The equations so far apply only to binary fluid mixtures. For more than two constituents, say K, there will exist K values of the mass ratios . These are not independent since the totals of the individual masses of constituents must add up to the mass of the mixture. Hence, because of the conservation of mass
There are also K different values of non-advective flux associated with the K constituents, and these are related by
which follows from the definition of velocity and the conservation of mass. Thus there are really only K-1 independent values of and , and the functions of state such as , , etc. depend therefore on K+1 variables, e.g.
There are K-1 virtual chemical potentials defined by
with the subscript signifying that all S except and are held constant. These potentials are related to the K Gibbsian partial molar chemical potentials as follows
where the are the molecular (or atomic) weights of the K different constituents. These relations state that the K Gibbs potentials are not all independent with the interdependence expressed by Eqs. (4.201-5) being consistent with the Gibbs-Duhem relation for systems of several components.