Partition coefficients

When a solute is shaken with two immiscible solvents (solvents that do not mix, e.g. ether and water) it distributes itself between them. At a given temperature, the ratio of its concentrations reaches a constant value — the partition (distribution) coefficient:

$K_{pc} = \frac{[\text{solute in solvent 1}]}{[\text{solute in solvent 2}]}$

Worked. If [X] in ether = 0.80 mol dm⁻³ and [X] in water = 0.20 mol dm⁻³: $K_{pc}\,(\text{ether/water}) = \frac{0.80}{0.20} = 4.0$

Kpc is usually quoted without units (the concentration units cancel). Keep the solvents in the order stated, or you will quote the reciprocal value.

Solvent extraction uses Kpc to predict how much solute moves into an extracting solvent. Set up concentrations as mass ÷ volume in each layer.

Single extraction. 1.00 g of Z in 100 cm³ water, shaken with 100 cm³ ether (Kpc ether/water = 3.0). Let x g enter the ether: $K_{pc} = \frac{x/100}{(1-x)/100} = \frac{x}{1-x} = 3.0 \Rightarrow x = 0.75\,\text{g}$

Two smaller extractions (2 × 50 cm³). First portion gives: $\frac{y/50}{(1-y)/100} = \frac{2y}{1-y} = 3.0 \Rightarrow y = 0.60\,\text{g}$ Then 0.40 g remains; the second 50 cm³ extracts: $\frac{2z}{0.40-z} = 3.0 \Rightarrow z = 0.24\,\text{g}$ Total extracted = 0.60 + 0.24 = 0.84 g — more than the single extraction's 0.75 g.

This is the key result: dividing the solvent into several smaller portions extracts more solute than using it all at once.

The partition coefficient is constant only if certain assumptions hold:

• The temperature is constant (Kpc varies with temperature).
• The solvents are immiscible.
• The solute is in the same molecular state in both solvents — it must not associate (e.g. dimerise) or dissociate (e.g. ionise) in one of them.

If the solute changes its molecular form in one solvent, the simple concentration ratio no longer holds. For example, a carboxylic acid that dimerises in an organic solvent, or ionises in water, will not give a constant Kpc unless the calculation accounts for the species actually present.