1. Chapter 6: The Thermodynamic Properties of Real Substances

2. Important Notation

3. Learning objectives After studying this chapter we should be able to:
Evaluate partial derivatives of thermodynamic variables;
Interrelate these partial derivatives;
Obtain volumetric equation of state parameters from critical properties;
Solve problems for real fluids using equations of state;
Construct tables and charts of thermodynamic properties.

4. Math preliminaries
In previous chapters, we identified the following variables in thermodynamic calculations:
For a single-phase pure substance, specifying two of them sets the state of the system.

5. Math preliminaries
Take as independent variables. Then, other molar properties will be written as functions of these two. For example:
Then, the change in molar internal energy can be calculated as function of changes in

6. Math preliminaries
It is essential to indicate the property (or properties) held constant in the differentiation.
If you do not do this, it won’t take long before you get lost in your derivations.

7. Math preliminaries
In open systems, the extensive properties will also depend on the amounts. For a pure substance:
Change in energy interval can be calculated as function of the changes in

8. Math preliminaries
Two independent variables are kept constant in each partial derivative. As before, it is essential to indicate the properties held constant in the differentiation.

9. Math preliminaries
There is a simple relationship between derivatives of extensive properties at constant mole numbers and similar derivatives involving molar properties. For example:

10. Math preliminaries
Pay attention to what is kept constant in the differentiation:

11. Example 1
Show that
Solution

12. Example 1 1
The term inside the rectangle is generally different from 0.

13. Math preliminaries In the previous slide, we used that:
For any property:

14. Math preliminaries
If is a dependent variable of independent variables :
If remains constant:

15. Mathematical preliminaries
If remains constant:
Interchanging the roles of dependent and independent variables:

16. Math preliminaries
Triple product rule (clock rule)

17. Thermodynamic partial derivatives
Change in molar internal energy expressed as function of the changes in the molar entropy and molar volume of a closed system, obtained in Chapter 4:
If is taken as a dependent property and and are taken as independent properties:

18. Thermodynamic partial derivatives
Compare:
These two expressions must always give identical results.

19. Thermodynamic partial derivatives
Compare:
These two expressions must always give identical results.

20. Thermodynamic partial derivatives
Remember that:
Classroom activity: for each of them, obtain the differential expression for a closed system and compare with the corresponding partial derivatives, as done in the molar internal energy in the preceding slides.

21. Thermodynamic partial derivatives
Some derivatives have a special name:

22. Thermodynamic partial derivatives
Maxwell relations
Starting from molar internal energy

23. Thermodynamic partial derivatives
Maxwell relations
Classroom activity
Obtain the Maxwell relations starting from the molar:
enthalpy
Helmholtz energy
Gibbs energy

24. Thermodynamic partial derivatives
With Maxwell relations, it is possible to evaluate changes in entropy, which cannot be measured directly, in terms of changes in measurable quantities, such as temperature, volume, and pressure.

25. Thermodynamic partial derivatives
A few words of advice
Section 6.2 of the textbook has many thermodynamic identities. Trying to memorize them is the wrong approach.
The right approach is to understand how they are derived. If you do, you will be able to derive these identities and many more.

26. Example 2 Derive the relationship between and . Solution
There are many ways to prove this. Compare this solution with that of Illustration in the textbook.
Let us begin with the definition of and the differential form of enthalpy as function of molar entropy and pressure:

27. Example 2 Derive the relationship between and . Solution
There are many ways to prove this. Compare this solution with that of Illustration in the textbook.
Let us begin with the definition of and the differential form of molar enthalpy as function of molar entropy and pressure:

28. Example 2
Now use the relationship between changes in molar entropy and in molar internal energy and molar volume:
Then:

29. Example 2
Now use the relationship between changes in molar internal energy and in temperature and molar volume. Then, will appear in the formula.
Then: 1

30. Example 2 Get rid of the molar energy derivative:
The Maxwell relation for the Helmholtz energy is:
Then:

31. Example 2 Let us now get rid of the molar energy derivative:
The Maxwell relation for the Helmholtz energy is:
Then:

32. Example 2

33. Example 2

34. Example 2
This is a famous thermodynamic identity that takes these many steps to prove.
More difficult than the number of steps is finding a suitable sequence of transformations.

35. Thermodynamic partial derivatives
There are more systematic approaches:
Jacobian transformations
Bridgman table (which results from Jacobian transformations)

36. Example 3
Estimate the Joule-Thomson coefficient of refrigerant HFC-134a at 140oC and 10 bar using its pressure-enthalpy diagram.
Solution 1
Using the triple product (clock) rule.
The pressure-enthalpy diagram does not give these derivatives directly, but it is possible to get numerical estimates.

37. f3_3_4
f3_3_4.jpg

38. Example 4 (dH/dP) at constant T -1 kJ/(kg.bar) (dH/dT) at constant P
T (deg C)
P (bar)
H (kJ/kg)
140 10
525
150
537 5
530
(dH/dP) at constant T -1
kJ/(kg.bar)
(dH/dT) at constant P
1.2
kJ/(kg.K)

39. Thermodynamic partial derivatives
Open systems of a pure substance
In Chapter 4, the following equation was derived relating changes in internal energy to changes in entropy, volume, and amount of a pure substance:
This gives us a hint that it might be interesting to treat the internal energy as:

40. Thermodynamic partial derivatives
Open systems of a pure substance

41. Thermodynamic partial derivatives
Open systems of a pure substance
Now, compare:

42. Thermodynamic partial derivatives
Open systems of a pure substance
Now, compare:
See equation b

43. Thermodynamic partial derivatives
Comparing the results for closed and open systems
A similar derivative did not appear during the analysis of closed systems.

44. Thermodynamic partial derivatives
Comparing the results for closed and open systems
A similar derivative did not appear during the analysis of closed systems.
Let us use this observation to discuss some misconceptions.

45. Thermodynamic partial derivatives
Clarifying some issues
Is a state property?

46. Thermodynamic partial derivatives
Clarifying some issues
Is a state property?
Yes, it is.
It can be written as function of other state properties, as in the derivative:

47. Thermodynamic partial derivatives
Clarifying some issues
The derivative appeared when analyzing an open system of a pure substance. Does this property exist for a closed system?

48. Thermodynamic partial derivatives
Clarifying some issues
The derivative appeared when analyzing an open system of a pure substance. Does it mean this property does not exist for a closed system?
To answer this question, let me first appeal to your previous knowledge and intuition.

49. Thermodynamic partial derivatives
Clarifying some issues
Suppose a closed rigid cylinder contains a gas in a given state. In this state, this gas has a certain pressure. But:

50. Thermodynamic partial derivatives
Clarifying some issues
Suppose a closed rigid cylinder contains a gas in a given state. In this state, this gas has a certain pressure. But:
However, the cylinder volume does not change. Is there a contradiction? Is something wrong?

51. Thermodynamic partial derivatives
Clarifying some issues
In thermodynamics, many of calculations are based on virtual processes.
In other words, think of a “What if” question.
We know the cylinder has rigid walls, but “what if its volume changes?”

52. Thermodynamic partial derivatives
Clarifying some issues – back to the formulated question
The derivative appeared when analyzing an open system of a pure substance. Does this property exist for a closed system?
It does exist. Although we know the system is closed, “what if there is a change in the number of moles”?

53. Changes in thermodynamic properties
Tables and diagrams of thermodynamic properties are available for few substances and mixtures.
Usually, these properties are estimated using models that describe the behavior of solids, liquids, and gases.
Many processes in chemical industries deal with fluids (liquid and/or gas): emphasis on studying their behavior. However, we will study other important problems containing solid phases (examples: adsorption, catalysis, electrochemistry).

54. Changes in thermodynamic properties
Necessary data:
Volumetric
Calorimetric

55. Volumetric equations of state
For pure substances, they are mathematical relationships among
For mixtures, the relationships also involve composition: later in the course.

56. Volumetric equations of state
The simplest equation of state (EOS):
Pros:
Simple;
Works well at high temperatures and low pressures.
Cons:
Wrong predictions at low temperatures or high pressures;
Even at low pressures, it fails for some types of substances, e.g., organic acids.

57. Volumetric equations of state
van der Waals EOS
The Nobel Prize in Physics 1910
Johannes Diderik van der Waals
The van der Waals EOS can describe the behavior of gases and liquids.
a and b are parameters that characterize each substance.

58. Volumetric equations of state
The Nobel Prize in Physics 1910
Johannes Diderik van der Waals

59. Example 5
Methane is stored in a 10 m3 tank at 300 K and 100 bar. Use the van der Waals EOS to estimate the molar volume of methane and then the number of moles of methane inside the tank.
methane
Write down the steps to solve this problem.

60. Volumetric equations of state
There are hundreds, possibly thousands, of EOS to describe fluid phase behavior.
Many (but not all) are empirical modifications of the van der Waals EOS.
Few are commonly used for chemical process design, e.g.:
Soave-Redlich-Kwong EOS
Peng-Robinson EOS
Patel-Teja EOS
D.B. Robinson
Amyn Teja

61. Volumetric equations of state
Several EOS are special cases of the general form:

62. t6_4_2
t6_4_2.jpg

63. Volumetric equations of state
The compressibility factor is defined as:
Isolating the molar volume:
Using this molar volume expression in the EOS:

64. Volumetric equations of state
After tedious algebra, it is possible to show that the EOS can be written in the form:

65. Volumetric equations of state
After tedious algebra, it is possible to show that the EOS can be written in the form:
An EOS that fits this format is called a cubic EOS.
Knowing {a, b, P, T}, the equation can be solved, giving three Z roots.

66. f6_4_3 Pressure-molar volume diagram for Oxygen
f6_4_3.jpg
Pressure-molar volume diagram for Oxygen
as predicted by the Peng-Robinson EOS.

67. Calorimetric information
Several approaches are possible but a key observation comes from the analysis of the following figure:

68. Calorimetric information
Several approaches are possible but a key observation comes from the analysis of the following figure:
For any path from state 1 to state 2, the change in pressure is the same and the change in temperature is also the same.

69. Calorimetric information
Several approaches are possible but a key observation comes from the analysis of the following figure:
For any path from state 1 to state 2, the change in any state property (e.g., internal energy, enthalpy, entropy) is the same.

70. Calorimetric information
Several approaches are possible but a key observation comes from the analysis of the following figure:
If the path from state 1 to state 2 does not matter to evaluate changes in state properties, choose a convenient path.

71. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
From state 1, go to a hypothetical ideal gas state at T1 and P1.
Why do we do this?
Changes in ideal gas properties are easy to evaluate.
Passing through a hypothetical state will not affect the final result.
State 1

72. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
From state 1, go to a hypothetical ideal gas state at T1 and P1.
From the hypothetical ideal gas state at T1 and P1, go to a hypothetical ideal gas state at T2 and P2.
State 1

73. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
From state 1, go to a hypothetical ideal gas state at T1 and P1.
From the hypothetical ideal gas state at T1 and P1, go to a hypothetical ideal gas state at T2 and P2.
From the hypothetical ideal gas state at T2 and P2, go to state 2.
State 1

74. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
State 1

75. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
State 1

76. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
Changes in ideal gas properties.
State 1

77. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
Departure (residual) function in state 1.
State 1

78. Calorimetric information
A convenient path
State 2
Real fluid at T1 P1
Real fluid at T2 P2
Ideal gas at T1 P1
Ideal gas at T2 P2
Departure (residual) function in state 2.
State 1

79. Calorimetric information
Changes in ideal gas properties.
Expressions derived in the previous chapters.

80. Calorimetric information
Departure (residual) functions

81. Calorimetric information
Departure (residual) functions
Real fluid at T, P
Ideal gas at T, P
Fluid at T, lim P->0
Similar paths can be used for the properties listed in the previous slide.

82. Calorimetric information
Departure (residual) functions
Come from the EOS used to model the fluid.

83. Calorimetric information
Departure (residual) functions
Similar operations can be done with other properties.
Another detail: the previous slide shows integrals in pressure. However, with equations of state such as van der Waals, Peng-Robinson, Soave, and most used in Chemical Engineering, the integration should be in molar volume. Therefore, it is necessary to change the integration variable.
The algebra involved in most cases is long and tedious. Please refer to the textbook for additional details.

84. f6_4_4 Pressure-molar enthalpy diagram for Oxygen
f6_4_4.jpg
Pressure-molar enthalpy diagram for Oxygen
as predicted by the Peng-Robinson EOS.

85. Principle of corresponding states
Empirical observation:
Different molecular species may be represented by a volumetric equation of state, e.g., the Peng-Robinson EOS does a good job at modeling hydrocarbons – what makes it a popular choice in the oil and gas industries.
Assumption
It should be possible to have generalized correlations applicable to many molecular species.
Interesting to examine the behavior predicted by the van der Waals equation of state.

86. f6_6_1
Critical point conditions for a pure substance
f6_6_1.jpg
At:

87. Principle of corresponding states
At the critical point, for the van der Waals equation of state :
Solving these two equations to find a and b, we obtain:

88. Principle of corresponding states
Now using:
in the van der Waals equation of state at the critical point :
We obtain:

89. Principle of corresponding states
The compressibility factor at the critical point is:
Then, it is possible to express a and b as functions of critical temperature and critical pressure:
Or as functions of critical volume and critical pressure:

90. t6_6_1
t6_6_1.jpg

91. Principle of corresponding states
Going back to the van der Waals equation of state:
But:

92. Principle of corresponding states
It follows that:
In these equations, we used the reduced properties, defined as:

93. Principle of corresponding states
Two fluids at the same values of reduced temperature and reduced pressure would have the same reduced volume: they are said to be in corresponding states.
A practical way to implement this idea is to express the compressibility factor as follows:

94. f6_6_2 van der Waals EOS is not totally accurate. Note:
f6_6_2.jpg
Between 0.23 and 0.31 for
real fluids

95. Principle of corresponding states
A practical alternative for the third parameter is to use the acentric factor:
In practice, property predictions using the principle of corresponding states are being gradually displaced by the use of more accurate methods.

96. Generalized equations of state
The algebraic steps used to set the relationship between a and b in the van der Waals equation of state and the critical properties can be repeated for other models. Some of them also use the acentric factor as an additional characterization parameter, as the Peng-Robinson EOS:

97. Estimation methods for critical and other properties
There is a huge number of experimental critical point data available in the literature.
But, from time to time, engineers need data for a substance whose information is not available:
New chemical process with unusual substances;
Pharmaceuticals;
Product engineering: finding substances with properties within a certain range for a specific application

98. Estimation methods for critical and other properties
Group contribution methods for property estimation
Assumption
Properties can estimated by adding the contribution of the chemical functions that constitute the molecule
Many such methods exist. The textbook describes the method of Joback, which is widely used.

99. The method of Joback

100. t6_9_1
t6_9_1.jpg

101. Example 7
Compute the properties of n-octane and ethylene glycol using the Joback method and compare them to experimental data, if available.

102. pg_258a
n-octane
pg_258a.jpg

103. pg_258b
ethylene glycol
pg_258b.jpg

104. Summary of the formulas

105. t6_2_1
t6_2_1.jpg

106. Recommendation
Read chapter 6 in the textbook and review all examples.