Single-component systems Gibbs phase rule

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

According to the Gibbs phase rule (number of degrees of freedom = number of components - number of phases + 2 see Atkins, 1998), for a system containing a single chemical distributed between two phases at equilibrium, there is only one... 

The Gibbs phase rule shows that specifying temperature and pressure for a two-component system at equilibrium containing a solid solute and a liquid solution fixes the values of all other intensive variables. (Verify this statement.) Furthermore, because the properties of liquids and solids are only slightly affected by pressure, a single plot of solubility (an intensive variable) versus temperature may be applicable over a wide pressure range. 

In 1926 Kohnstamm extended the Gibbs phase rule to encompass the appearance of critical points in one- and multicomponent systems. He assumed that the critical point may be considered as a specific, additional phase. If / phases coexist, and next become critical, p l meniscuses disappear in a solution consisting of c components. Hence, the Gibbs phase mle supplemented by the critical phase has the following form =c- p + p-l) + 2 = c-2p + > Consequently, for / -critical point at least c = 2p- component system is required. This yields for = 0, i.e. a single critical point, following conditions ... 

Gibbs phase rule for a single-component system... 

In order to understand this isotherm, it is worthwhile to apply Gibbs phase rule (93) which, at constant temperature and pressure, states that the number of degrees of freedom (F) in a system at equilibrium is equal to the number of components (C) in the system minus the number of phases (solid, liquid, gas) present in the system, or F — C — P. Crisp (94) has derived a two-dimensional phase rule to apply to a single plane surface containing q surface phases. The rule predicts that the number of degrees of Freedom (F) will he F = C — Ph — q — ), where C is the total number of components in the system, F is the number of bulk phases, and q is the number of surface phases. In the case of deoxycholic acid spread on aqueous substrate the number of components (C) can be considered to be two, the water of the aqueous phase and the deoxycholic acid. The number of bulk phases, that is the substrate, can be 1 or 2 and the number of surface phases can be 1 or 2. When the area per molecule is very large, for instance 10,000 A- molecule (right side of Fig. 11), the surface pressure is very low (>0.1 dynes/cm) but... 

Single-component systems are useful for illustrating some of the concepts of equilibrium. Using the concept that the chemical potential of two phases of the same component must be the same if they are to be in equilibrium in the same system, we were able to use thermodynamics to determine first the Clapeyron and then the Clausius-Clapeyron equation. Plots of the pressure and temperature conditions for phase equilibria are the most common form of phase diagram. We use the Gibbs phase rule to determine how many conditions we need to know in order to specify the exact state of our system. 

In Chapter 6, we introduced some important concepts that we can apply to systems at equilibrium. The Clapeyron equation, the Clausius-Clapeyron equation, and the Gibbs phase rule are tools that are used to understand the establishment and changes of systems at equilibrium. However, so far we have considered only systems that have a single chemical component. This is very limiting, because most chemical systems of interest have more than one chemical component. They are multiple-component systems. 

In Chapter 6, we introduced the Gibbs phase rule for a single component. Recall that the phase rule gives us the number of independent variables that must be specified in order to know the condition of an isolated system at equilibrium. For a single-component system, only the number of stable phases in equilibrium is necessary to determine how many other variables, or degrees of freedom, are required to specify the state of the system. 

Equation 7.3 is the more complete Gibbs phase rule. For a single component, it becomes equation 6.19. Note that it is applicable only to systems at equilibrium. Also note that although there can be only one gas phase, due to the mutual solubility of gases in each other, there can be multiple liquid phases (that is, immiscible liquids) and multiple solid phases (that is, independent, nonalloyed solids in the same system). 

In dealing with a single component system, the Gibbs phase rule tells us the melting point is a function of pressm-e only. Since most of the processes we will be dealing with take place at ambient pressrue we will concern oruselves with T, the temperature at which a plane front solid can remain indefinitely with its melt. 

Equation (15) is called the general Gibbs—Duhem relation. Because it tells us that there is a relation between the partial molar quantities of a solution, we will learn how to use it to determine a Xt when all other X/ il have been determined. (In a two-component system, knowing Asolvent determines Asolute.) This type of relationship is required by the phase rule because, at constant T, P, and c components, a single-phase system has only c — 1 degrees of freedom. 

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