Real Solutions

1. Valentim M. B. Nunes ESTT - IPT April 2019 Real Solutions

2. For liquid solutions is advantageous to describe the solutions as ideal, and deviations from the ideality through excess functions. To keep the formalism used for ideal solutions, we define the chemical potential of a component 1 in a real solution as follows: were a1 is the activity of component 1 (ideal solutions: a1 = x1).

3. The excess functions are thermodynamic properties of solutions that are in excess relative to an ideal solution or ideally diluted, in the same conditions of temperature, pressure and composition. Relationship between thermodynamic properties are preserved :

4. The partial derivatives of extensive properties are also maintained.

5. The partial molar excess functions are defined similarly. If M is an extensive thermodynamic property, then the corresponding partial molar property, mi comes: By the Euler theorem:

6. The activity coefficient of component i is given by: At constant T and p:

7. Fundamental results Since, then:

8. Symmetric convention: Anti-symmetric convention:

9. Deviations to ideality: At low pressures:

10. Calculation of i

11. Binary mixtures At moderate or low pressures (away from the critical point) the effect of pressure in GE is low. It is necessary to correlate GE with the composition of the mixture. The simplest expression is: A =A(T)  Empirical constant with units of energy, depending on the temperature, but independent of the composition.

12. Two suffix Margules equations: Simple Mixtures Similar molecules in size Shape Chemical structure

13. Variation of activity coefficients: ln  A/RT 0.5 x1

15. Redlich-Kister expansion: Margules's equation is very simple. Generally it is required a more complex equation to represent adequately the excess Gibbs energy.

16. The number of parameters needed to represent the activity coefficients gives an indication of the complexity of the solution. If the number of parameters is large (4 or more) the solution is complex. If you only need one parameter the solution is simple. Most frequently solutions used in chemical engineering require two or three parameters in the expansion of Redlich-Kister. For simple solutions, B = C = D =0 ln(1/2) 0 x1

17. Gibbs-Duhem equation applied to i At constant p and T, Binary mixture

18. The expansion of Wohl is a general method to express the excess Gibbs energy in terms of parameters with physical meaning. were,

19. Parameters q – Represents actual volumes, or sections of molecules and measures the "sphere of influence" of molecules in solution. Larger molecules have higher values of q's. In solutions containing non-polar molecules: Parameters a – Represent interaction parameters. a12 represents a 1-2 interaction; a112, interaction between 3 molecules, etc.

20. van Laar equation It is an application of the previous expansion. For binary solutions of two components, not very chemically dissimilar and with different molecular sizes (e.g. benzene and isooctane) the coefficients a112, a122, etc., can be ignored:

21. From the van Laar equation we obtain : were A’ = 2q1a12 and B’ = 2q2a12

22. This equation is useful for more complex mixtures. If A’ = B‘ then we obtain the Margules's equation.

23. NRTL – “non-random two liquid”

24. UNIQUAC – “Universal quasi-chemical theory” UNIFAC – “Universal quasi-chemical functional group activity coefficients”

25. VLE calculations (Bubble point, dew point and Flash)

27. Azeotropy Disregarding the non-ideality in the vapor phase:

28. Until now we admitted total liquid phase miscibility. Let's now consider the cases where the liquids are only partially miscible. At constant p and T, the condition for stability is equivalent to a minimum Gibbs energy.

29. It occurs partial miscibility when, xx1 ou x2 For a ideal solution, GE = 0, then we never have phase separation!

30. Considering GE = Ax1x2 The lower value that satisfies the inequality is A = 2RT

31. The separation between stability and instability of a liquid mixture is called incipient instability. The condition of instability is dependent on non-ideality and temperature. From previous equation, the critical temperature of (solution) solubility comes: By the Margules's equation Tc is always a maximum!

35. Theories of solutions When two or more liquids are mixed to form a liquid solution the purpose of theories of solutions is to express the properties of liquid mixture in terms of Intermolecular forces and liquid structure. It is intended to predict the values of the coefficients of activity in terms of meaningful, molecular properties calculated from the properties of pure components. Precursors: van der Waals  van Laar.

36. Scatchard – Hildebrand Theory Van Laar recognized correctly that simple theories could be built if we considered cases in which VE and SE ~ 0 Later, Hildebrand has verified experimentally that many solutions were in agreement with those assumptions, designated regular solutions. It is defined cohesive energy density as follows : Uvap– energy of complete vaporization of saturated liquid to the ideal gas state (infinite volume)

37. It is also defined, for a binary mixture, the volume fraction of 1 and 2: For UE we obtain: For molecules for which the dominant forces are London dispersion forces:

38. Introducing the concept of solubility parameter, : We obtain:

39. if SE = 0, then we obtain the regular solutions equation: 1 e 2 are function of temperature, but 1 - 2 it’s almost temperature independent! The difference between the solubility parameters of a mixture gives a measure of the non-ideality of the solution.