The general thermodynamic schemes for predicting the amount of dye on the fiber which have been used by other workers are outlined. A new approach is then developed for the complex system of vat dyes on cotton, and the affinities of 14 vat dyes are calculated. The theory is extended to mixtures, and the variables of temperature, amount of salt added, amount of dye added, and amount of reducing agent added required to give a known color are predicted from the theory. -- AATA