What this calculator does
This tool converts a concentration written as % w/v (weight/volume — grams of solute per 100 mL of solution) into % w/w (weight/weight — grams of solute per 100 g of solution). The two scales only line up when you know the density of the solution, because % w/v is measured against volume while % w/w is measured against mass.
For dilute water-based solutions the density is close to 1.00 g/mL, so the two numbers are nearly identical. For concentrated syrups, acids, brines or organic mixtures the density can sit well above or below 1.00, and the gap between % w/v and % w/w becomes significant.
How to use it
Enter the concentration you already know in % w/v (grams of solute per 100 mL). Then enter the density of the finished solution in grams per millilitre (g/mL). Press calculate and the tool returns the equivalent % w/w, along with the solute mass and solution mass on a 100 mL basis so you can check the arithmetic yourself.
If you do not have a measured density, use 1.00 g/mL for a rough estimate on dilute aqueous solutions, but remember the answer is then only approximate.
The formula explained
Work on a 100 mL basis. By definition, a concentration of C in % w/v means there are C grams of solute in that 100 mL. The mass of the whole solution is its volume times its density, that is 100 times rho grams, where rho is the density in g/mL. The % w/w is the solute mass divided by the solution mass:
$$\%\,w/w = \frac{ \text{solute mass} }{ \text{solution mass} } \times 100 = \frac{ C }{ 100\,\rho } \times 100$$The factors of 100 cancel, leaving the compact rule:
$$\%\,w/w = \frac{ \%\,w/v }{ \rho }$$Here rho is the solution density in g/mL. When rho is greater than 1 the % w/w comes out smaller than the % w/v; when rho is less than 1 the % w/w comes out larger.
Worked example
Suppose you have a 20% w/v solution whose measured density is 1.25 g/mL. On a 100 mL basis there are 20 g of solute, and the solution mass is 100 times 1.25 = 125 g. So the % w/w is 20 divided by 125, times 100, which is 16% w/w. Using the shortcut directly, 20 divided by 1.25 gives the same 16% w/w.
Frequently asked questions
Why do I need the density? Percent w/v is grams per 100 mL of solution while percent w/w is grams per 100 g of solution. Converting between a volume basis and a mass basis requires the density that links millilitres to grams.
What density should I use if I only have water? Pure water is about 1.00 g/mL near room temperature, so for dilute aqueous solutions % w/w is roughly equal to % w/v. For accuracy with concentrated solutions, measure or look up the real density.
Can the % w/w be larger than the % w/v? Yes. If the solution is less dense than 1.00 g/mL, as with many alcohol-based mixtures, dividing by a density below 1 makes the % w/w larger than the % w/v.
