Equilibrium univariant

The consequences of the stability condition are clearly demonstrated by considering the univariant equilibrium... 

Uniform positive background, 255 United atom model, 276 Univariant, equilibrium, 19, 23 system, 91... 

It should be noted that we have here considered the system at constant pressure. If we are not considering the system at isobaric conditions, the invariant equilibrium becomes univariant, and a univariant equilibrium becomes divariant, etc. A... 

As we have already stated, the Gibbs free energy modification connected with a polymorphic transition is zero. The univariant equilibrium along the transition curve is described by the Clapeyron equation ... 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

Figure 5.71A shows univariant equilibrium curves for various molar amounts of ferrous component in the orthopyroxene mixture. The P-T field is split into two domains, corresponding to the structural state of the coexisting quartz (a and j3 polymorphs, respectively). If the temperature is known, the composition of phases furnishes a precise estimate of the P of equilibrium for this paragenesis. Equation 5.277 is calibrated only for the most ferriferous terms, and the geobarometer is applicable only to Fe-rich rocks such as charnockites and fayalite-bearing granitoids. 

The Clausius-Clapeyron equation describes the univariant equilibrium between crystal and melt in the P-Tfield. Because molar volumes and molar entropies of molten phases are generally greater than their crystalline counterparts, the two terms and AFfusion both positive and we almost invariably observe an... 

Univariant equilibrium for which there is one degree of freedom, represents the equilibrium between two co-existing phases. Since there is only one degree of freedom, choosing a value for one external variable, e.g. temperature, determines the remaining variable in a dependent manner, and the locus of points represented on the phase diagram for univariant behavior must lie on a line or curve. Thus the curves on the unary phase diagram represent solid-liquid, solid-vapor, solid-solid, and liquid-vapor equilibrium. 

There are other types of equilibria, in addition to the invariant type, which can be deduced from Eq. 2.5. For example, when three phases of a two-component system are in equilibrium, such as with a closed vessel containing hydrogen gas in equilibrium with a metal and the metal hydride, immersed in a water bath, it is possible to change the value of just one variable (temperature or pressure or composition) without changing the number of phases in equilibrium. This is called univariant equilibrium (/ = 1). If the composition is held constant, temperature and pressure will have a fixed relationship in a univariant system. Hence, if the pressure of hydrogen gas in the vessel is increased slightly, the temperature of its contents remains the same as heat escapes through the vessel walls to the water bath. 

From the phase rule, this is seen to be a univariant equilibrium (there is 1 degree of freedom). If x is constant and the pressure is 1 atm, the metal and its oxide may be considered to be in their standard states, so their activities, aMxo2 and aM> respectively, are unity hence,... 

The common feature of these phase diagrams is that the melt is in univariant equilibrium with one crystalline phase of a variable composition. A complete series of solid solutions are formed. 

Figure 14.1 The univariant equilibrium curve for the anorthite breakdown reaction, Sanorthite = grosssular + 2 kyanite +quartz. Calculated with the univariant curve option in SUPCRT92. The effect of adding Mg to the grossular is to lower the equilibrium pressure, as shown by the data of Hensen et al. (1975).

At any point along the liquidus curves TA-eAB and Te-eAB, there exist two phases-either solid A -i- melt, or solid B -i- melt Consequently, the degree of freedom F is 1 that is, either the temperature or the composition can be altered without changing the number of phases present (univariant equilibrium). At the eutectic point Oab the two solid phases A and B are in equilibrium with the melt Thus, the number of phases is P = 3, and F = 0, since any variation of the temperature or the composition will invariably displace the system from point Oab at which two solubility (liquidus) curves intersect (invariant equilibrium). 

Using the foregoing compositions, the univariant equilibrium garnet to spinel Iherzolite in the quaternary system has the following coefficients ... 

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