — that is the definition of the unit. The number on the table is 12.011 because natural carbon is not all carbon-12. About 1.1% of it is carbon-13, which is heavier, and the tabulated figure is the abundance-weighted average across the mixture. The isotopes guide works that average out arithmetically.

Three consequences follow that catch people out:

• No individual atom has the tabulated mass. There is no carbon atom weighing 12.011 u. The average describes a population, not a particle.
• A few elements have no single agreed weight at all. For lead, argon, hydrogen, sulfur, boron and several others, natural isotope ratios vary enough between sources that IUPAC publishes an interval rather than one value. Lead from different ore bodies genuinely has different atomic weights, because it is the end point of different radioactive decay chains.
• Elements with no stable isotope have no meaningful average. Technetium and promethium are quoted as the mass number of their longest-lived isotope, in brackets, which is a different kind of number wearing the same clothes.

How many figures you are allowed to keep

A molar mass computed from five-figure atomic weights is good to five figures. That does not license a five-figure answer to the problem it feeds.

Suppose you have 4.6 g of calcium nitrate. Then:

n = 4.6 g ÷ 164.086 g/mol = 0.028035... mol

The mass was given to two significant figures, so the answer is 0.028 mol. Writing 0.028035 mol claims a precision the balance never had.

The rule to internalise is that the molar mass is almost never the limiting quantity — measurements are. Carry the molar mass at full precision through the working and let the measured value decide the figures in the final line. Rounding 164.086 to 164 at the start, then quoting three figures at the end, produces an answer whose last digit is decorative.

When you cannot compute it, you measure it

Every calculation above assumes you already know the formula. The historically important case is the reverse: you have an unknown substance and want its molar mass in order to find the formula.

The gas-phase route uses the ideal gas law rearranged. Density ρ, pressure P and temperature T give

M = ρRT / P

A gas with a density of 1.96 g/L at 273.15 K and 1 atm gives M = 1.96 × 0.08206 × 273.15 / 1 = 43.9 g/mol, which identifies it as carbon dioxide (44.01) and rules out propane (44.10) only if the measurement is much better than this one is.

For solids, the classical methods are colligative: dissolving a known mass in a known solvent depresses the freezing point by an amount proportional to the number of dissolved particles, so the measured depression gives moles, and mass divided by moles gives molar mass. Modern laboratories use mass spectrometry instead, which counts far more directly. All three answer the same question the periodic table cannot: how heavy is a mole of something whose formula nobody has written down yet.

Hydrates: the water counts

A crystal that carries water in its structure has that water in its formula, joined by a raised dot, and the water is part of the mass you weigh out. Ignoring it is one of the largest errors anyone makes in this topic, because the water is often more than a third of the sample.

Copper(II) sulfate pentahydrate, CuSO₄·5H₂O, works like this. The anhydrous part comes to 63.546 + 32.06 + (4 × 15.999) = 159.602. One water molecule is (2 × 1.008) + 15.999 = 18.015, so five of them add 90.075. The total is 249.677 g/mol.

The dry salt and the blue crystals are the same compound in the sense that matters for a reaction, but 10 g of the pentahydrate contains far less copper sulfate than 10 g of the anhydrous powder — the water accounts for 36.08% of the crystal's mass. Working out that share properly is a percent composition problem, and the number is exactly why a question will specify which form it means.

Ions, and the electron you are entitled to ignore

Strictly, Na⁺ is lighter than Na by the mass of one electron: 0.000549 u, or about 24 parts per million. Every ionic molar mass in every textbook ignores this, and so should you — it sits far below the precision of the atomic weights themselves.

The one context where it stops being ignorable is mass spectrometry of highly charged ions, where the missing electrons are a measurable fraction of the total. For a sulfate ion in a homework question, SO₄²⁻ has the same molar mass as a neutral SO₄ group would: 96.06 g/mol.