Mass percent and density: converting a w/w solution to molarity

Mass percent on a w/w basis

Mass percent w/w is the mass of solute divided by the mass of the whole solution, multiplied by 100. A solution made from 10 g of solute and 90 g of solvent has a total mass of 100 g, so its concentration is 10% w/w. The denominator is the solution mass, not the solvent mass, and that distinction is the first place hand calculations go wrong.

The value is independent of sample size: 10 g in 100 g and 25 g in 250 g are both 10% w/w. That is useful when the actual batch size is set by the glassware available rather than by the recipe, and it is why mass percent is a natural unit for concentrated stock solutions.

From mass percent to molarity: worked example

Take a 10.0% w/w NaCl solution with a density of 1.07 g/mL. One liter of this solution has a mass of 1070 g, and 10.0% of that mass is solute: 0.100 × 1070 = 107 g of NaCl. Sodium chloride has a molar mass of 58.44 g/mol, so the amount of solute per liter is 107 / 58.44 = 1.83 mol.

The molarity is therefore 1.83 mol/L. The shortcut is C = (w/w% × ρ × 10) / M, with the percentage as a number, the density in g/mL and the molar mass in g/mol: (10.0 × 1.07 × 10) / 58.44 = 1.83 M. The factor of 10 combines the percent-to-fraction conversion and the g/mL-to-g/L conversion.

Why density is the bridge

Mass percent has no volume in it, while molarity is defined per liter of solution. Density converts one to the other: multiply the density in g/mL by 1000 to get g/L, apply the mass percent to find the solute mass per liter, and divide by the molar mass to reach moles per liter. The same density in SI units is 1070 kg/m³, which is numerically the same as 1070 g/L.

The conversion also shows why w/w and w/v percentages are not interchangeable. A 10% w/v solution contains 10 g of solute per 100 mL of solution, while a 10% w/w solution contains 10 g per 100 g, and only a density of exactly 1.00 g/mL would make the two identical.

Rearranging the same relationship converts molarity back to mass percent: w/w% = (C × M) / (10 × ρ). Solving for the density instead gives ρ = (C × M) / (10 × w/w%), which is a convenient way to check that a reported density is consistent with a labeled concentration.

Preparing a mass-percent solution

To prepare 250 g of a 10% w/w solution, weigh 0.100 × 250 = 25 g of solute and add 225 g of solvent; the total is 250 g. If the recipe specifies a volume instead, convert it first with the density. For 250 mL at 1.07 g/mL, the solution mass is 267.5 g, the solute mass is 26.75 g and the solvent mass is 240.75 g.

Weighing is usually more accurate than measuring a volume for a w/w recipe, especially when the density differs from 1 g/mL. For small volumes the density also does not vary much with concentration, but for concentrated solutions the density must be taken at the intended composition and temperature.

A practical shortcut for small batches is to prepare a larger mass of solution and take a portion by mass, because splitting a weighed solution preserves the concentration exactly. Mixing must still be thorough before a sample is taken.

Temperature, purity and rounding

Density changes with temperature, so a mass-percent solution has a different molarity at a different temperature even though its w/w composition is unchanged. Use a density measured at the temperature where the solution is used; a density taken from a different temperature shifts the molarity by roughly the thermal expansion of the solution.

If the solute has a stated assay below 100%, divide the required mass by the purity fraction. Round the final concentration to match the significant figures of the density and the balance, and remember that ideal-volume assumptions do not correct for volume contraction when unlike liquids are mixed.