Relations concerning partial molar quantities

The extensive quantities are functions of the temperature, pressure, and mole numbers, and, consequently, so are the partial molar quantities, so that Xk = Xk(T, P, n). The differential is then written directly as 

Various expressions for the differential of the partial molar quantities can be obtained by the use of these expressions. 

The extensive properties are homogenous functions of the first degree in the mole numbers at constant temperature and pressure. Then, the partial molar quantities are homogenous functions of zeroth degree in the mole numbers at constant temperature and pressure that is, they are functions of the composition. We use the mole fractions here, but we could use the molality, mole ratio, or any other composition variable that is zeroth degree in the mole numbers. For mole fractions, Xk = Xk(T, P, x) and the differential of Xk may be written as 

The sum in the last term of this equation is taken over all of the components of the system except one. The sum of the mole fractions must be unity, and consequently for C components in the system there are (C — 1) independent mole fractions or for S species, there are (S — 1) independent mole fractions in terms of the species. The proof of the identity of Equations (6.3) and (6.7) is not difficult. 

One of the most important applications of Equation (6.7) is to the chemical potentials. From their definition in terms of the Gibbs energy we know that they are functions of the temperature, pressure, and the mole fractions. The differential of the chemical potential of the fcth component is then given by 

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