What you'll be able to do: convert between grams, moles and numbers of particles for any substance whose formula you know
A chemical formula counts particles, but a balance measures mass. The mole is the unit that connects those two worlds: one mole is a fixed count of particles (6.022 × 10²³ of them), and the molar mass of a substance tells you how many grams that count weighs. Because atoms and molecules are far too small and too numerous to count directly, essentially every quantitative calculation in chemistry passes through the mole at some point.
These are recommended, not required. You can start this lesson at any time.
A single water molecule has a mass of about 3 × 10⁻²³ g, far too small for any balance. A drop of water contains roughly 10²¹ molecules, far too many to count.
Chemists solve both problems at once by working with a huge, fixed-size group of particles called the mole, in the same way a stationery shop counts paper in reams rather than sheets.
Reactions happen particle by particle, so their recipes are written in particles; laboratory work happens gram by gram. The mole is the translation between those two languages.
Since the 2019 revision of the SI, the mole is defined exactly: one mole contains exactly 6.02214076 × 10²³ elementary entities. That number is the fixed numerical value of the Avogadro constant, NA, expressed in mol⁻¹.
In this course you will normally round it to 6.022 × 10²³ mol⁻¹.
The definition says nothing about which particles they are, so you must always state the entity: one mole of CO₂ means 6.022 × 10²³ CO₂ molecules, while one mole of oxygen atoms means 6.022 × 10²³ O atoms.
1 mol = 6.022 × 10²³ entities (atoms, molecules, formula units, ions, electrons)
Molar mass, symbol M, is the mass of one mole of a substance, in grams per mole.
Its numerical value is the same as the atomic or formula mass in atomic mass units, which is why you can read it straight off the periodic table: carbon has an average atomic mass of 12.01 amu, so its molar mass is 12.01 g/mol.
Periodic-table values are weighted averages over the naturally occurring isotopes, so molar masses are rarely whole numbers.
M(element) = average atomic mass from the periodic table, in g/mol
For a compound, add the molar mass of every atom in one formula unit. Subscripts multiply, and a subscript outside parentheses multiplies everything inside them. Setting the work out as a small table makes missing atoms obvious.
For calcium nitrate, Ca(NO₃)₂, one formula unit contains 1 Ca, 2 N and 6 O.
Ca(NO₃)₂: 1 × 40.08 + 2 × 14.01 + 6 × 16.00 = 40.08 + 28.02 + 96.00 = 164.10 g/mol
Almost every mole problem is a walk along this two-step map. Molar mass connects grams and moles; the Avogadro constant connects moles and particles. Moles sit in the middle, so a grams-to-particles question is never one step. Write each conversion as a fraction whose bottom unit cancels the unit you already have; if the units cancel to leave the unit you want, the setup is right.
The reverse route uses the same two bridges in reverse order. For 3.011 × 10²³ CO₂ molecules, divide by 6.022 × 10²³ molecules/mol to obtain 0.5000 mol, then multiply by 44.01 g/mol to obtain 22.01 g. In scientific notation, divide the numerical parts and subtract the exponents: 10²³/10²³ = 1. Keep the entity label attached throughout; the molar mass of CO₂ cannot convert a count of isolated O atoms directly into grams of CO₂.
grams ÷ M → MOLES × NA → particles (reverse: × M and ÷ NA)
The subscripts in a formula are themselves mole ratios. One mole of Al₂(SO₄)₃ contains 2 mol of Al atoms, 3 mol of S atoms and 12 mol of O atoms.
That extra ratio is a third conversion step, applied after you reach moles of the compound: multiply by the subscript to get moles of the element, then by NA if the question asks for atoms.
For ionic compounds, the same idea gives moles of ions: 1 mol of Mg₃(PO₄)₂ contains 3 mol Mg²⁺ ions and 2 mol PO₄³⁻ ions.
1 mol Al₂(SO₄)₃ → 2 mol Al, 3 mol S, 12 mol O
Multiplication and division keep the smallest number of significant figures among the measured values, so 8.80 g divided by 44.01 g/mol gives 0.200 mol, three significant figures. Treat molar masses and NA as precise enough not to limit the answer, and round only at the end.
Then sanity-check the size: an amount smaller than one mole must give fewer than 6.022 × 10²³ particles, and a mass smaller than the molar mass must give less than one mole.
A mole is a count, so write what is being counted beside every number. For 0.250 mol Ca(NO₃)₂, there are 0.250 mol formula units, 0.250 mol Ca²⁺ ions and 0.500 mol NO₃⁻ ions if the salt dissociates completely. There are also 1.50 mol oxygen atoms in those formula units. The different answers are all valid because they count different entities. In a mass-to-particles problem, first convert grams of compound to moles of compound, then use the formula subscripts to switch entities, and only then multiply by Nₐ. This order prevents confusing molecules, formula units, ions and atoms.
| Quantity in 1 mol of Ca(NO₃)₂ | Amount counted |
|---|---|
| Ca(NO₃)₂ formula units | 1 mol |
| Calcium ions upon complete dissociation | 1 mol |
| Nitrate ions upon complete dissociation | 2 mol |
| All dissolved ions together | 3 mol |
| Oxygen atoms within the nitrate ions | 6 mol |
Nitrate remains a polyatomic ion when this salt dissolves: the six oxygen atoms are not six separate oxide ions in solution. Also distinguish molar mass in g/mol from the mass of one particle in grams: dividing 44.01 g/mol by NA gives about 7.31 × 10⁻²³ g per CO₂ molecule. That microscopic mass is why laboratory measurements use moles.
The SI unit of amount of substance. One mole contains exactly 6.02214076 × 10²³ elementary entities; the entity must always be named.
The mass of one mole of a substance, in g/mol. Numerically equal to the atomic or formula mass in amu.
Grams and particles are never converted directly into each other. Both conversions pass through moles.
A subscript tells you how many moles of that element are in one mole of the compound; a subscript outside parentheses multiplies every atom inside.
Periodic-table masses are isotope-weighted averages for a normal terrestrial sample. A problem about one specific isotope (for example deuterium) must use that isotope's mass instead.
n = m / M
N = n × NA
M(compound) = Σ (subscript × atomic mass of each element)
What is the molar mass of calcium nitrate, Ca(NO₃)₂, and how many moles are in a 25.0 g sample? (Atomic masses: Ca 40.08, N 14.01, O 16.00)
Try it first: Count every atom in one formula unit first, remembering that the subscript 2 applies to the whole nitrate group. Then decide whether you divide or multiply by the molar mass.
0 of 4 steps revealed.
How many CO₂ molecules are in 8.80 g of carbon dioxide? (Atomic masses: C 12.01, O 16.00)
Try it first: This is a grams-to-particles question, so plan two steps before calculating anything. Which quantity sits between them?
0 of 4 steps revealed.
Challenge: How many oxygen atoms are in 12.0 g of aluminium sulfate, Al₂(SO₄)₃? (Atomic masses: Al 26.98, S 32.06, O 16.00)
Try it first: This needs three steps, not two. Where does the formula's subscript enter the chain?
0 of 4 steps revealed.
How many nitrate ions are produced when 0.250 mol of Ca(NO₃)₂ dissolves completely?
Try it first: Count nitrate groups per formula unit before using Avogadro's constant.
0 of 2 steps revealed.
| Mistake | Why it's wrong | Check instead |
|---|---|---|
| Treating one mole as a fixed mass, for example assuming 1 mol always weighs the same or that 1 mol = 1 g. | A mole is a count of particles. One mole of He weighs 4.00 g while one mole of Pb weighs 207.2 g, yet both contain 6.022 × 10²³ atoms. | Ask what is being counted, then look up the molar mass of that specific substance before converting. |
| Multiplying by molar mass when converting grams to moles (or dividing when converting moles to grams). | The units do not cancel: 25.0 g × 164.10 g/mol gives g²/mol, which is not an amount. | Write each conversion as a fraction and confirm the unwanted unit cancels before you press a calculator key. |
| Applying an outside subscript to only the first element inside the parentheses, for example counting 3 O in Ca(NO₃)₂. | The subscript multiplies every atom inside the parentheses, so Ca(NO₃)₂ has 2 N and 6 O. | Expand the formula atom by atom in a short list before adding masses. |
| Reporting molecules when the question asks for atoms, for example saying 2.50 mol CO₂ contains 1.51 × 10²⁴ atoms of oxygen when that value is the molecule count times one. | Each CO₂ molecule contains 3 atoms, 2 of which are oxygen, so the atom count is a subscript multiple of the molecule count. | Underline the entity in the question (molecules, formula units, atoms of a named element, ions) and apply the subscript ratio if it names an element. |
| Multiplying grams directly by 6.022 × 10²³. | The Avogadro constant converts moles, not grams; skipping molar mass makes the answer wrong by a factor equal to M. | Draw the map grams → moles → particles and confirm you have used two different conversion factors. |
| Letting molar mass or NA set the number of significant figures in the answer. | Those are reference values known to far more digits than a typical laboratory measurement. | Count significant figures only in the measured quantities given in the problem. |
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The mole is a counting unit; molar mass converts that count into a mass you can measure. Together they let you move between grams, moles and particles for any substance.
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