Maximum Number of Phases

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 

We must have certain properties to distinguish the phases, e.g., density, composition. These properties must differ for different phases. Otherwise, we could not state that there are certain phases present. We consider the case of one component, and we have two intensive parameters, the temperature T and the pressure p to characterize such properties that we address as 77. There should exist a state function of the form 

Since at maximum three phases should coexist with properties different from each other, these phases should have the properties 77i, 772, 77 for a certain pair of T, p. Therefore, the state function has a solution of 

if we could expand the state function in a polynomial, then the degree of the polynomial is at minimum three. 

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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. 

In the case of a unary or one-component system, only temperature and pressure may be varied, so the coordinates of unary phase diagrams are pressure and temperature. In a typical unary diagram, as shown in Figure 3.11, the temperature is chosen as the horizontal axis by convention, although in binary diagrams temperature is chosen as the vertical axis. However, for a one-component system, the phase rule becomes F=l-P+2 = 3-P. This means that the maximum number of phases in equilibrium is three when F equals zero. This is illustrated in Figure 3.11 which has three areas, i.e., solid, liquid, and vapour In any... 

An interesting extension of the original methodology was proposed by Lopes and Tildesley to allow the study of more than two phases at equilibrium [21], The extension is based on setting up a simulation with as many boxes as the maximum number of phases expected to be present. Kristof and Liszi [22, 23] have proposed an implementation of the Gibbs ensemble in which the total enthalpy, pressure and number of particles in the total system are kept constant. Molecular dynamics versions of the Gibbs ensemble algorithm are also available [24-26]. 

The phase diagram of an electrode material may be determined from the slope of the coulometric titration curve. An electrode of N components shows activities which are independent of the composition as long as the maximum number of N phases are in equilibrium with each other. Relative changes in the amounts of the different phases do not change the activities of the components and therefore keep the cell voltage constant. This causes voltage plateaux for any region of the equilibrium of the maximum number of phases. 

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. 

The system is not strictly ternary, because of the limited solubility of AI2O3 in diopside, but we will neglect this minor complication here. Because the maximum number of phases in the system is three (diopside plus plagioclase mixtures plus liquid) and the components also number three, the minimum variance of the system is 1, at constant P. 

The first representation (Figure 46a) shows a portion of Al-Si-K space where H O is present in great abundance and where pH is determined by the phases present. The mineral assemblages are determined by the relative proportions of Al, Si and K, which are extensive variables. No intensive variables are considered. The maximum number of phases which can be present at a given composition is three three phases will be present in a phase-field. 

According to this equation the maximum number of phases that can be in equilibrium in a binary system is = 4 (F= 0) and maximum number of degrees of freedom needed to describe the system = 3 (n=l). This means that all phase equilibria can be represented in a three-dimensional P,T,x-space. At equilibrium every phase participating in a phase equilibrium has the same P and T, but in principle a different composition x. This means that a four-phase-equilibrium (F=0) is given by four points in P, 7, x-space, a three-phase equilibrium (P=l) by three curves, a two-phase equilibrium (F=2) by two planes and a one phase state (F= 3) by a region. The critical state and the azeotropic state are represented by one curve. 

What is the maximum number of phases we can expect in this intermediate system By comparing with the earlier simplified model where C = 5, we might suggest that the maximum F would be seven, adding one new phase for each new component. In fact, if we assume seven phases we find F = 8 + 2— 7 = 3, and again the best independent variables would be p, T, and [C1 ]. 

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 JV = 2, the phase rule becomes F = 4 - it. Since there must be at least one phase (it = 1), the maximum number of phase-rule 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... 

Similarly, in two-component condensed systems, the condensed-phase rule restricts the maximum number of phases that can co-exist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible (Table 11.2), each of which maintain the number of equilibrium phases at three, and keep/equal to zero. The same terms given in Table 11.2 ate also applied to the structures of the phase mixmres. 

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... 

The phase rule answers the question as to the maximum number of phases which can co-exist in equilibrium in a system with a prescribed number of components. The variance (degrees of freedom) of the system is given by... 

This says that the maximum number of phases to be expected in any naturally occurring system is given by the number of components in that system. This is called the Mineralogical Phase Rule, and was pointed out by Goldschmidt in 1911. In open systems,... 

Thus the maximum number of phases to be expected in a natural system is decreased by one for each environmental component, compared to the same system having that component as a system component. In other words, systems having / < 2 will lose one phase for every component that is given an arbitrary potential. 

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. 

We thus obtain the conclusion that the number of phases cannot exceed the number of components by more than two.f For example, in a single component system the maximum number of phases in equilibrium together is three, as occurs at the triple point. 

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. 

In a two-component system, what is the maximum number of phases that can be in equilibrium ... 

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. 

This value of variance is obtained from Equation 4.6. The maximum number of phases, when F = 0, is obtained from Equation 4.7. 

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