Atoms to Moles Calculator

What a mole counts, and why the number is exact

A dozen is a way of counting eggs without writing twelve every time. A mole does the same job for atoms and molecules, except the number is 6.02214076 x 10^23. That figure is the Avogadro constant, and it stopped being a measured value in 2019, when the redefinition of the SI fixed it exactly. NIST Special Publication 330 puts it plainly: one mole contains exactly 6.02214076 x 10^23 elementary entities, and the seven defining constants of the SI carry no uncertainty. The vote that locked the value came in November 2018, with the new definitions taking effect on 20 May 2019, as the American Chemical Society's account of the redefinition records.

The word entities does the heavy lifting. A mole counts whatever you point it at: atoms in a piece of copper, molecules in a cup of water, ions in a salt crystal, or electrons drifting through a wire. The count is the same in every case. What changes is what one unit of the substance is made of.

That is why this page separates two jobs. Going between a particle count and moles is a single multiplication or division, and no chemistry is involved. Going between grams and moles needs a molar mass, because a scale weighs matter and a mole counts pieces. Mixing the two jobs up is the most common way a homework answer lands one power of ten away from the right one.

The two formulas that cover every conversion

Everything on this page comes from one fixed number and two equations:

• Moles = particles / 6.02214076 x 10^23
• Particles = moles x 6.02214076 x 10^23

Moles are written mol, and a particle count is a plain number with no unit. That lack of a unit unsettles people at first, since every other quantity in a chemistry problem carries one. Think of a count of eggs again: 24 eggs is 24, not 24 egg. The Avogadro constant is the exchange rate between the two, and it is written per mole because one mole contains that many particles.

The fourth row is there for round trips. If a problem gives you grams and asks for atoms, you take two steps: grams to moles with the molar mass, then moles to particles with the constant. The calculator on this page does both steps at once when you pick the grams mode.

Worked example: the atoms in a one carat diamond

A metric carat is 200 milligrams, so a one carat diamond weighs 0.200 grams. Diamond is pure carbon, and the conventional atomic weight of carbon is 12.011 grams per mole, the single value the IUPAC commission on atomic weights recommends for ordinary materials in place of its isotope range. The first step is mass to moles:

0.200 / 12.011 = 0.016651 mol of carbon.

The second step is moles to atoms, and here the count per formula unit is 1, because carbon in diamond is a lattice of single atoms rather than molecules:

0.016651 x 6.02214076 x 10^23 comes out to 1.0028 x 10^22 atoms.

Ten thousand million million million atoms, in a stone you could lose down a drain. Run the same two steps on a one gram bag of graphite and you get 5.0139 x 10^22 atoms, five times as many, because five times the mass at the same molar mass means five times the count.

The reverse trip checks the work. Take 1.0028 x 10^22 atoms and divide by 6.02214076 x 10^23 and you land back on 0.016651 mol, which is 0.200 grams once you multiply by 12.011. Every conversion on this page can be audited that way, and the calculator shows both directions in the working line under the results.

Worked example: molecules and atoms in a glass of water

Water is the case that trips people up, because a molecule of water holds three atoms. A 250 milliliter glass holds about 250 grams of water at room temperature, and water has a molar mass of 18.015 grams per mole. One division gives the moles:

250 / 18.015 = 13.8773 mol of water.

Multiply by the Avogadro constant and you get 8.3571 x 10^24 water molecules. That is the molecule count, and it is the number a chemist means when they say how much water is in the glass.

The atom count is three times larger, because every molecule carries two hydrogen atoms and one oxygen atom. 8.3571 x 10^24 x 3 is 2.5071 x 10^25 atoms. If a question asks for the number of hydrogen atoms on its own, the multiplier is 2, not 3.

Now compare the glass with a teaspoon of water, roughly 5 grams. The teaspoon holds 0.277546 mol, which is 1.6714 x 10^23 molecules, and the glass holds 50 times that. Nothing about the water changed between the two containers. Only the amount did, and the count scales with it in a straight line.

Molecules or atoms: read the question twice

Half the errors in this topic come from answering a slightly different question than the one asked. Three counts show up in homework and in lab work, and they are not interchangeable:

• Formula units, the number of molecules or ion pairs. This is what one mole of the substance contains.
• Atoms, the number of individual atoms, which is the formula unit count times the atoms in each unit.
• Moles of atoms, the atom count divided by the Avogadro constant, which is not the same as moles of the compound.

One mole of water holds 1 mol of molecules and 3 mol of atoms. One mole of table salt holds 1 mol of formula units and 2 mol of atoms, since each unit is one sodium ion and one chloride ion. Get the multiplier wrong and the answer is off by a factor of two or three, which is easy to spot; use the wrong count per mole and the answer is off by 10^23, which is harder to notice in a hurry.

The table also shows why a gram is a poor way to compare samples. A gram of gold holds 3.0574 x 10^21 atoms and a gram of water holds 3.3428 x 10^22 units, ten times more, because gold atoms are heavy and water molecules are light.

Reading the answer in powers of ten

Particle counts run from a handful in a single cell to 10^24 in a spoonful, so answers belong in scientific notation. The calculator writes them as a number between 1 and 10 times a power of ten, the same form your textbook uses. Read 1.0028 x 10^22 as 1.0028 followed by 22 places, which is 10,028,000,000,000,000,000,000 atoms.

Chemistry also uses prefixed moles, and each prefix is a factor of 1,000. The particle count moves with it:

Notice that a millimole still holds 602,210,000,000,000,000,000 particles. Small units in chemistry are never small counts. That is the reason the mole exists: it turns an invisible number into a number a person can weigh.

When you read a result, check the exponent before you check the digits. Going from 1 mol to 1 mmol drops the exponent by 3, from 23 to 20, and any answer that keeps the same exponent while changing the prefix is wrong.

Small samples: a tablet and a lungful of air

Two short examples show the range. A standard 200 milligram ibuprofen tablet is C13H18O2, which works out to 206.285 grams per mole: 13 carbons at 12.011, 18 hydrogens at 1.008 and 2 oxygens at 15.999. Divide the mass by the molar mass and you get 9.6953 x 10^-4 mol, or 0.9695 millimoles, a number small enough to sit comfortably in a blood test report.

Multiply that by the Avogadro constant and the tablet holds 5.8387 x 10^20 ibuprofen molecules. Each molecule carries 33 atoms, so the atom count is 1.9268 x 10^22. A single tablet, dissolved in a stomach, delivers about 19 thousand million million million atoms of the drug.

Now run it backwards. Suppose a problem hands you 2.0 x 10^21 molecules of carbon dioxide and asks for the mass. Divide by the constant first: 2 x 10^21 / 6.02214076 x 10^23 is 0.003321 mol. Then multiply by the molar mass of CO2, which is 44.009 grams per mole, and you get 0.146157 grams. That is the volume of gas in one exhaled breath, near enough, and it weighs less than a small paper clip.

Five places the power of ten goes missing

These are the mistakes that show up again and again in graded work, and each one changes the answer by a clean factor of ten or more.

• Dividing when you should multiply. To go from particles to moles you divide. Doing it the other way turns 1.0028 x 10^22 atoms into 6 x 10^45 mol.
• Using the constant upside down. The value per mole is 6.02214076 x 10^23. A negative exponent, 6.02214076 x 10^-23, is the reciprocal and describes a different quantity entirely.
• Forgetting the units inside the unit. Two moles of atoms is not two moles of a compound with four atoms in it. Multiply by the atoms per formula unit before you report an atom count.
• Feeding grams into the particle formula. Grams and particles never meet directly. There is always a division by molar mass in between.
• Rounding the mantissa too early. Keeping two digits, 1.0 x 10^22, is fine for a headline, but carry four through the steps and round at the end, or the small digits will drift.

A quick order of magnitude check catches all five. One mole is 6 x 10^23, so a few grams of anything with a molar mass near 20 should land near 10^23 particles, not 10^20 and not 10^26. If your answer is a power of ten or more away from that guess, go back and look at which way you moved the decimal point.

Two habits make the check fast. Write the unit on every number, and say the operation out loud as you do it: grams divided by grams per mole has to leave moles, because the grams cancel. If the units do not cancel to what you want, the formula is upside down.