Coexisting phases maximum number

From this equation one concludes that the maximum number of phases that can coexist in a oiie-component system (d = 1) is tliree, at a unique temperature and pressure T = 0). When two phases coexist F= 1), selecting a temperature fixes the pressure. Conclusions for other situations should be obvious. 

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. 

Determine the maximum number of phases which can coexist in equilibrium in a ternary system. 

The phase rule(s) can be used to distinguish different types of equilibria based on the number of degrees of freedom. For example, in a unary system, an invariant equilibrium (/ = 0) exists between the liquid, solid, and vapor phases at the triple point, where there can be no changes to temperature or pressure without reducing the number of phases in equilibrium. Because / must equal zero or a positive integer, the condensed phase rule (/ = c — p + 1) limits the possible number of phases that can coexist in equilibrium within one-component condensed systems to one or two, which means that other than melting, only allotropic phase transformations are possible. Similarly, in two-component condensed systems, the condensed phase rule restricts the maximum number of phases that can coexist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible, each of which maintains the number of equilibrium phases at three and keeps / equal to zero (L represents a liquid and S, a solid) ... 

The minimum number of degrees of freedom for any system is zero. When F = 0, the system is invariant, and Eq. (2.12) becomes ir = 2 + N. This value of tt is the maximum number of phases which can coexist at equilibrium for a system containing N chemical species. When N - 1, this number is 3, and we have a triple point For example, the triple point of water, where liquid, vapor, and the common form of ice exist together in equilibrium, occurs at 0.01°C and 0.00610 bar. Any change from these conditions causes at least one phase to disappear. 

When N = 2, the phase mle becomes F = 4 — n. Since there must be at least one phase (n= 1), the maximum number of phase-mle variables which must be specified to fix the intensive state of the system is three namely, P, T, and one mole (or mass) fraction. All equilibrium states of the system can therefore be represented in three-dimensional P-T-composition space. Within this space, the states of pairs of phases coexisting at equilibrium (F=4 — 2 = 2) define surfaces. A schematic three-dimensional diagram illustrating these surfaces for VLE is shown in Fig. 10.1. 

Further examples. Three independent components. Consider a system composed of water and two soluble salts which have a common ion, and which form a double salt, e.g. magnesium chloride and potassium chloride. These two salts can combine according to the equation MgCl2+2KCl = MgCl2.2KC1 (carnallite). We have thus four different types of molecules and one chemical equation connecting them, hence three independent components. The maximum number of coexistent phases is therefore five, e.g. the three solid salts, saturated solution, and water vapour. The coexistence of these phases is, however, only possible at one temperature, namely, the transition temperature of the double salt. 

In other words, whenever two phases of the same single substance are in equilibrium, at a given temperature and pressure, the molar free energy is the same in each phase. This conclusion can be extended to three phases, which is the maximum number that can coexist in equilibrium for a system of one component. 

This means that, in this four-component system, the maximum number of phases that can coexist at equilibrium isF=C — F = 4 — 0 = 4. In other words, the three minerals in equilibrium with solution produce an invariant point with F = 0 (the intersection of... 

From the phase rule, we can easily obtain the maximum number of coexisting phases as we set the number of freedoms equal to zero. Thus, the maximum number of phases is... 

The maximum number of phases that can coexist is three (e.g. solid, liquid and gas). In this case, P = 3, and Equation (SI.4) gives F = 1, and there is only one degree of freedom available to the system. Three phases will coexist along a line in the phase diagram. On a phase diagram drawn for one atmosphere pressure, three phases occur at a point. 

This shows that the maximum number of phases that can coexist at equilibrium in a binary system at an arbitrarily chosen pressure (or temperature) is three (/ = 3 for c = 2, / = 0), which is consistent with our observations. 

The value of r cannot exceed r ax for a given system because then f would be negative, which is unphysical. Eor example, for a one-component system, the maximum number of phases that can coexist is rmax = 1 + 2 = 3. Therefore, four-phase coexistence in a system with only one component is impossible, although it can occur in a two-component system, where rniax is equal to 4. 

What is the maximum number of phases that can coexist in a system with five components present What is the dimensionality of the three-phase coexistence region in a mixture of Al, Ni, and Cu What type of geometrical region does this define Liquid-liquid two-phase coexistence in pure (that is, one-component) systems is extremely rare. Using library and Internet resources, find two examples of liquid-liquid coexistence in a pure material. 

Equation (1.72) is the unary Gibbs phase rule. It indicates that the maximum number of phases which can coexist in a unary system is 3 and this results in an invariant equilibrium (f = 0). Note that the equilibria in each type of phase diagram in Figure 1.4 satisfy this condition. 

We also have electroneutrality, so the number of components is C = 7 — 3 — 1=3. The maximum number of phases that can coexist in equilibrium (for E=0)isC+2 = 5. Here we have used up one degree of freedom by specifying the temperature, so the maximum number of phases that can coexist is four. 

In general there are K components in FI different phases (solid, liquid, gas,...) at constant T and f. We may ask What is the maximum number of coexisting phases at equilibrium Or to be pictorial, is the situation in Fig. 3.2 possible, where a one-component system contains four coexisting phases— Gas , Liquid , Solid , and Flubber ... 

Applied to our above system we find 1 - 4 - - 2 < 0. This means that four phases cannot coexist simultaneously in a one-component system. The maximum number of coexisting phases in a one-component system is three— but thermodynamics does not specify which three phases. Relation Eq.(3.12) is Gibbs phase rule. 

If we consider A and A3 as components, then the phase rule Eq.(3.12) allows up to four coexisting phases. However, we have an additional equilibrium constraint imposed by Eq. (3.69), reducing the degrees of freedom by one and the maximum number of coexisting phases to three. The modified phase rule therefore is... 

For polymorphic systems of a particular material we are interested in the relationship between polymorphs of one component. A maximum of three polymorphs can coexist in equilibrium in an invariant system, since the system cannot have a negative number of degrees of freedom. This will also correspond to a triple point. For the more usual case of interest of two polymorphs the system is monovariant, which means that the two can coexist in equilibrium with either the vapour or the liquid phases, but not both. In either of these instances there will be another invariant triple point for the two solid phases and the vapour on the one hand, or for the two solid phases and the liquid on the other hand. These are best understood in terms of phase diagrams, which are discussed below, following a review of some fundamental thermodynamic relationships that are important in the treatment of polymorphic systems. 

The program VDWMIX is used to calculate multicomponent VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either IPVDW or 2PVDW see Sections 3.3 to 3.5 and Appendix D.3). The program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which the calculations are to be done (for as many sets of calculations as the user wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the /f constants of the PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and the vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply binary interaction parameteifs) for each pair of components in the multicomponent mixture. These interaction parameters can be... 

SV, SL, LV lines of Figure 1), nor as a point with zero degrees of freedom (the triple point of Figure 1). With two components and zero degrees of freedom, a maximum of four phases can coexist at a quadruple point, e.g., solid-liquid one-liquid two-vapor (SLLV) equilibrium with three components a maximum of five phases can coexist, etc. The phase rule demonstrates that increasing the number of components substantially increases the possible complexity of the phase behavior. 

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