Calculating Molarity in Solution

Molarity is the concentration of a solution expressed as the number of moles of solute per liter of solution: M = n / V. In brief: dissolving 10.0 g of NaOH (40.00 g/mol) in enough water to make 250.0 mL of solution gives n = 0.250 mol and a molarity of 0.250 mol / 0.2500 L = 1.00 M.

Molarity is the most common concentration unit in chemistry, since the number of moles is what matters in a reaction. It makes it simple to calculate how much of a reagent to weigh out or how much solution is needed to supply a given number of moles. This article gives the formula, explains how to prepare a solution of a given molarity and solves several calculations, including the molarity of commercial concentrated acids.

What is molarity?

The molarity (or molar concentration) is written as M, with the unit mol/L, read “molar”. A 2.0 M solution of HCl contains 2.0 mol of HCl per liter of solution. The symbol [X] means the molarity of the species X, so [Na⁺] = 0.10 M means that the solution contains 0.10 mol of sodium ions per liter.

The volume in the formula is the volume of the finished solution, not of the solvent alone. Since volume changes with temperature, molarity also changes slightly: a solution measured at 25 °C has a slightly different molarity at 60 °C.

How do you calculate molarity?

1. Convert the mass of solute to moles: n = m / M_molar.
2. Convert the volume of solution to liters.
3. Divide: M = n / V.

The formula can be rearranged to find the number of moles, n = M × V, or the volume, V = n / M. Combining n = m / M_molar gives the mass needed: m = M × V × M_molar.

How do you prepare a solution of a given molarity?

1. Calculate the mass of the solute needed.
2. Weigh the solute and transfer it into a volumetric flask of the correct size.
3. Add distilled water until the flask is about half full and swirl until the solute dissolves.
4. Add water until the bottom of the meniscus touches the calibration mark.
5. Close the flask and invert it several times to mix.

The solute should not be dissolved in the full volume of water directly, because the final volume would be uncertain.

Worked examples

Example 1: molarity from mass and volume

Find the molarity of a solution of 10.0 g of NaOH in 250.0 mL of solution. M_molar(NaOH) = 22.99 + 16.00 + 1.008 = 40.00 g/mol.

1. n = 10.0 / 40.00 = 0.250 mol.
2. V = 0.2500 L.
3. M = 0.250 / 0.2500 = 1.00 M.

Example 2: mass needed for a given solution

How many grams of copper(II) sulfate pentahydrate, CuSO₄·5H₂O (249.69 g/mol), are required to make 500.0 mL of a 0.200 M solution?

1. n = 0.200 mol/L × 0.5000 L = 0.100 mol.
2. m = 0.100 mol × 249.69 g/mol = 25.0 g.

The hydrate must be used in the calculation, since the weighed solid includes the water of crystallization.

Example 3: concentration of ions

What are the ion concentrations in 0.150 M CaCl₂? Each formula unit gives one Ca²⁺ and two Cl⁻, so [Ca²⁺] = 0.150 M and [Cl⁻] = 2 × 0.150 = 0.300 M.

Example 4: finding the volume

What volume of 0.250 M KOH contains 0.0500 mol of KOH? V = n / M = 0.0500 / 0.250 = 0.200 L, or 200 mL.

Molarity of concentrated reagents

For a commercial solution, the molarity can be calculated from the mass percent (w) and density (ρ in g/mL) using M = (w × ρ × 1000) / M_molar, where w is a fraction. For example, concentrated HCl is 37 % by mass with a density of 1.18 g/mL: M = 0.37 × 1.18 × 1000 / 36.46 = 12.0 M.

The exact percentage and density depend on the supplier, so the label on the bottle should always be used.

Where does the unit come from?

Volumetric analysis, in which the volume of a solution of known concentration is used to determine an unknown quantity, was developed in the late 18th and early 19th centuries, with important improvements by Gay-Lussac. The “normal” solution, containing one gram-equivalent per liter, was used for a long time. The molar solution became the standard as the concept of the mole developed in the 20th century. The capital letter M for molar was a common shorthand, though IUPAC recommends writing mol/L or mol dm⁻³ in formal use.

Which mistakes are common?

• Using the volume of the solvent: use the volume of the solution.
• Forgetting to convert mL to L: 250 mL is 0.250 L.
• Ignoring water of hydration: include it in the molar mass of the weighed solid.
• Confusing molarity with molality: molality uses kilograms of solvent.
• Forgetting the number of ions per formula unit: the concentration of Cl⁻ in CaCl₂ is twice that of the salt.