1. Chemical Thermodynamics 2013/2014 11 th Lecture: Thermodynamics of Simple Mixtures Valentim M B Nunes, UD de Engenharia

2. 2 Introduction In this lecture we will begin the study of simple non-reactive mixtures. To do that we will introduce the concept of partial molar properties. At this stage we deal mainly with binary mixtures, that is: We shall also consider mainly non-electrolyte solutions, where the solute is not present as ions.

3. 3 Partial Molar Volume To understand the concept of partial molar quantities, the easiest property to visualize is the “volume”. When we mix two liquids A and B, we have three possibilities: Volume contraction Volume dilatation No total volume change. In the last case if we mix, for instance, 20 cm 3 of liquid A and 80 cm 3 of liquid B, we will obtain 100 cm 3 of solution. This mean that A…B interactions are equal to A…A or A…B molecular interactions! partial molar volume In most cases that is not going to happen! For instance the molar volume of water is 18 cm 3.mol -1. But if we add 1 mol of water to a huge amount of ethanol the volume will only increase by 14 cm 3. The quantity 14 cm 3 is the partial molar volume of water.

4. 4 Partial Molar Volume The partial molar volumes change with composition because molecular environment also change. For water/ethanol system we have: The partial molar volume of a substance is defined as: Once we know the partial molar volume of two components of a binary mixture we can calculate the total volume:

5. 5 Measuring Partial Molar Volume One method is measuring the dependence of volume on composition and determine the slope dV/dn:

6. 6 Method of Intercepts Dividing both members of last equation (slide 4) by the total number of moles, n, we obtain: Differentiating: Substituting: See next part of lecture!

7. 7 Method of Intercepts Finally we obtain:

8. 8 Partial Molar Gibbs Function chemical potential. Another partial molar property, already introduced, is the partial molar Gibbs function or the chemical potential. Then, the total Gibbs function of a mixture is: But we have seen before that (recall the fundamental equations for open systems):

9. 9 Gibbs – Duhem Equation Since G is a state function the previous equations must be equal, and then we obtain: Generalizing: Gibbs – Duhem Equation This is the Gibbs – Duhem Equation. It tell us that partial molar quantities cannot change independently: in a binary mixture if one increases the other should decrease! It can also be applied to the partial molar volume.

10. 10 Thermodynamics of Mixing Perfect Gases We already now that systems evolves spontaneously to lower Gibbs energy. If we put together two gases they will spontaneously mix, then G should decrease. Let us consider two perfect gases in two containers at temperature T and pressure p. The initial total Gibbs energy is:

11. 11 Gibbs energy of Mixing After mixing we have: The difference is the Gibbs energy of mixing and therefore: Using Dalton´s Law, y i =p i /p, we finally obtain: Since molar fractions are always lower than 1, the Gibbs energy of mixing is always negative!

12. 12 Other Thermodynamic mixing functions Sincewe easily obtain: Now since ΔH = ΔG + TΔS: And since for perfect gases ΔG mix is independent of pressure, then: Purely entropic

13. 13 Summary

14. 14 Chemical potential of liquids In order now to study the equilibrium properties of liquid solutions we need to calculate the chemical potential of a liquid. To do this, we will use the fact that the chemical potential of a substance present as a dilute vapor must be equal to the chemical potential of the liquid, at equilibrium. Important note: We will denote quantities relating to pure substances by the superscript “*” so the chemical potential of a pure liquid A it will be written as μ* A (l). Remember also that it is usual, in order to characterize a given solution, to distingue between the solvent (usually the substance in bigger quantity or in the same physical state of solution) and solute (in lower quantity or in a different state of aggregation)

15. 15 Ideal Solutions For a pure liquid in equilibrium with the vapor we can write: If another substance is present (solution!) then the chemical potential is: Combining the two equations leads to: ?

16. 16 Ideal Solutions French chemist François Raoult found that the ratio p A /p A * is equal to mole fraction of A in the liquid, that is: Raoult´s Law This equation is known as the Raoult´s Law. It follows that for the chemical potential of liquid: ideal solutions This important equation is the definition of ideal solutions. Alternatively we may also state that an ideal solution is the on in which all the components obey to the Raoult´s Law.

17. 17 Raoult`s Law

18. 18 Examples

19. 19 Ideal Dilute Solutions In ideal solutions the solute, as well as the solvent, obeys Raoult´s Law. But in some cases, in non-ideal solutions, the partial pressure of solute is proportional to the mole fraction but the constant of proportionality is not the pure component vapor pressure but a constant, known as the Henry´s constant, K B : This is known as the Henry`s Law. Mixtures obeying to the Henry`s Law are called ideal dilute solutions.

20. 20 Example

21. 21 Ideal Mixture of Liquids When two liquids are separated we have: When they are mixed we have: As a consequence: This equation is the same as that for two perfect gases! All the other conclusions are equally valid. The “driving” force for mixing is the increase of entropy.

22. 22 Excess Functions Real solutions have different molecular interactions between A…A, A…B and B…B particles. As a consequence ΔH≠0 and ΔV≠0. For instance if ΔH is positive (endothermic) and ΔS negative (“clustering”) ΔG is positive and the liquids are immiscible. Alternatively the liquids can be only partially miscible. excess functions Thermodynamic properties of real solutions can be expressed in terms of excess functions (G E, S E, etc.). For example for entropy: In this context the concept of regular solutions is very important. In this solutions H E ≠0 but the S E =0.

23. 23 Examples These figure shows experimental excess functions at 25 ºC. H E values for benzene/ cyclohexane show that the mixing is endothermic (since ΔH=0 for an ideal solution). V E for tetrachloroethene/cyclopentane shows that there is a contraction at lower mole fractions of tetrachloroethene but expansion at higher mole fractions (because ΔV=0 for an ideal solution)

24. 24 Solutions containing non-volatile solutes Let us now consider a solution in which the solute is non-volatile, so it does not contribute to the vapor pressure, and it does not dissolve in solid sovent. Recalling that the chemical potential of liquid A in solution is: colligative propertiesvapor pressure lowering We can conclude that the chemical potential in the liquid solvent is lower that the chemical potential of pure liquid (remember that x A <1). These give origin to a set of properties of this solutions that are called colligative properties. For instance the vapor pressure lowering: or

25. 25 Colligative Properties Boiling point elevationFreezing point depressionOsmotic pressure All colligative properties have in common that they only depend on the amount of solute but not on the particular substance. The other colligative properties are: Boiling point elevation, Freezing point depression and Osmotic pressure. Graphically we can observe the first two properties: Effect of the solute

26. 26 The elevation of boiling point The heterogeneous equilibrium that maters when considering boiling is the solvent vapor and the solvent in the solution. The equilibrium condition is:

27. 27 The elevation of boiling point This rearranges to: When x B =0, the boiling point is that of pure liquid, Tb, and: The difference between the two equations is

28. 28 The elevation of boiling point Assuming now dilute solutions, that is, x B <<1, then ln(1-x B ) ≈ -x B, so: Since T ≈ T b we can also write: Which gives: This means that the elevation of boiling point, ΔT, depends on x B (no reference to the identity of solute, the other properties are related to the solvent!)

29. 29 The elevation of boiling point We can also calculate the elevation of boiling point in terms of the molality of the solute, m B. Since the mole fraction of solute is small n A >> n B, so: Therefore we obtain: Boling point elevation can now be calculated by: or molal ebullioscopic constant K b is the molal ebullioscopic constant.

30. 30 The depression of freezing point The equilibrium of interest now is between the pure solid solvent and the solution with dissolved solute: At freezing point the equilibrium condition is:

31. 31 The depression of freezing point All the previous calculations are equal, so we can write directly: or molal cryoscopic constant K f is the molal cryoscopic constant.

32. 32 Constants of Common Liquids

33. 33 Osmosis osmosis We have shown that boiling point and freezing point depends on the equilibrium between the solvent in solution and in the solid or vapor state. The last possibility is to have an equilibrium between the solvent in solution and the pure solvent. This is the basis of the phenomenon of osmosis. This consists in the passage of a pure solvent into the solution separated from it by a semipermeable membrane. h μ A *(p,T)μ A (p+π,T) osmotic pressure Π = ρhg is the osmotic pressure

34. 34 Osmotic pressure The equilibrium condition is: From these equation we can easily obtain: For dilute solutions ln x A = ln (1-x B ) ≈ -x B, so, assuming that V m is constant: Now, since x B ≈ n B /n A and n A V m = V, we finally obtain: van`t Hoff equation This is the van`t Hoff equation for osmotic pressure. Note the remarkable similitude with the perfect gas equation!

35. 35 Applications Van`t Hoff equation can be rewritten as: This is a very useful equation for the determination of molar mass of polymers and other macromolecules, assuming dilute solutions and incompressible solutions. inverse osmosis Other application is the reverse or inverse osmosis, If we apply from the solution side a pressure bigger than the osmotic pressure we will reverse the passage of water molecules. This can be used to obtain fresh water from sea water ( π≈ 30 to 50 atm)

36. 36 Ideal solubility of solids Although is not strictly a colligative property the solubility of a solid can be estimated by the same techniques of previous slides. In contact with a liquid the solid will dissolve until the solution is saturated. The equilibrium condition is: From this starting point and using similar deduction we will obtain: Being the properties related only to the solute.