how-to · 5 min read
Moles, molar mass and molarity: a practical guide
Most introductory chemistry problems start on a balance and end in a volumetric flask. The balance gives grams, the flask holds a volume, and a balanced equation speaks in moles. Three calculations bridge those worlds: molar mass turns a formula into grams per mole, mole conversions move between mass, moles and particle counts, and molarity and dilution arithmetic link moles to the volume of a solution.
This guide works through each step with worked examples and states the assumptions behind the numbers. The molar mass, mole, molarity and dilution calculators on this site perform the same arithmetic in your browser and show the working, so a calculation can be followed line by line instead of taken on trust.
Reading a formula correctly
A formula is shorthand for a ratio of atoms. Element symbols are case-sensitive: Co is cobalt while CO is carbon monoxide, and that difference changes the molar mass completely. A subscript applies only to the symbol immediately before it, so H2SO4 contains two hydrogen atoms, one sulfur atom and four oxygen atoms.
Brackets group a repeating unit and the number after the bracket multiplies everything inside. Ca(OH)2 is one calcium plus two OH groups, which means two oxygen and two hydrogen atoms; Al2(SO4)3 is two aluminum, three sulfur and twelve oxygen atoms. In hydrate notation such as CuSO4·5H2O, the dot separates the salt from water of crystallization and the coefficient multiplies the water part, so the formula unit contains one CuSO4 and five H2O units.
- H2SO4: 2 H, 1 S, 4 O
- Ca(OH)2: 1 Ca, 2 O, 2 H
- Al2(SO4)3: 2 Al, 3 S, 12 O
- CuSO4·5H2O: 1 Cu, 1 S, 9 O, 10 H
From formula to molar mass
Molar mass is the sum of the relative atomic masses of every atom in the formula unit, expressed in grams per mole. Standard values are quoted to several decimal places, which is why H is 1.008, C is 12.011, O is 15.999 and S is 32.06 rather than whole numbers. Textbooks often round them, so a result can differ in the second or third decimal place without being wrong.
For sulfuric acid, H2SO4: 2 × 1.008 + 1 × 32.06 + 4 × 15.999 = 98.072 g/mol. For aluminum sulfate, Al2(SO4)3: 2 × 26.9815 + 3 × 32.06 + 12 × 15.999 = 342.131 g/mol. Hydrates add the water: CuSO4·5H2O is 63.546 + 32.06 + 4 × 15.999 + 5 × (2 × 1.008 + 15.999) = 249.677 g/mol, of which about 36 percent is water.
Moles, mass and particles
The mole is the amount of substance that contains 6.02214076 × 10²³ elementary entities, the Avogadro constant. Dividing a mass by the molar mass gives moles: n = m / M. Ten grams of sodium chloride, molar mass 58.44 g/mol, is 10 / 58.44 = 0.1711 mol. Multiplying by the Avogadro constant gives 1.030 × 10²³ formula units.
The conversion runs in both directions. Half a mole of water is 0.5 × 18.015 = 9.008 g and contains 3.011 × 10²³ water molecules. State the entity when quoting a particle count: one mole of sodium chloride contains one mole of formula units but two moles of ions, while one mole of water contains one mole of molecules and three moles of atoms.
Molarity and dilution
Molarity is the number of moles of solute per liter of solution, M = n / V. Dissolving 0.1711 mol of sodium chloride in enough water to make 500 mL of solution gives 0.1711 / 0.5 = 0.3422 mol/L. The volume is the final solution volume: a solid can change the volume slightly when it dissolves, so solutions are made up to the mark in volumetric glassware rather than by adding a measured volume of water.
Dilution keeps the amount of solute constant while the volume changes, so C1V1 = C2V2. To prepare 250 mL of 0.1 mol/L solution from a 1 mol/L stock, V1 = C2V2 / C1 = (0.1 × 0.25) / 1 = 0.025 L, or 25 mL of stock made up to 250 mL. Mass concentrations such as g/L and mg/mL follow the same equation, but converting between a mass concentration and molarity requires the molar mass.
Accuracy, units and common mistakes
Most errors are unit errors. Molarity needs liters, so 250 mL must become 0.25 L; a volume in cubic centimeters is the same number of milliliters, and a cubic decimeter is a liter. Check whether a question gives the solvent volume or the final solution volume, and remember that molarity changes with temperature because the solution expands while the amount of solute does not.
The arithmetic here assumes an ideal solution: the solute does not react, volumes behave additively and activity coefficients are ignored. At high concentrations or in non-aqueous solvents those assumptions break down. Round only at the end, keep an extra digit through the working, and quote an answer with a sensible number of significant figures based on the least precise measurement.
- Convert milliliters to liters before using M = n / V.
- Use the final solution volume, not the volume of solvent added.
- Do not mix molar and mass concentration units in one equation.
- Check which entity is being counted in a particle conversion.