Principle: Nernst distribution law — a solute distributes between two immiscible solvents such that the ratio of concentrations is constant at a given temperature:

K = C(organic phase) / C(aqueous phase) — the partition coefficient

If K > 1 for ether/water, the compound prefers the organic phase → extraction is favorable.

Mathematical proof that multiple small extractions are better than one large extraction:

If total volume V of organic solvent is used, and the aqueous phase has volume W, with initial mass m of solute:

After one extraction with V mL: mass remaining = m x W/(W + KV)

After n extractions with V/n mL each: mass remaining = m x [W/(W + KV/n)]^n

Since [W/(W + KV/n)]^n < W/(W + KV) for n > 1, multiple extractions always leave less compound behind.

Practical rule: 3 extractions with 1/3 volume each recover significantly more than 1 extraction with full volume.