Phase Equilibrium

Phase equilibrium is the study of the equilibrium which exists between or within different states of matter namely solid, liquid and gas. Equilibrium is defined as a stage when chemical potential of any component present in the system stays steady with time. Phase is a region where the intermolecular interaction is spatially uniform or in other words physical and chemical properties of the system are same throughout the region. Within the same state, a component can exist in two different phases such as allotropes of an element. Also, two immiscible compounds in same liquid state can coexist in two phases.

Phase equilibrium has wide range of applications in industries including production of different allotropes of carbon, lowering of freezing point of water by dissolving salt (brine), purification of components by distillation, usage of emulsions in food production, pharmaceutical industry etc. Solid-solid phase equilibrium has a special place in metallurgy and is used to make alloys of different physical and chemical properties. For instance, melting point of alloys of copper and silver is lower than melting point of either copper or silver.

Phase diagrams
Phase diagrams are used to understand the relationship between different phases and are usually represented as the change in the phase of a system as a function of temperature, pressure or composition of the components in a system. The system exists in a phase where Gibbs free energy of the system is least. At equilibrium, temperature, pressure and chemical potential of constituent component molecules in the system have to be same throughout all the phases. Figure 1 gives a general schematic of phase diagram of a single component system  (Lue, 2009).

The curves shown in the figure represent the coexistence of two phases. Melting curve is the curve in the phase diagram along which solid and liquid phase of a system stays in equilibrium. Liquid and gas phase of a system stay in equilibrium along the vaporization curve while sublimation curve represents the equilibrium stage between solid and gas phase. Triple point is point on the graph where all the three states coexist and is unique for every component. If α and β are any two phases in which a component can exist, by using first and second law of thermodynamics, the slope of any of the curves in figure 1 can be represented by

                                                                                                                     
 (1)
 
where P and T are the pressure and temperature of the component, respectively. ΔS and ΔV are the changes in molar entropy and molar volume, respectively of the component when it changes its phase from α to β at equilibrium.
Equation (1) can also be written as

 

 
   
                                                                                                     (2)
 
and is referred as Clausius Clapeyron equation. Vapor pressure in equilibrium means number of molecules evaporating is equal to the number of molecules condensing on the water surface. According to Gibbs phase rule about multi-component systems, the number of intensive degrees of freedom in a non-reactive multi-component system, F at equilibrium is given by
                                                                   (3)

where C is the number of non-reactive components in a system and n is the number of phases. For example, for two-component and two-phase systems there are two intensive degrees of freedom namely either temperature, pressure or mole fraction. In other words, in case of two-component and two-phase system at equilibrium, there are only two intensive variables needed to uniquely determine the thermodynamic state of system.
There are different kinds of equilibrium that are studied in detail namely liquid-vapor equilibrium, liquid-liquid equilibrium, solid-liquid equilibrium, solid-solid equilibrium (alloys or allotropic forms) etc. Increase in number of components, chemical reactions, presence of surfactants and deviation from ideal behavior are some of the causes of enhanced complexity of phase diagrams. Here, we have discussed a special case of vapor-liquid equilibrium which is commonly used in distilleries i.e. ethanol-water mixtures.

Vapor-Liquid equilibrium


Raoult’s law states that partial vapor pressure exerted by a component in an ideal solution is the product of its mole fraction and vapor pressure of pure component. Thereby, total vapor pressure exerted by an ideal solution with k components, P is given by
 
(4)
 where pi is the vapor pressure of the pure component and xi is the respective mole fraction. For instance, if in a brine solution (two-component mixture), one component(common salt) has negligible vapor at the specified temperature, therefore vapor pressure of the mixture is simply product of the mole fraction of water and vapor pressure of water at specified temperature. Raoult’ law is only valid for ideal solutions which assume no intermolecular interaction between different components while in most of the cases dealt with in industries there are hardly any ideal solutions. Modified Raoult’s law is used for non-ideal solutions(J.M. Smith, 2005); accordingly total pressure exerted, P is expressed as
 
(5)
 
where yi is the activity coefficient for the ith component.

Figure 2: Vapor-Liquid equilibrium diagram for ethanol-water system
 
Figure 2 shows the phase diagram for ethanol-water mixtures(Dortmund Data Bank). Similar graphs are used in distilleries to calculate the desired temperatures for specific outlet compositions. It contains a vapor curve and a liquid curve that is used to estimate the vapor or liquid compositions. The vapor curve is also called as dew point curve while the liquid curve is also called as bubble point curve. Interestingly, the curves meet each other at their lowest point suggesting that if an ethanol-water mixture is boiled further it will have same composition in the vapor phase. Therefore, the maximum ethanol concentration obtained by simple distillation process in an ethanol-water mixture would be 95.5%.These mixtures which have the lowest/highest boiling point on the phase diagram and cannot be purified further are called as Azeotropes. Azeotropes have higher or lower boiling point than either of its constituent components depending on if the mixtures have positive or negative deviation from Roult’s law, respectively. In case of three-component or three-phase systems,  triangular graphs are used with three end points of triangle representing the three pure components (phases).
References
Dortmund Data Bank.
J.M. Smith, H. V. (2005). Introduction to Chemical Engineering Thermodynamics. New York: McGraw Hill.
Lue, L. (2009). Chemical Thermodynamics. Leo Lue & Ventus publishing APS.

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