Concepts: concentration terms and their units

2.1 What a solution is

A solution forms when two or more substances mix evenly at the particle level. The dispersed ions or molecules are small enough that they never settle out, so the mixture looks uniform. The component present in the larger amount is called the solvent and the others are solutes. Salt in water, sugar syrup, air and brass are all solutions, because the mixed particles are of molecular or ionic size and do not scatter a light beam.

2.2 Types of solutions

Solutes and solvents can each be solid, liquid or gas, and the common combinations are worth knowing because the solubility rules change with the state.

2.3 Mass percentage and ppm

Mass percentage tells how many grams of solute are present in 100 g of the solution, not in 100 g of the solvent. A 10 percent w/w glucose solution contains 10 g of glucose in 90 g of water. Parts per million applies the same idea to one million grams of solution and is used for traces, such as a contaminant in drinking water. Because both are mass ratios, neither changes when the solution is heated.

2.4 Mole fraction

The mole fraction of a component is its amount in moles divided by the total amount in moles. It is unitless, a whole mixture gives fractions that add to one, and it is the quantity that appears in Raoult's law and in Henry's law. For a two component solution of A and B, x_A plus x_B is exactly one.

2.5 Molarity

Molarity counts moles of solute in one litre of the finished solution. The volume in the denominator belongs to the solution, not the solvent, so two solutions with the same amount of solute can have different molarities if the solutes pack differently. Heating expands the solution, the volume rises and the molarity falls, which is why molarity is described as a temperature dependent unit.

2.6 Molality

Molality counts moles of solute in one kilogram of solvent. Neither the amount of solute nor the mass of the solvent changes with a modest change of temperature, so molality stays fixed. That is the reason colligative properties are written with m and not with M. A solution does not have molality until the mass of solvent is known separately.

2.7 Converting between units

Start from a convenient sample, usually 100 g of solution for a mass percentage, and take the density only when a volume is needed. To go from a mass percentage to mole fraction, convert the 100 g sample into moles of each component. To go from mole fraction to molality, take the moles of solvent, multiply by its molar mass and divide by 1000 to reach kilograms. To go from molarity to molality, find the mass of one litre of solution from its density, subtract the mass of solute to obtain the solvent mass and then divide.

2.8 Worked examples

Example 1: nitric acid from a bottle. Concentrated nitric acid is 68 percent by mass and its density is 1.504 g mL^-1. Take one litre of solution. Its mass is 1504 g, so the acid in it is 0.68 times 1504, which is 1022.7 g. Dividing by the molar mass of 63 g mol^-1 gives 16.2 mol. Therefore the molarity is 16.2 M. The same sample contains 1504 minus 1022.7, or 481.3 g of water, so the molality is 16.2 divided by 0.4813, which is 33.7 mol kg^-1.

Example 2: glucose solution. A 10 percent w/w glucose solution is labelled, and the density is 1.2 g mL^-1. In 100 g of solution there are 10 g of glucose, which is 0.0556 mol, and 90 g of water, which is 0.090 kg. Molality is therefore 0.5556 divided by 0.0900, or 0.617 mol kg^-1. The mole fractions are 0.0556 divided by the total of 5.0556 for glucose and 5.0000 divided by 5.0556 for water, giving 0.0110 and 0.9890. The volume of 100 g of solution is 100 divided by 1.2, which is 83.3 mL, so molarity is 0.0556 divided by 0.0833, or 0.667 M.

Example 3: mole fraction to molality. An aqueous solution has a urea mole fraction of 0.05. Take 0.05 mol of urea and 0.95 mol of water. The water mass is 0.95 times 18 g, which is 17.1 g or 0.0171 kg. Molality is 0.05 divided by 0.0171, or 2.92 mol kg^-1.

2.9 Common mistakes

Using 100 g of solvent in place of 100 g of solution for a mass percentage is the most common slip. Another is dividing by the mass of the solution when molality asks for the mass of the solvent. Mixing grams with kilograms inside the same substitution loses marks even when the arithmetic is right. A mole fraction must lie between zero and one, so an answer larger than one signals that moles and masses have been swapped.

A final check on every conversion: the number of moles is the bridge between all five units. Find the moles of solute and the mass of solvent first, and the other units follow from the definitions without memorising a separate formula for each pair.