The mole and molar mass

A mole is a number: exactly 6.022 140 76 × 10²³ of whatever you are counting, atoms, molecules or ions. Chemists use it because atoms are too small to count one by one but easy to weigh by the trillion trillion, and the molar mass, the mass of one mole, turns a reading on a balance into a count. Water's molar mass is 18.015 g/mol, so 18.015 g of water holds one mole of molecules.^[1]

Everything else on this site rests on that exchange. A balanced equation is written in molecules; a laboratory measures grams. Between them sits the mole, and the bookkeeping of chemistry is mostly a matter of converting back and forth without losing track. This chapter is about the exchange itself: what a mole is, where its absurd size comes from, why each substance has its own rate, and how many digits the rate deserves.

Counting by weighing§

Imagine you need ten thousand identical washers. You would not count them. You would weigh ten, find that one washer weighs 2.0 g, and pour washers onto the scale until it reads 20 kg. Weighing counts for you, as long as every washer weighs the same.

Atoms of one element are close enough to identical for the same trick to work, and they come in such numbers that no other method is practical. A single water molecule has a mass of about 3.0 × 10⁻²³ g. A balance that reads to a tenth of a milligram is a fine instrument, and a tenth of a milligram of water is about 3 × 10¹⁸ molecules: three million million million. The only sensible unit for a chemist is a vast, fixed pile of particles, and that pile is the mole.

The figure at the top of this page shows the trick as three rulers lined up under each other. The top ruler is what a balance reads. The middle ruler is the count in moles, and the bottom ruler is the same count written out as a number of molecules. A vertical line through all three is one sample, read in three currencies. Moving it along the gram scale moves it along the others in proportion: double the grams, double the moles, double the molecules.

Every substance has its own exchange rate§

The gram ruler is the same for everything. The mole ruler is not. Its tick marks sit at multiples of the substance's molar mass: at 18.015 g, 36.030 g, 54.045 g and so on for water, but at 196.97 g for gold and at 2.0160 g for hydrogen gas. The bottom ruler, the actual count of particles, is the mole ruler multiplied by the same constant every time. That is the whole content of the idea:

Here m is the mass in grams, M the molar mass in grams per mole, n the amount in moles, N the number of particles and N_A = 6.022 140 76 × 10²³ mol⁻¹ the Avogadro constant.^[2] The first equation depends on the substance. The second does not: one mole of anything has the same number of particles, which is why chemists can compare amounts of different substances at all.

So the answer to the question is the hydrogen, and by a wide margin. The intuition that a heavier sample holds more of something is right about matter and wrong about particles. Chemistry happens between particles: one oxygen molecule meets two hydrogen molecules, never one gram meets two grams. That is why chemists convert to moles before they compare anything.

Where the number 6.022 × 10²³ comes from§

The size of the mole is an accident of history, chosen to make molar masses come out equal to the familiar atomic weights. From 1971 to 2019 the mole was defined as the number of atoms in 0.012 kg of carbon-12.^[4] With that definition, a substance whose molecules weigh 18 times as much as one-twelfth of a carbon-12 atom has a molar mass of 18 g/mol, and the table of atomic weights doubles as a table of molar masses. The price was that the number of particles in a mole had to be measured, and it was known only as well as the measurements allowed.

The most precise of those measurements counted atoms directly, in a way. Teams at national metrology institutes polished spheres of nearly pure silicon-28 crystal, measured their diameter and mass, measured the spacing of the atoms in the crystal lattice by X-ray interferometry, and from those worked out how many atoms each sphere held.^[5] Results like these let the General Conference on Weights and Measures fix the Avogadro constant at its meeting in November 2018, in a revision of the SI that came into force on 20 May 2019. Since then the mole is defined as exactly 6.022 140 76 × 10²³ entities, with no reference to carbon at all.^[1] established

One consequence is subtle and almost never matters. The molar mass of carbon-12 used to be exactly 12 g/mol by definition; now it is a measured quantity, equal to 12 g/mol within a relative uncertainty of 4.5 × 10⁻¹⁰ when the new definition was adopted.^[1] Every calculation on this site treats the conversion from atomic weight to grams per mole as exact. The error that introduces is smaller than the last digit of any atomic weight by a factor of more than twenty thousand (the closest case is niobium, 92.906 ± 0.001: a relative 1 × 10⁻⁵ against 4.5 × 10⁻¹⁰).

Molar mass: adding up the atomic weights§

The molar mass of a compound is the sum of the atomic weights of its atoms, in grams per mole. Carbon dioxide, CO₂, has one carbon and two oxygens:^[2]

Brackets multiply what is inside them. Calcium hydroxide, Ca(OH)₂, has one calcium, two oxygens and two hydrogens: 40.078 + 2 × 15.999 + 2 × 1.0080 = 74.092 g/mol. A hydrate's dot adds whole water molecules: copper(II) sulfate pentahydrate, CuSO₄·5H₂O, is one CuSO₄ plus five H₂O, 249.68 g/mol. The calculator below does this for any formula you type, and shows each element's share of the mass.

The calculator is strict on purpose. Typed as co2, a formula could mean carbon dioxide or a molecule of two cobalt atoms (Co₂). It will not choose: it says why it stopped and offers both readings. The same goes for a charge written without a caret: Fe3+ could be an iron ion with a 3+ charge or Fe₃ with a 1+ charge, so the calculator asks you to write Fe^3+ (or Fe³⁺) and offers both. Its full rules are on the Molar mass calculator page.

Why atomic weights are not whole numbers§

A carbon-12 atom weighs exactly 12 units by definition, and other atoms weigh close to whole numbers of units, because protons and neutrons each weigh about one. Yet chlorine's atomic weight is 35.45, which no single atom has. The reason is that atomic weight is an average over the atoms you actually meet.

Atoms of one element can differ in their number of neutrons. These variants are isotopes. Chlorine has two stable ones: chlorine-35, with a relative atomic mass of 34.968 853, and chlorine-37, at 36.965 903.^[6][7] A typical natural sample is about 75.76 % chlorine-35 and 24.24 % chlorine-37 by number of atoms.^[6] The atomic weight is the abundance-weighted mean:

A weighted mean has a picture that a bookkeeper or a carpenter would recognise. Hang each isotope's share on a beam at its mass, and the beam balances at the average. A heavier share on the right pulls the balance point right.

Why some atomic weights are ranges§

The share of each isotope is not quite the same everywhere. Chlorine from different minerals, seawater and industrial sources varies between about 75.5 % and 76.1 % chlorine-35.^[8] For a handful of elements this natural variation is larger than the precision of the measurements, so since 2009 the Commission on Isotopic Abundances and Atomic Weights (CIAAW) of IUPAC has given their standard atomic weights as intervals. Chlorine's is [35.446, 35.457]; hydrogen's is [1.007 84, 1.008 11]; carbon's is [12.0096, 12.0116].^[9][10] Fourteen elements are listed this way in the current table: hydrogen, lithium, boron, carbon, nitrogen, oxygen, magnesium, silicon, sulfur, chlorine, argon, bromine, thallium and lead. established

Nobody wants to add intervals to work out a molar mass, so CIAAW also publishes an abridged table, quoted to five significant figures where natural variation allows, each value with an uncertainty that covers the full interval: chlorine 35.45 ± 0.01, hydrogen 1.0080 ± 0.0002, carbon 12.011 ± 0.002.^[11] Every calculation on this site uses the 2024 edition of that abridged table, which incorporates the 2021 report and the 2024 revisions for gadolinium, lutetium and zirconium. Lead is the extreme case: its isotopic composition depends so much on the ore it came from that its abridged value is 207.2 ± 1.1.^[11]

Some elements have no standard atomic weight at all. Technetium, promethium, and the elements from polonium to actinium and from neptunium on have no stable isotopes with a characteristic natural composition, so the mass of a sample depends on which isotope it contains.^[11] Many periodic tables print a mass number in brackets for them, such as [98] for technetium. That is the mass number of one isotope, not an atomic weight, and the calculators here refuse those elements rather than guess which isotope you have.

How many digits a molar mass deserves§

Adding atomic weights is addition, and the rule for adding measured numbers is that the sum is known only to the last decimal place that every term carries.^[2] Sodium's weight is 22.990 (three decimal places) and chlorine's is 35.45 (two), so sodium chloride is 22.990 + 35.45 = 58.44 g/mol, not 58.440. Multiplying by a count of atoms adds no uncertainty, since the count is exact. Each calculator here keeps the exact sum and rounds once, at the end, to the decimal place of the least precise atomic weight in the formula.

Other tables sometimes differ from these in the last digit. PubChem lists oxygen gas as 31.999 g/mol and calcium carbonate as 100.09.^[12] Both are consistent with the numbers here once the uncertainty of the atomic weights is counted: oxygen's abridged value is 15.999 ± 0.001, so two oxygen atoms can be anywhere from 31.996 to 32.000. A disagreement in the last digit between two careful sources is a sign of different conventions, not of an error.

Ions, electrons and the last digit§

An ion has gained or lost electrons, and electrons have mass: 0.000 548 6 in atomic weight units each.^[13] By the usual convention, a molar mass is the sum of atomic weights and ignores them, so sulfate, SO₄²⁻, is given the same 96.06 g/mol as a neutral SO₄. For almost every ion the electrons change nothing at the precision of the table. The exception is the lightest: a hydrogen ion, H⁺, would be 1.0075 rather than 1.0080 if the missing electron were subtracted. The calculator follows the convention and reports the electron-corrected value beside it.

Where this goes next§

The mole lets you weigh out a count. The next step is to know what count you need, and for that you need a balanced equation, which tells you how many of each particle take part. Balancing equations is about writing one and why it must balance. Stoichiometry and the limiting reactant then puts the two together: grams to moles, moles through the equation, and moles back to grams.