Expressing Concentration of Solutions

Everyday products come labelled with a concentration -- a chlorhexidine mouthwash might say 0.2% (w/v), commercial hydrogen peroxide is typically 3% (w/v), tap-water purity is reported in ppm (parts per million), and laboratory reagents are labelled molar or normal. In every case, concentration expresses the amount of solute present in a given quantity of solvent (or solution).

Different situations call for different concentration units, because each unit is convenient for a different kind of calculation:

• Molar (M) solutions are used where reactions proceed in a known, fixed mole ratio -- e.g. complexometric titrations with EDTA, where EDTA and the metal ion react 1:1.
• Normality (N) is used in redox and acid-base (neutralisation) titrations, where the "reacting unit" (equivalent) matters more than the whole molecule.
• Mole fraction (x) is used to calculate the partial pressure of a gas in a mixture, and the vapour pressure of a solution (sections 9.5-9.7).
• Percentage units (w/w, w/v, v/v) are used to state the active-ingredient strength of a therapeutic or commercial product.
• ppm is reserved for solutes present in very small (trace) amounts, such as dissolved solids in drinking water.

The eight concentration terms (Table 9.2), with a worked illustration for each:

1. Molality (m) $=\dfrac{\text{moles of solute}}{\text{mass of solvent in kg}}$. Example: dissolving 45 g glucose (molar mass 180 g/mol, so 0.25 mol) in 2 kg water gives $m=\dfrac{0.25}{2}=0.125$ mol/kg.
2. Molarity (M) $=\dfrac{\text{moles of solute}}{\text{volume of solution in L}}$. Example: 5.845 g NaCl (molar mass 58.45, so 0.1 mol) made up to 500 mL gives $M=\dfrac{0.1}{0.5}=0.2$ M.
3. Normality (N) $=\dfrac{\text{gram equivalents of solute}}{\text{volume of solution in L}}$. Example: 3.15 g oxalic acid dihydrate (equivalent mass 63, so 0.05 equivalents) made up to 100 mL gives $N=\dfrac{0.05}{0.1}=0.5$ N.
4. Formality (F) $=\dfrac{\text{formula weights of solute}}{\text{volume of solution in L}}$ -- used for ionic compounds that do not exist as discrete molecules. Example: 5.85 g NaCl (formula weight 58.5) made up to 500 mL gives $F=\dfrac{5.85}{58.5\times0.5}=0.2$ F.
5. Mole fraction (x) $=\dfrac{\text{moles of one component}}{\text{total moles of all components}}$. For a two-component solution of A and B with $n_A$, $n_B$ moles respectively, $x_A=\dfrac{n_A}{n_A+n_B}$ and $x_B=\dfrac{n_B}{n_A+n_B}$, and always $x_A+x_B=1$. Example: mixing 0.5 mol ethanol with 1.5 mol water gives $x_{ethanol}=\dfrac{0.5}{2.0}=0.25$ and $x_{water}=1-0.25=0.75$.
6. Mass percentage (% w/w) $=\dfrac{\text{mass of solute (g)}}{\text{mass of solution (g)}}\times100$. Example: 300 mg (0.3 g) neomycin sulphate in 30 g of ointment gives $\dfrac{0.3}{30}\times100=1\%$ w/w.
7. Volume percentage (% v/v) $=\dfrac{\text{volume of solute (mL)}}{\text{volume of solution (mL)}}\times100$. Example: 10 mL benzoin in 50 mL tincture of benzoin gives $\dfrac{10}{50}\times100=20\%$ v/v.
8. Mass by volume percentage (% w/v) $=\dfrac{\text{mass of solute (g)}}{\text{volume of solution (mL)}}\times100$. Example: 3 g paracetamol in 60 mL suspension gives $\dfrac{3}{60}\times100=5\%$ w/v. …