Phase one-component system

If we express the composition of a phase in terms of the mole fractions of all the components, then (C 1) intensive variables are needed to describe the composition, if every component appears in the phase, because the mole fractions must sum to 1. In a system of p phases, p(C — 1) intensive variables are used to describe the composition of the system. As was pointed out in Section 3.1, a one-phase, one-component system can be described by a large number of intensive variables yet the specification of the values of any two such variables is sufficient to fix the state of such a system. Thus, for example, two variables are needed to describe the temperature and pressure of each phase of constant composition or any alternative convenient choice of two intensive variables. Therefore, the total number of variables needed to describe the state of the system is... 

The state of a single-phase, one-component system may be defined in terms of the temperature, pressure, and the number of moles of the component as independent variables. The problem is to determine the difference between the values of the thermodynamic functions for any state of the system and those for the chosen standard state. Because the variables are not separable in the differential expressions for these functions, the integrations cannot be carried out directly to obtain general expressions for the thermodynamic functions without an adequate equation of state. However, each of the thermodynamic functions is a function of the state of the system, and the changes of these functions are independent of the path. The problem can be solved for specific cases by using the method outlined in Section 4.9 and illustrated in Figure 4.1. 

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,... 

Figure 14.2. A two-phase, one-component system in a gravitational field.

Further insight regarding the concept of the chemical potential may be obtained as follows Consider a two-phase, one—component system at fixed temperature and pressure for which G - niPi + n p, and suppose that at some instant > px. The system can then not be at equilibrium instead, some spontaneous process must occur which ultimately results in the equalization of and. At constant T and P this can occur only by a transfer of matter from one phase to the other. Let there be a transfer of - dnx - + dn > 0 moles from phase 1 to phase 2 then dG — (p — p1)dn1, where we have set dnx = dnx. Since we assumed p > the preceding relation shows that dG < 0 for this case i.e., the transfer of matter from the phase of higher chemical potential to the phase of lower chemical potential occurs spontaneously. Thus, a difference in chemical potential represents a driving force for transfer of chemical... 

If we have a two-phase one-component system and we carry out an equilibrium change during the course of which the masses of the two phases remain unchanged, then the change is called a constant-mass equilibrium change. 

We shall now explain in greater detail how knowledge of the thermodynamic potential function of a single-phase, one-component system makes it possible to determine the conditions under which the system breaks up into different phases. We start from the stability condition and show the manner in which the thermodynamic potential function of the two-phase equilibrium can be constructed. As an example, let us select energy as the thermodynamic potential function for such a system that can only exchange heat with its surroundings. The respective boundary... 

Even for a one-phase, one-component system (molten Si, e.g.) the concept of the chemical potential can be applied. G should be the molar free energy of pure i at the temperature T and the pressure P of the system. The free energy is an extensive quantity, hence the total free energy of a one-component, one-phase system is... 

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