text: Lawrence Stamper Darken has shown that the Gibbs-Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential  of only one component (here component 2) at all compositions. He has deduced the following relation
  x, amount (mole) fractions of components.
  Making some rearrangements and dividing by (1 – x) gives:
  or
  or 
  : as formatting variant
  The derivative with respect to one mole fraction x is taken at constant ratios of amounts (and therefore of mole fractions) of the other components of the solution representable in a diagram like ternary plot.
  The last equality can be integrated from  to  gives:
  Applying LHopital's rule gives:
  This becomes further:
  Express the mole fractions of component 1 and 3 as functions of component 2 mole fraction and binary mole ratios:
  and the sum of partial molar quantities
  gives 
   and  are constants which can be determined from the binary systems 1_2 and 2_3. These constants can be obtained from the previous equality by putting the complementary mole fraction x = 0 for x and vice versa.
  Thus 
  and 
  The final expression is given by substitution of these constants into the previous equation: