Mass Fraction (Mass Percent)

The mass fraction w is the quotient of the mass of the dissolved substance and the total mass of the solution: $w =\frac{ m_{\mathrm{solute}}}{m_{\mathrm{solution}}}$. The total mass is the sum of solute and solvent. Multiplying w by 100 gives the mass percent. Example: if you dissolve 30 g of salt in 270 g of water, the total mass is 300 g and $w =\frac{ 30}{300} = 0.10$, that is 10 %. It is important to really use the mass of the whole solution in the denominator, not just that of the solvent. The mass fraction is dimensionless and independent of temperature, because masses do not change on heating. This makes it a very robust content quantity for recipes and labels.

The mass fraction w compares masses: the mass of the dissolved substance to the total mass of the solution, given dimensionless or in percent. The molar concentration c, by contrast, compares the amount of substance in moles with the volume of the solution, given in $\frac{mol}{L}$. The key difference is the reference: w is mass-based and temperature-independent, c is volume-based and changes with temperature because volumes expand. You can convert via the density ρ and the molar mass M with the relation $c =\frac{ w\cdot{}\rho{}}{M}$. In practice the mass fraction is often used for solids and commercial products, the molar concentration for reactions and in volumetric analysis, where particle ratios matter.

For the conversion you need two additional quantities: the density ρ of the solution and the molar mass M of the dissolved substance. The relation is $c =\frac{ w\cdot{}\rho{}}{M}$. The idea behind it: the mass fraction w times the density gives the mass of the substance per litre of solution, and this divided by the molar mass yields the amount of substance per litre. Example: a sulfuric acid with $w = 0.10$, density $\rho{} =\frac{ 1066 g}{L}$ and $M =\frac{ 98 g}{mol}$ has $c =\frac{ 0.10\cdot{}1066}{98} \approx{}\frac{ 1.09 mol}{L}$. Watch for consistent units, in particular the density in $\frac{g}{L}$. Without the density the conversion is impossible, because the mass fraction contains no information about the volume.

The label 0.9 % denotes the mass fraction and means that 100 g of solution contain 0.9 g of salt, the rest is water. In one litre of this solution, which because of the low concentration weighs approximately 1000 g, about 9 g of sodium chloride is dissolved. This concentration is called physiological or isotonic saline, because it has the same osmotic pressure as blood plasma. That is why it can be given as an infusion without red blood cells swelling or shrinking. Converting to $\frac{mol}{L}$ gives, with $M =\frac{ 58.44 g}{mol}$, a concentration of about $\frac{0.154 mol}{L}$. So the seemingly small percentage has a precisely tuned medical meaning.

Mass percent (wt-%) refers to masses, volume percent (vol-%) to volumes. The mass fraction $w =\frac{ m_{\mathrm{solute}}}{m_{\mathrm{solution}}}$ is dimensionless and temperature-independent. The volume fraction $\varphi{} =\frac{ V_{\mathrm{solute}}}{V_{\mathrm{solution}}}$, by contrast, is used for mixtures whose components are liquid, such as alcohol in drinks. An important point: volumes do not always add up exactly on mixing, so the reference for the volume fraction must be defined carefully. Both quantities can differ markedly when the densities of the components differ. Thus 40 vol-% ethanol is not the same as 40 wt-%, because ethanol has a lower density than water. When calculating you must therefore always know which percentage is meant and must not equate them.