Number of particles from moles:
formula and worked example

From a chemist's unit to an actual particle count

A mole is a counting unit, like a dozen, only far larger. It exists so that macroscopic quantities of matter can be discussed without writing absurdly long numbers. Sooner or later, though, a question asks for the number of molecules in a drop of water or the number of ions in a pinch of salt, and then the mole has to be converted back into a count.

This conversion is a single multiplication by Avogadro's constant, the number of particles in one mole. It appears in gas-law problems, in questions about how many atoms a sample contains, and in any comparison between a bulk quantity and the microscopic scale at which reactions actually happen.

Seeing the count helps build intuition about why the mole was defined at all. A single drop of water holds on the order of a sextillion molecules, far too many to enumerate, but a few thousandths of a mole is easy to measure. Moles let you reason about amounts in manageable numbers, and Avogadro's constant is the exchange rate back to individual particles when you need it.

The relationship and its inputs

The number of particles equals the amount in moles times the number of particles per mole.

• Moles: the amount of substance, normally obtained from a mass divided by a molar mass, or from a concentration times a volume.
• Avogadro: the particles-per-mole constant, about 6.022 x 10^23 per mole. Since the 2019 redefinition of SI units it is a fixed exact value, so more digits are only a matter of how precisely you need to quote it.
• Result: a pure count, with the particle type stated in words, such as molecules, atoms or formula units.

In scientific notation 6.0221e+23 means 6.0221 x 10^23. Enter it that way in a calculator rather than typing 6.0221 and forgetting the exponent.

Worked example: half a mole

1. 1Take the amount: 0.5 mol of a molecular substance.
2. 2Take the constant: 6.0221 x 10^23 particles per mole.
3. 3Multiply the coefficients: 0.5 x 6.0221 = 3.01105.
4. 4Keep the power of ten: 3.01105 x 10^23, quoted as 3.0111 x 10^23 particles.

Halving a mole halves the count, so the answer being roughly half of Avogadro's number is exactly the sanity check you want.

Making sense of a very large number

Results in this calculation are always huge, so the exponent is what carries the meaning. Compare answers by their power of ten first and the leading digits second. A result of 10^22 for a gram-scale sample of a compound is plausible; 10^15 would suggest the amount was in micromoles or the exponent was dropped.

Also state the particle. One mole of a substance contains one mole of its formula units, but the number of atoms depends on the formula. Half a mole of water has 3.01 x 10^23 molecules, and since each has three atoms, about 9.03 x 10^23 atoms in total. Half a mole of sodium chloride gives the same number of formula units but twice that many ions.

Counts like these are exact in principle, yet limited in practice by the precision of the moles you started with. A result quoted to five digits from a moles value known to two is false precision.

What a change in moles does

A 10% increase in moles gives a 10% increase in particles, 3.0111e+23 rising to 3.3122e+23. There is no hidden curve here; the constant never changes, so the relationship is perfectly linear. That makes this a good place to check your intuition on scale: a few tenths of a mole is already hundreds of sextillions of particles, which is why chemistry works with moles rather than counts.

Counting mistakes to avoid

• Giving atoms when the question asked for molecules, or the reverse. Multiply by the number of atoms per molecule only when the atom count is requested.
• Treating an ionic compound's formula unit as a molecule, or forgetting that NaCl yields two ions per formula unit.
• Typing 6.0221 without the e+23 exponent and reporting a result that is smaller by twenty-three orders of magnitude.
• Using 6.02 x 10^23 for a calculation that needs four or more significant figures, which quietly truncates the answer.

It is also easy to pass the wrong quantity into the formula. Putting a mass in grams where moles belong gives a number that looks like a particle count but is meaningless. Always convert to moles first, and write the unit next to every number as you go so that a stray gram or litre is visible.

A routine for particle-count questions

Establish the amount in moles, decide which particle the question wants, multiply by the constant, then multiply again by atoms or ions per formula unit if needed. Finish by checking the exponent against the size of the sample.

For example, suppose a question asks how many chloride ions are in 0.20 mol of calcium chloride, CaCl2. The formula units number 0.20 times the constant, about 1.2 x 10^23. Each formula unit releases two chloride ions, so the chloride count doubles to about 2.4 x 10^23. Writing out that extra multiplication step explicitly is what separates a correct answer from a near miss.