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
Pressure, psia Z
a=RT/P * (1-z)
1.0
3.8
100
0.9462
3.923
200
0.8873
4.110
300
0.8221
4.324
400
0.7787
4.582
500
0.6621
4.929

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 =0.771 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. Use a fugacity coefficient table like Figure 7.1
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. Mixtures of Ideal Gases
True only for ideal gases
But R, T and P are constant, so… 1
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

51. Example 7.4
At 1 atm pressure the ethanol-water azeotrope has a composition of:
10.57 mole% water
89.43 mole% ethanol
The composition is the same in the vapor and liquid phases
The temperature is C

52. Example 7.4 cont
At this temperature the pure component vapor pressures are:
Water atm
Ethanol atm
Estimate the fugacity and activity coefficients in each phase

53. For an azeotrope… xethanol=yethanol and xwater=ywater
At one atm it is a reasonable assumption that both the water and ethanol behave as ideal gases
Thus the fugacity in the gas phase is equal to the partial pressure

54. Because we are at equilibrium the gas phase fugacity equals the liquid phase fugacity
If this was an ideal solution, a would be equal to x – but it’s not, so…

55. We use the pure component vapor pressure as the standard fugacity
If this was an ideal solution, a would be equal to x – but it’s not, so…
Remember, I’m using Pi0 to indicate the pure component vapor pressure – but Dr. deNevers is using p

56. Thus

57. Thus
This solution is not ideal, as shown by the fact that the activity coefficients are not 1

58. Fugacity coefficient, Poynting Factor and Alternative Notation
The ratio of fugacity to pressure is called the fugacity coefficient
Often the symbol, n, is used
Older literature
Usually refers only to pure component fugacities and pressures
Sometimes the symbol, f, is used
Newer literature
Can refer to both pure components and mixtures

59. Older usage
g, the activity coefficient corresponds to deviation from ideality due to mixing in the gas or liquid
n, the fugacity coefficient, corresponds to deviation from ideal gas behavior
And since
a, the activity

60. Modern usage
For pure species fi is exactly the same as ni, and accounts only for the departure from ideal gas behavior
For mixtures accounts for not only the deviation from ideal gas behavior, but also for nonideal mixing behavior.

61. Modern Usage In older usage, , is called the fugacity of component i.
In more modern usage it is often represented as , and called the partial fugacity – like partial pressure
Obviously, it can’t be a partial molal property, because fugacity is not an extensive property

62. Poynting Factor

63. Estimating Fugacities of Individual Species in Gas Mixtures
If you don’t have an ideal gas – how do you estimate the fugacity of a gas?
Use reliable PvT data
Use an equation of state
Example 7.5

64. Example 7.5
Table 7.F presents Volume residuals (a) for a mixture of methane with n-butane, at 220 F
Use this data to find the fugacity of methane and the fugacity of n-butane in a mixture that is:
78.4 mole% methane
21.6 mole% n-butane
This corresponds to a mixture that is 50 wt% methane

65. We’ll need the partial molal volume residual,
Find its value at each of the pressures in table 7. F, which will allow us to integrate using the trapezoid rule

66. At 220 F and 100 psia
Use the method of tangent intercepts
6.6 ft3/lbmole
This is the partial molal volume residual of methane
0.6 ft3/lbmole
Tangent Line, at 78.4% methane

67. Repeat the process at the other pressures, to create Figure 7.10
aavg=0.290

68. Substitute

69. Fugacities from an EOS for Gas Mixtures
PvT data are only available for a small number of gas mixtures
For other cases, using a reliable EOS is a good (though not as accurate) alternative

70. What equation to use?
This is easier for the graphical procedure, used with PvT data
This is easier if we are going to use an EOS

71. But we have a problem
The equations of state from chapter 2 are for singe pure species
For mixtures we need a mixing rule
Usually semi-empirical
For a mixture of a and b, at some T and P, we can find the pure component values of z and use them in our mixing rule to get a combination value of z

72. Any mixing rule will have the form…
One possibility is…

73. Lewis and Randall fugacity rule
If n=1
which is just a weighted average
It is equivalent to an ideal solution of non-ideal gases
This is not rigorously correct, but is a useful approximation, especially at pressures less than a few atmospheres

74. Example 7.6
Compares the Lewis Randall fugacity rule results to those found with PvT data
I’ll let you work through it on your own

75. Lewis Randall Rule Used because it is simple
It is the next step in complexity after the ideal gas law
Unfortunately, sometimes gases exist in states for which we can not compute ni

76. This makes finding the fugacity coefficient hard
Gases can exist as a mixture at conditions where they would normally be liquids if they were pure
This makes finding the fugacity coefficient hard
You’ll need to use an equation of state, because you can’t compare it to some non-existent gas
See Example 7.7

77. Other mixing rules
The Lewis Randall rule is only one of many possibilities
They’ll be introduced in Chapters 9 and 10

78. Summary Fugacity was invented because chemical potential is awkward
For pure gases we correlate and compute f/P based on either measured PvT data, or an appropriate EOS

79. Summary cont
For pure liquids and solids we usually compute the fugacity from the vapor pressure. The effect of increases above the vapor pressure are small.
Ideal liquids are like gases, an approximation
Activity and activity coefficient are nondimensional representations of fugacity

80. Summary Cont
Fugacity, activity, and activity coefficient are calculated – they can not be measured
For mixtures, we calculate fugacity directly from PvT data, or from an EOS
When we use an EOS we need to use a mixing rule

81. Summary cont.
The simplest mixing rule is the Lewis and Randall rule – an ideal solution of nonideal gases
For mixtures of liquids we’ll have to wait for chapters 8 and 9 for appropriate mixing rules