What molarity measures
Molarity — amount concentration in IUPAC's language — is the number of moles of solute in one litre of solution. A 0.100 mol/L sodium chloride solution contains 0.100 mol of NaCl in every litre you pour, no matter how much you make.
It is the concentration unit almost all solution chemistry runs on, for one reason: reactions happen mole for mole, so a concentration expressed in moles lets you compute a reaction directly from a volume you can measure with a pipette. A titration is exactly this — you measure a volume, multiply by a molarity, and you have moles.
The definition contains a trap worth stating at the start. The volume in the denominator is the volume of the finished solution, not the volume of solvent you added. You dissolve the solid in a part of the solvent and then make up to the mark, because dissolving changes the volume. For dilute aqueous solutions the difference is small; for concentrated acids and for anything in an organic solvent it is not.
The equation and its three rearrangements
Start from C = n ÷ V. Substituting n = m ÷ M gives the working form C = m ÷ (M × V), which links the three things you can actually control: the mass you weigh, the molar mass of what you weighed, and the volume you make up to.
Each rearrangement answers a different question. C = m ÷ (M·V) tells you what you have made. m = C·V·M tells you what to weigh, which is the form you use ninety percent of the time at the bench. V = m ÷ (M·C) tells you how far to dilute a fixed mass — useful when a vial contains a stated mass of an expensive reagent and you want a particular working strength.
The molar mass term is where hydrates bite. Copper(II) sulfate pentahydrate is 249.68 g/mol against the anhydrous salt's 159.60 g/mol. Use the wrong one and your solution is out by 56%, and nothing in the appearance of the solution will tell you. Enter the formula exactly as the bottle labels it; the molar mass calculator explains the parsing rules in full.
Note also that molarity is temperature-dependent, because volume is. A solution made up at 20 °C and used at 40 °C is slightly more dilute than its label says, since water expands by about 0.6% over that range — its density falls from 0.99821 to 0.99222 g/mL. For work where that matters, chemists switch to molality — moles per kilogram of solvent — which has no volume in it and so does not drift with temperature.
Worked example: 250 mL of 0.100 mol/L sodium chloride
You need a quarter-litre of 0.100 mol/L saline for a calibration. Work out the weighing.
- Molar mass. Na is 22.990 and Cl is 35.45, so NaCl is 58.44 g/mol.
- Moles needed. n = C × V = 0.100 mol/L × 0.250 L = 0.0250 mol.
- Mass to weigh. m = n × M = 0.0250 mol × 58.44 g/mol = 1.461 g.
- Make it up. Weigh 1.461 g into a beaker, dissolve in roughly 150 mL of water, transfer quantitatively to a 250 mL volumetric flask, rinse the beaker into the flask twice, then fill to the graduation mark and invert twenty times.
Check the result the other way. 1.461 g ÷ 58.44 g/mol = 0.02500 mol, and 0.02500 mol ÷ 0.250 L = 0.1000 mol/L. The concentration by mass is 0.100 × 58.44 = 5.844 g/L, which is the number you would use to compare against a label quoted in g/L.
If your balance actually settles on 1.457 g rather than 1.461 g, do not discard it — record it. The real concentration is 1.457 ÷ 58.44 ÷ 0.250 = 0.09973 mol/L, a known value rather than a nominal one, and for standardisation work a known value is worth more than a round one.
How to judge the number you get
Compare against solubility before you trust a high concentration. Saturated sodium chloride is only about 5.4 mol/L at 25 °C — 26.4% by mass at a density near 1.20 g/mL — so a calculated 8 mol/L recipe will simply leave solid at the bottom of the flask and give you a saturated solution of unknown strength. The calculator warns above 6 mol/L for exactly this reason, but the real limit is substance-specific — check a solubility table for yours.
Check the weighing is within your balance's useful range. Below about 20 mg, a four-place balance contributes more than half a percent of relative error, which is larger than most people's tolerance for a standard. The fix is always the same: make a more concentrated stock from a comfortable weighing and take it down with the dilution calculator.
Finally, be explicit about what dissolves into what. For a salt that dissociates, the ion concentrations are not the molarity of the salt: 0.100 mol/L CaCl₂ is 0.100 mol/L in calcium ions but 0.200 mol/L in chloride. Ionic strength, conductivity and activity calculations all need the ion concentrations, not the formula concentration.
Grams needed per litre of solution at common concentrations
| Solute | Molar mass (g/mol) | 0.010 mol/L | 0.100 mol/L | 1.000 mol/L |
|---|---|---|---|---|
| Sodium chloride, NaCl | 58.440 | 0.584 g | 5.844 g | 58.44 g |
| Sodium hydroxide, NaOH | 39.997 | 0.400 g | 4.000 g | 40.00 g |
| Potassium hydroxide, KOH | 56.105 | 0.561 g | 5.611 g | 56.11 g |
| Sodium hydrogen carbonate, NaHCO₃ | 84.006 | 0.840 g | 8.401 g | 84.01 g |
| Calcium chloride, CaCl₂ | 110.978 | 1.110 g | 11.098 g | 110.98 g |
| Potassium permanganate, KMnO₄ | 158.032 | 1.580 g | 15.803 g | 158.03 g |
| Glucose, C₆H₁₂O₆ | 180.156 | 1.802 g | 18.016 g | 180.16 g |
| Copper(II) sulfate pentahydrate | 249.677 | 2.497 g | 24.968 g | 249.68 g |
| Silver nitrate, AgNO₃ | 169.874 | 1.699 g | 16.987 g | 169.87 g |
| EDTA disodium dihydrate, C₁₀H₁₄N₂Na₂O₈·2H₂O | 372.238 | 3.722 g | 37.224 g | 372.24 g |
Scale linearly for other volumes: half a litre needs half the mass. Not every entry is soluble at 1 mol/L — check solubility before scaling up.
Mistakes that give a wrong concentration
- Adding solvent to a mark instead of making up to a mark. Dissolve first in part of the solvent, then top up. Adding one litre of water to the solid gives more than one litre of solution.
- Using the anhydrous molar mass for a hydrate. The waters are part of what you weighed and must be part of the molar mass.
- Ignoring reagent purity. A solid assayed at 97% needs the calculated mass divided by 0.97 to deliver the intended amount of substance.
- Confusing molarity with normality. A 1 mol/L sulfuric acid solution is 2 normal, because each molecule supplies two protons. Old procedures often quote N, not M.
- Treating concentrated acid bottle strengths as molarity. Concentrated hydrochloric acid is labelled 37% by mass; you need its density to convert that to about 12 mol/L.
- Assuming molarity is temperature-independent. It is not, because volume expands with temperature. Use molality when the temperature will change.
Molarity among the other concentration units
Molality is moles of solute per kilogram of solvent. It is temperature-independent and is the unit colligative-property equations require, which is why boiling-point elevation and freezing-point depression use molality rather than molarity. In dilute aqueous solution the two are numerically close, since one litre of water weighs almost exactly one kilogram.
Mass percent and ppm divide solute mass by total solution mass, with no reference to a formula. They are the natural units for a mixture whose molar mass is not defined, and for trace analysis where mol/L would be an awkward power of ten.
Normality counts reacting equivalents rather than moles — protons for an acid, electrons for an oxidant. It survives in water treatment and in older analytical procedures. Sulfuric acid at 1 mol/L is 2 N for acid–base purposes, so misreading one for the other doubles or halves your titre.
Once a stock exists, the arithmetic of getting to a working strength is conservation of solute, C₁V₁ = C₂V₂, which the dilution calculator handles. And if the solution is going into a reaction, the mole figure here feeds directly into the limiting reagent calculator and the mole-ratio calculator.