# 9: Phase Equilibria

• 9.1: Thermodynamics of Mixing
A natural place to begin a discussion of mixtures is to consider a mixture of two gases.
• 9.2: Partial Molar Volume
he partial molar volume of compound A in a mixture of A and B can be defined using the total differential of V.
• 9.3: Chemical Potential
The chemical potential tells how the Gibbs function will change as the composition of the mixture changes. And since systems tend to seek a minimum aggregate Gibbs function, the chemical potential will point to the direction the system can move in order to reduce the total Gibbs function.
• 9.4: The Gibbs-Duhem Equation
The Gibbs-Duhem equation relates how the chemical potential can change for a given composition while the system maintains equilibrium. So for a binary system, consisting of components A and B (the two most often studied compounds in all of chemistry)
• 9.5: Non-ideality in Gases - Fugacity
The relationship for chemical potential was derived assuming ideal gas behavior. But for real gases that deviate widely from ideal behavior, the expression has only limited applicability. In order to use the simple expression on real gases, a “fudge” factor is introduced called fugacity. Fugacity is used instead of pressure.
• 9.6: Colligative Properties
Colligative properties are important properties of solutions as they describe how the properties of the solvent will change as solute (or solutes) is (are) added.
• 9.7: Solubility
The maximum solubility of a solute can be determined using the same methods we have used to describe colligative properties. If this chemical potential is lower than that of a pure solid solute, the solute will dissolve into the liquid solvent (in order to achieve a lower chemical potential!) So the point of saturation is reached when the chemical potential of the solute in the solution is equal to that of the pure solid solute.
• 9.8: Non-ideality in Solutions - Activity
The bulk of the discussion in this chapter dealt with ideal solutions. However, real solutions will deviate from this kind of behavior. So much as in the case of gases, where fugacity was introduced to allow us to use the ideal models, activity is used to allow for the deviation of real solutes from limiting ideal behavior.
• 9.9: Prelude to Phase Equilibrium
From the very elementary stages of our journey to describe the physical nature of matter, we learn to classify mater into three (or more) phases: solid, liquid, and gas. This is a fairly easy classification system that can be based on such simple ideas as shape and volume.
• 9.E: Mixtures and Solutions (Exercises)
Exercises for Chapter 7 "Mixtures and Solutions" in Fleming's Physical Chemistry Textmap.
• 9.E: Phase Equilibrium (Exercises)
Exercises for Chapter 8 "Phase Equilibrium" in Fleming's Physical Chemistry Textmap.
• 9.S: Mixtures and Solutions (Summary)
Summary for Chapter 7 "Mixtures and Solutions" in Fleming's Physical Chemistry Textmap.
• 9.10: Single Component Phase Diagrams
The stability of phases can be predicted by the chemical potential, in that the most stable form of the substance will have the minimum chemical potential at the given temperature and pressure.
• 9.11: Criterion for Phase Equilibrium
The thermodynamic criterion for phase equilibrium is simple. It is based upon the chemical potentials of the components in a system. For simplicity, consider a system with only one component. For the overall system to be in equilibrium, the chemical potential of the compound in each phase present must be the same.
• 9.12: The Clapeyron Equation
Based on the thermodynamic criterion for equilibrium, it is possible to draw some conclusions about the state variables p and T and how they are related along phase boundaries. This results in the Clapeyron equation.
• 9.13: The Clausius-Clapeyron Equation
The Clapeyron equation can be developed further for phase equilibria involving the gas phase as one of the phases. This is the case for either sublimation (solid → gas) or vaporization (liquid → gas).

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