1. Lecture 18Multicomponent Phase Equilibrium1 Theories of Solution The Gibbs energy of mixing is given by: And the chemical potential is: For ideal gases, the partial pressures are given by: And the activity coefficients are: Substitution 4 into 1 gives; Thus: 1- Ideal Gas Mixtures 1 2 3 4

2. Lecture 18Multicomponent Phase Equilibrium2 Ideal Gas Mixtures GMGM 0 XBXB 1 For systems with zero enthalpies of mixing, which we generally call ideal mixtures, the entropy of mixing completely determines G of mixing. SMSM 0 XBXB 1 HMHM 0 XBXB 1

3. Lecture 18Multicomponent Phase Equilibrium3 Ideal Gas Entropy of Mixing For a binary ideal gas mixture we can plot the entropy as a function of composition as shown on the right (in units of R). The ideal entropy of mixing: In the case of a binary becomes: The slope of the entropy of mixing curve is given by: This implies that it is impossible to completely purify a material!

4. Lecture 18Multicomponent Phase Equilibrium4 Raoult’s Law If upon forming a mixture the partial pressures of the vapor in equilibrium above the mixture is such that: Then the pressure of the vapor will be a weighted sum of the partial pressures, where the weights are the mole fractions of each component: The solution is thus said to be Raoutian or Ideal and P(X i ) for a Raoutian solution is plotted on the right. Also note that for a Raoutian solution: Since the activity and mole fraction are equal, we have the same thermodynamics as the ideal gas mixture. That is: V V S S PA0PA0 0 XBXB 1 PB0PB0 P Ideal a i >1 a i <1

5. Lecture 18Multicomponent Phase Equilibrium5 Dilute Solutions For a dilute solution, the X i of one component is very small, while X i for the other component is nearly one. In this case, the activity coefficient  i of the dilute component should be composition independent since the component’s environment is constant (it is surrounded by the other component). As the dilute component is added, the probability of it having a like neighbor is small and so its activity is constant over a range of dilute concentrations. For this case, the partial pressure of the dilute component is proportional to the amount of that component. This is known as Henry’s Law: 1 0 XBXB 1 a

6. Lecture 18Multicomponent Phase Equilibrium6 Dilute Solutions Note that using the Gibbs Duhem equation for the partial molar G, it can be shown that when B obeys Henry’s law, A obeys Raoult’s Law. Henry’s Law for B dilute 1 0 XBXB 1 a For A-B binary The infinitesimal change in the partial molar Gibbs free energy Raoult’s Law for A:

7. Lecture 18Multicomponent Phase Equilibrium7 Excess Functions Remember that: For an ideal solution: We define the excess function as the difference between the actual value of the mixture and the value for an ideal mixture:

8. Lecture 18Multicomponent Phase Equilibrium8 Excess Functions The entropy of mixing is usually assumed to be ideal so that the excess Gibbs free energy of mixing is the excess enthalpy of mixing The Gibbs free energy of mixing is then The excess enthalpy of mixing minus T times the ideal entropy of mixing. Let’s take a closer look at the Gibbs free energy of mixing using the concept of excess mixing functions:

9. Lecture 18Multicomponent Phase Equilibrium9 Regular Solutions The Regular Solution Model is a simple example of a non-ideal solution. Recall that for a mixture: The partial molar Gibbs free energy of mixing (the difference between component i’s contribution to G in the mixture versus pure i) is related to the activity. The Gibbs free energy of mixing is the weighted sum of the contributions from each component. The Gibbs free energy of mixing is then related to the activities as shown. In the ideal case the activities were just the mole fractions: The excess Gibbs free energy of mixing is the difference between the non-ideal and ideal G of mixing:

10. Lecture 18Multicomponent Phase Equilibrium10 The Regular Solution Model The Regular Binary Solution is defined as one which has the following form for the activity coefficients: Of course because the mole fraction of component A is just one minus the mole fraction of B we have: And substituting the activity relationships for the Regular solution gives: This can be manipulated to find: The excess Gibbs free energy of mixing is: The excess Gibbs free energy of mixing of the Regular Binary Solution.

11. Lecture 18Multicomponent Phase Equilibrium11 Regular Solutions And substituting the Regular Solution excess G of mixing: Notice that the first two terms are the negative ideal entropy of mixing multiplied by T: The Gibbs free energy of mixing is the sum of the excess and ideal Gibbs free energies of mixing : Thus, the last term is the enthalpy of mixing (and also the excess enthalpy of mixing since the ideal enthalpy of mixing is just zero): The enthlapy of mixing of the Regular Binary Solution with  = 10 J/mol.

12. Lecture 18Multicomponent Phase Equilibrium12 Regular Solutions Regular Solutions with  =10000J/mol.

13. Lecture 18Multicomponent Phase Equilibrium13 Regular Solutions Regular Solutions at T=300K

14. Lecture 18Multicomponent Phase Equilibrium14 Regular Solutions: Atomistic Interpretation The enthalpy of mixing is related to the interactions between the atoms that make up the mixture. If the solid has bond energies as follows: E AA E AB E BB The enthalpy of mixing is given by: Z is the coordination number N T the total number of atoms N A the number of A atoms

15. Lecture 18Multicomponent Phase Equilibrium15 Regular Solutions: Atomistic Interpretation The enthalpy of mixing is determined as the sum of the total interactions between the like and unlike atoms in the mixture: E AA E AB E BB Then the enthalpy of mixing is: Z is the coordination number N T the total number of atoms N A the number of A atoms AA BB BB  A (in A)  B (in B)

16. Lecture 18Multicomponent Phase Equilibrium16 Regular Solutions: Atomistic Interpretation Notice that the number of A atoms divided by the total number of atoms is just the mole fraction of A. Substituting in the mole fractions gives: This is the correct form for the Regular solution where we make the definition: The enthalpy of mixing of the Regular Binary Solution is determined by the difference between the AB bond energy and the average of the AA and BB bond energies. Various more complex models of solutions have been developed with more complicated expressions for the enthalpy of mixing, including: next nearest neighbor interactions, non-ideal entropies of mixing, etc. E R AA BB ABAB

17. Lecture 18Multicomponent Phase Equilibrium17 Solution of Defects We could extend the principles of the thermodynamics of mixtures between atoms to mixtures between atoms and defects, such as vacancies Vacancies increase energy because they result in broken bonds and the decrease in energy due to the entropy they contribute from the uncertainty of their placement in the solid.