If a solute dissociates into ions or associates into larger units, the number of particles in solution differs from the number of formula units dissolved. The van’t Hoff factor i corrects the colligative formulae for this.
The colligative formulae assume that each formula unit of solute dissolved gives one particle in solution. For some solutes, a molar mass calculated from a colligative measurement differs greatly from the molar mass calculated from the formula. This is called an abnormal molar mass.
The measurement is not at fault. The assumption is. If the solute dissociates (splits into ions) or associates (joins into larger units), the number of particles differs from the number of formula units dissolved. Colligative properties measure the number of particles, so the calculated molar mass is off.
Ionic compounds separate into ions when they dissolve in water. One mole of KCl gives one mole of K⁺ ions and one mole of Cl⁻ ions: two moles of particles.
So a KCl solution lowers the freezing point of water by close to twice the amount expected for one mole of particles. The calculation assumes one particle per formula unit, so it concludes there are twice as many formula units as there really are. The calculated molar mass is about half the true value, 74.5 g/mol.
More particles than formula units gives a calculated molar mass that is too low.
Some molecules join together in certain solvents. In benzene, ethanoic acid molecules form pairs, called dimers, held together by two hydrogen bonds. If nearly all the molecules pair up, two moles of ethanoic acid give about one mole of particles.
The colligative effect is about half the expected value, so the calculated molar mass is about double the true value of 60 g/mol.
Fewer particles than formula units gives a calculated molar mass that is too high.
Benzoic acid in benzene gives a calculated molar mass of about 244 g/mol. The formula C₆H₅COOH gives 122 g/mol. What is the benzoic acid doing in solution?
It is associating into dimers.
The calculated molar mass is twice the true value, so the colligative effect was half the expected value. That means there are about half as many particles as formula units. Each particle must be two benzoic acid molecules.
Like ethanoic acid, benzoic acid forms hydrogen-bonded dimers in benzene, a solvent that cannot form hydrogen bonds with it.
Jacobus van’t Hoff introduced a factor, i, to correct for this:
i = normal molar mass / abnormal molar mass = observed colligative effect / calculated colligative effect
Both ratios equal the number of particles actually present divided by the number of formula units dissolved.
Each colligative formula is multiplied by i:
ΔTb = i Kb m ΔTf = i Kf m Π = i C R T
Example: 0.10 mol of NaCl in 1 kg of water, with i ≈ 1.9. ΔTf = 1.9 × 1.86 × 0.10 ≈ 0.35 K. The solution freezes at about −0.35°C. Without the factor, the prediction would be −0.19°C.
For a very dilute solution of K₂SO₄, i approaches 3. Why 3, and why does it only approach that value?
One formula unit of K₂SO₄ separates into three ions: 2 K⁺ and 1 SO₄²⁻. So complete dissociation gives 3 particles per formula unit, and i = 3.
In a solution of finite concentration, oppositely charged ions attract one another. Some spend time close together and act partly as one particle. The effective number of particles is then a little less than 3, so i is a little less than 3.
As the solution is diluted, the ions are further apart on average, these attractions matter less, and i gets closer to 3.
Concentration can be expressed in several ways; molarity depends on temperature and the others do not. Solubility depends on temperature, and for gases on pressure through Henry’s law. Raoult’s law gives the vapour pressure of a solution, and deviations from it show whether unlike molecules attract more or less strongly than like ones. The lowering of vapour pressure leads to the four colligative properties, and the van’t Hoff factor corrects them for solutes that dissociate or associate.
1An abnormal molar mass from a colligative measurement usually means that the solute has…
2When one mole of KCl dissolves in water, it gives…
3Dissociation makes the molar mass calculated from a colligative property…
4Ethanoic acid in benzene gives a calculated molar mass about twice the true value because it…
5The van’t Hoff factor i is less than 1 when the solute…
6For a very dilute solution of K₂SO₄, i approaches…
7Including the van’t Hoff factor, the freezing-point depression is…
That completes the unit on solutions.
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