Three-phase, one-component systems

A one-component system that exists in three phases is indifferent and has no degrees of freedom. In order to define the state of the system then, three extensive variables must be used. We choose for discussion the enthalpy, volume, and number of moles of the components. The enthalpy of the system is additive in the molar enthalpies of the three phases, as is the volume. We can then write three equations  

For a given set of values for H, V, and n, values of n, n , and n can be calculated if the molar enthalpies and molar volumes of the three phases can be determined. The molar volumes can be obtained experimentally, but the absolute values of the molar enthalpies are not known. In order to solve this problem, we make use of the concept of standard states. We choose one of the three phases and define the standard state to be the state of the system when all of the component exists in that phase at the temperature and pressure of the triple point. If we choose the triple-primed phase as the standard phase, we subtract nH from each side of Equation (8.50) and obtain 

This equation gives the enthalpy of the system relative to the standard state, and the independent variable would now be (H — nH ) rather than H itself. The quantities (H — H ) and (ft — H ) are the changes of enthalpy when the state of aggregation of 1 mole of the component is changed from the triple-primed state to the primed state and to the double-primed state, respectively, at the temperature and pressure of the triple point. These quantities can be determined experimentally or from the Clapeyron equation, as discussed in Section 8.2. The three simultaneous, independent equations can now be solved, provided values that permit a physically realizable solution have been given to (H — nH ), V, and n. If such a solution is not obtained, the system cannot exist in three phases for the chosen set of independent variables. Actually, the standard state could be defined as one of the phases at any arbitrarily chosen temperature and pressure. The values of the enthalpy and entropy for the phase at the temperature and pressure 

The values of the other thermodynamic functions are readily obtained for these independent variables. The chemical potential is identical in each phase, and consequently G = np. If we choose the same standard state for the Gibbs energy as we did for the enthalpy, we have 

The equations developed in Section 8.1 for single-phase, one-component systems are all applicable to single-phase, multicomponent systems with the condition that the composition of the system is constant. The dependence of the thermodynamic functions on concentration are introduced through the chemical potentials because, for such a system, 

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