Integration of Equation 6.12 for a binary system

However, in order to obtain the integral, the integration constant or the values of the lower limits must be known. 

No difficulty exists for gases and for liquid and solid solutions that have no region of immiscibility or limited solubility. In this case dV2/dxl)TiP (Eq. (6.65)) or dVJdxf)TP (Eq. (6.66)) can be determined over the whole range of composition. The lower limit of integration of Equation (6.65) is = 1, for which Vl = V, and the lower limit of integration of Equation (6.66) is xY = 0, for which V2 = V2. 

The problem discussed here is common to all partial molar quantities. It is for this and similar reasons, which we have already indicated, that reference and standard states are defined (see Chapter 8). 

The conditions of equilibrium expressed by Equations (5.25)—(5.29) and (5.46) involve the temperature, pressure, and chemical potentials of the components or species. The chemical potentials are functions of the temperature, pressure or volume, and composition, according to Equations (5.54) and (5.56). In order to study the equilibrium properties of systems in terms of these experimentally observable variables, expressions for the chemical potentials in terms of these variables must be obtained. This problem is considered in this chapter and in Chapter 8. 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

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