Fugacity, Ideal Solutions, Activity, Activity Coefficient

1. Fugacity, Ideal Solutions, Activity, Activity Coefficient Chapter 7

2. Why fugacity? • Equality of chemical potential is the fundamental criterion for phase and chemical equilibrium. • It is difficult to use, because it approaches -∞ as the concentration approaches 0. • Fugacity approaches 0 as the concentration approaches 0.

3. As chemical potential approaches -∞, RT ln(f) must also approach -∞ Since RT is a constant, ln(f) must approach -∞, and so f must approach 0

4. The units of chemical potential are kJ/mole – so RT was needed to make the units work out, since natural logs are dimensionless Most practical calculations do not include B(T)

6. We know that if phase 1 and phase 2 are in equilibrium then: Our new criterion for physical equilibrium

7. Pure substance fugacities For pure substances…

8. Take the derivative And recall that… At constant temperature….

9. Combine and rearrange And for an ideal gas… But…

10. Which implies that for an ideal gas, f=p For all gases, as the pressure approaches zero, the gas approaches ideal behavior, so…

11. You can check out the rest of the math in Appendix C, however… The * indicates an ideal gas

12. For an ideal gas, a=0 Z=1 for an ideal gas, so 1-z=0

13. So what is the fugacity? • For an ideal gas it is identical to the pressure • It has the dimensions of pressure • Think of it as a corrected pressure, and use it in place of pressure in equilibrium calculations • Calculations often include the ratio of fugacity to pressure, f/P – called the fugacity coefficient

14. Notice that f/P approaches 1 as the value of Pr goes down, and the value of Tr goes up—in other words, as the gas approaches ideality

15. Fugacity of liquids and solids • Follows the same equations as gases, but in practice are calculated differently • The fugacity of gases and liquids is usually much less than the pressure

16. The Fugacity of Pure gases Illustrate this process with example 7.1, page 134 • Estimate the fugacity of propane gas at 220 F and 500 psia • The easiest way is to use figure 7.1

17. We need Pr and Tr

19. Now let’s calculate f, instead of using a chart • If you have reliable PvT measurements, like the steam tables, you could use them – which would give the most reliable values • For this problem, PvT measurements were used to create Figure 7.2 – a plot of z values. (Pv=zRT) • The data are also presented in Table 7.A, so we don’t have to read a graph

21. Table 7.A – Volume residuals of Propane at 220 F

22. Recall that… I shamefully left out units, in the interest of space – see page 135 a changes with pressure, but not as strongly as z, so we can approximate a as a constant

23. We could also use an equation of state (EOS) to find the fugacity • The ideal gas law would tell us that pressure and fugacity are equal – which is obviously a really bad estimate for this case • Let’s try the “little EOS” from Chapter 2 • Equations 2.48 – 2.50

24. Little EOS Pv=zRT Eq 2.48 Eq 2.49 Eq 2.50

25. Substitute into Eq. 7.9 Do the integration Now we can substitute in the values of Pr and Tr =0.771

26. One more approach…Use Thermodynamic tables Recall that… Rewrite this equation for two different states (If you keep the temperature constant, and vary the pressure)

27. Divide both sides by P1P2 and rearrange At very low pressures, f and P are equal But only at very low pressures

28. Now plug in values from a table of thermodynamic values – pg 137 Choose P1 as 1 psia

29. There are obviously several ways to find fugacity, depending on what you know and how accurate you need to be • Use a fugacity coefficient table like Figure 7.1 • Use PvT data, to find z , then use equation 7.9 • Use an equation of state • Use thermodynamic tables and equation 7.11

30. Fugacity of Pure Liquids and Solids • We could compute the fugacity just like we did in the previous example • It is impractical for mathematical reasons • See examples 7.2 and 7.3

31. Fugacity of Solids and Liquids • Does not change appreciably with pressure • Normally we approximate the fugacity of pure solids and liquids as the pure component vapor pressure • We’ll return to this in Chapter 14 when we study cases where this approximation is no longer valid

32. Fugacities of species in mixtures See Appendix C

33. Fugacities of species in mixtures See Appendix C These equations are the same as the pure component equations, except that P has been replaced with the partial pressure, xiP

34. 1 Mixtures of Ideal Gases True only for ideal gases But R, T and P are constant, so… and

35. All of which leads to… For mixtures of ideal gases, the fugacity of each species is equal to the partial pressure of that species

36. Can we extend this concept to ideal solutions? • Ideal solutions are like ideal gases • Neither exist in nature – real gases and solutions are much more complicated • Many gases and solutions exhibit practically ideal behavior • It is often easier to work with deviations from ideal behavior, rather than work directly with the property of interest • The compressibility factor z is an example • Activity coefficient is similar for liquids

37. Ideal solutions • The definition of an ideal solution is that … Usually fi0 is defined as the pure component fugacity – though this is not the only choice

38. Any ideal solution has the following properties True for gases, liquids and solids

39. Any ideal solution has the following properties Tells us there is no volume change with mixing

40. Any ideal solution has the following properties Tells us that there is no heat of mixing

41. Any ideal solution has the following properties Which means there is a Gibbs free energy change with mixing There is always an entropy change associated with mixing

42. Activity and Activity Coefficients • Fugacity has dimensions of pressure • This can cause problems if we don’t pay strict attention to units • Often we want a non-dimensional representation of fugacity – which leads to the activity • When we deal with non-ideal solutions we’ll want a measure of departure from ideal – like z – which leads to the activity coefficient

43. Activity Activity is defined as the ratio of the fugacity of component i, to it’s pure component fugacity Recall that for an ideal gas… Recall that for an ideal solution… Usually – but not necessarily, chosen as the pure component fugacity

44. Activity Rearrange to give… For an ideal solution of either gas or liquid, activity is equivalent to mole fraction

45. Activity Coefficient The activity coefficient is just a correction factor on x, which converts it to a

46. Why bother? • Both are dimensionless • They lead to useful correlations of liquid-phase fugacities. • The normal chemical equilbrium statement – the law of mass action – is given in terms of activities • (The law of mass action is the definition of the equilibrium constant, K)

47. For now… • Activity is rarely used for phase equilibria • It will show up again in Chapters 12 and 13 • Activity coefficient is useful to us now!!

48. For a pure species or for ideal solutions • Activity = mole fraction and.. • Activity coefficient, g =1 • We could redefine an ideal solution as one where g equals 1 • This is like defining an ideal gas as one where z=1

49. Activity coefficients • For real solutions activity coefficients can be either greater or less than 1 • Typically range between 0.1 and 10

50. From Appendix C (the general case) Eq. 7.31 For an ideal solution Eq. 7.32