Two-phase, one-component systems

A one-component system that exists in two phases is univariant and hence indifferent. In order to define the state of the system, at least two of the independent variables must be extensive. We choose the temperature, volume, and number of moles as the independent variables, and consider the enthalpy, entropy, and Gibbs energy as functions of these variables. The two condition equations 

This equation makes it possible to determine the number of moles of the component in each phase or to use n as an independent variable rather than V, if we so choose. The molar volumes are properties of the separate phases, and consequently can be considered as functions of the temperature and pressure. However, the system is univariant, and consequently the pressure is a function of the temperature when the temperature is taken as the independent variable. The relation between the temperature and pressure may be determined experimentally or may be determined by means of the Clapeyron equation. The differential of each molar volume may then be expressed by 

With this introduction, we consider the enthalpy of the system as a function of the temperature, volume, and number of moles. The differential of the enthalpy is given by 

The evaluation of (dH/dV)Tn is also based on Equation (8.31). At constant temperature both dH and dH are zero, and consequently 

That is, the change of enthalpy of a closed system for a change in volume at constant temperature is simply the molar change of enthalpy for the change of phase multiplied by the number of moles that undergo the change of phase. The equation for the derivative may be written either as 

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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. 

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... 

As a first example for saturated phases, we consider one phase of a two-phase, single-component system that is closed. The molar enthalpy, and hence the molar heat capacity, of a phase is a function of the temperature and pressure. However, the pressure of the saturated phase is a function of the temperature because, in the two-phase system, there is only one degree of freedom. The differential of the molar enthalpy is given by... 

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... 

When, for a one-component system, one of the two phases in equilibrium is a sufficiently dilute gas, i.e. is at a pressure well below 1 atm, one can obtain a very usefiil approximate equation from equation (A2.1.52). The molar volume of the gas is at least two orders of magnitude larger than that of the liquid or solid, and is very nearly an ideal gas. Then one can write... 

Wlien 2 g > (Eaa BB binary alloy corresponds to an Ismg ferromagnet (J> 0) and the system splits into two phases one rich in A and the other rich in component B below the critical temperature T. On the other hand, when 2s g < (Eaa+ bb > system corresponds to an antiferromagnet the ordered phase below the critical temperature has A and B atoms occupying alternate sites. 

With a further increase in the temperature the gas composition moves to the right until it reaches v = 1/2 at the phase boundary, at which point all the liquid is gone. (This is called the dew point because, when the gas is cooled, this is the first point at which drops of liquid appear.) An unportant feature of this behaviour is that the transition from liquid to gas occurs gradually over a nonzero range of temperature, unlike the situation shown for a one-component system in figure A2.5.1. Thus the two-phase region is bounded by a dew-point curve and a bubble-point curve. 

A one-component system (C = 1) has two independent state variabies (T and p). At the tripie point three phases (soiid, iiquid, vapour) coexist at equiiibrium, so P = 3. From the phase ruie f = 0, so that at the tripie point, T and p are fixed - neither is free but both are uniqueiy determined. If T is free but p depends on T (a sloping line on the phase diagram) then f = 1 and P = 2 that is, two phases, solid and liquid, say, co-exist at equilibrium. If both p and T are free (an area on the phase diagram) F = 2 and P = 1 only one phase exists at equilibrium (see Fig. A1.18). 

In Fig. 8.8, we see that sulfur can exist in any of four phases two solid phases (rhombic and monoclinic sulfur), one liquid phase, and one vapor phase. There are three triple points in the diagram, where various combinations of these phases, such as monoclinic solid, liquid, and vapor or monoclinic solid, rhombic solid, and liquid, coexist. However, four phases in mutual equilibrium (such as the vapor, liquid, and rhombic and monoclinic solid forms of sulfur, all in mutual equilibrium) in a one-component system has never been observed, and thermodynamics can be used to prove that such a quadruple point cannot exist. 

Mist flow, one component In a one-component system with finely dispersed drops in the mist flow, the mass transfer between phases over a large interfacial area has to be considered. For the compression wave the frozen state can be assumed to be subcooled liquid, superheated vapor conditions generated by the wave are fairly stable, and the expressions for the two-component system are valid (Henry, 1971) ... 

Liquid chromatography (LC) and, in particular, high performance liquid chromatography (HPLC), is at present the most popular and widely used separation procedure based on a quasi-equilibrium -type of molecular distribution between two phases. Officially, LC is defined as a physical method... in which the components to be separated are distributed between two phases, one of which is stationary (stationary phase) while the other (the mobile phase) moves in a definite direction [ 1 ]. In other words, all chromatographic methods have one thing in common and that is the dynamic separation of a substance mixture in a flow system. Since the interphase molecular distribution of the respective substances is the main condition of the separation layer functionality in this method, chromatography can be considered as an excellent model of other methods based on similar distributions and carried out at dynamic conditions. 

Figure 3.3 outlines the similarities in those methods in which the polarity of the various test substances is an important consideration. All of these methods are essentially chromatographic, a word coined originally by Tswett in 1906 which now implies the separation of the components of a mixture by a system involving two phases, one of which is stationary and the other mobile. 

Temperature and pressure are the two variables that affect phase equilibria in a one-component system. The phase diagram in Figure 15.1 shows the equilibria between the solid, liquid, and vapour states of water where all three phases are in equilibrium at the triple point, 0.06 N/m2 and 273.3 K. The sublimation curve indicates the vapour pressure of ice, the vaporisation curve the vapour pressure of liquid water, and the fusion curve the effect of pressure on the melting point of ice. The fusion curve for ice is unusual in that, in most one component systems, increased pressure increases the melting point, whilst the opposite occurs here. 

An interesting example of a one-component systems is SiOa, which can exist in five different crystalline forms or as a liquid or a vapor. As C = 1, the maximum number of phases that can coexist at equilibrium is three. Each phase occupies an area on the T P diagram the two-phase equilibria are represented by curves and the three-phase equilibria by points. Figure 13.1 (2, p. 123), which displays the equUi-brium relationships among the sohd forms of Si02, was obtained from calculations of the temperature and pressure dependence of AG (as described in Section 7.3) and from experimental determination of equUibrium temperature as a function of equilibrium pressure. 

Thus whenever two phases of the same single substance (one component system) are in equilibrium, at a given temperature and pressure, the molar free energy is the same for each phase. 

Figure 1. Phase diagram for a one-component system with two solid phases.

Several studies attempted to relate the partition coefficient P of a solute in a liquid chromatographic or a gas chromatographic system with the composition of the two phases, one of which has a varying composition [19-23]. Tijssen et al. [24] and Schoenmakers [25] derived a relation between the partition coefficient and a binary mobile phase in reversed-phase HPLC from the solubility parameter theory of Hildebrand et al. [26]. Similarly, a relation can be derived for liquid-liquid extraction with extraction liquids composed of three components ... 

So, for a one-component system containing two phases in equilibrium, we have three thermodynamic conditions of equilibrium [Eqs. (2.14)-(2.16)] and four unknown parameters, Ta, Pa, Pp, and Pp. If we arbitrarily assign a value to one of the parameters, we can solve for the other three (three equations, three unknowns). 

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