Chapter 11 Solution Thermodynamics: Theory

1
Chapter 11Solution Thermodynamics Theory

• Chapter 6 treats the thermodynamic properties of
  pure species or constant-composition fluids.
  However, the preceding chapter demonstrates that
  applications of chemical engineering
  thermodynamics are often to systems wherein
  composition is a primary variable.

2
Chapter 11Solution Thermodynamics Theory

• In the chemical, petroleum, and pharmaceutical
  industries multicomponent gases or liquids
  commonly undergo composition changes as the
  result of mixing and separation processes, the
  transfer of species from one phase to another, or
  chemical reaction.

3
Chapter 11Solution Thermodynamics Theory

• Because the property of such systems depend
  strongly on composition as well as on temperature
  and pressure, our purpose in this chapter is to
  develop the theoretical foundation for
  application of thermodynamics to gas mixture and
  liquid solution.

4
Chapter 11Solution Thermodynamics Theory

• The theory is introduced through derivation of a
  fundamental property relation for homogeneous
  solution of variable composition. Convenience
  here suggests the definition of a fundamental
  new property called the chemical potential, upon
  which the principles of phase and chemical
  reaction equilibrium depend.

5
Chapter 11Solution Thermodynamics Theory

• This leads to the introduction of a new class of
  thermodynamic properties known as partial
  properties. The mathematical definition of these
  quantities allows them to be interpreted as
  properties of the individual species as they
  exist in solution.
• For example, in a liquid solution of ethanol and
  water the two species have partial molar
  properties whose values are somewhat different
  from the molar properties of pure ethanol and
  pure water at the same temperature and pressure.

6
Chapter 11Solution Thermodynamics Theory

• Property relations for mixtures of ideal gases
  are important as references in the treatment of
  real-gas mixtures, and they form the basis for
  introduction of yet another important property,
  the fugacity. Related to the chemical potential,
  it is vital in the formulation of both phase and
  chemical reaction equilibrium relations.

7
Chapter 11Solution Thermodynamics Theory

• Finally, a new class of solution properties is
  introduced. Known as excess properties, they are
  based on an idealization of solution behavior
  called the ideal solution. Its role is like that
  of the ideal gas in that it serves as a reference
  for real-solution behavior. Of particular
  interest is the excess Gibbs energy, a property
  which underlies the activity coefficient.

8
Chapter 11Solution Thermodynamics Theory

• Properties depend strongly on composition,
• T and P.
• Gas mixtures and liquid solutions
• New terms chemical potential, partial
  properties, fugacity, and excess property (e.g.
  excess Gibbs energy).
• Why we have to study Solution Thermodynamics?....
  ..
• The prediction of the equilibrium existing
  between phases, and to understand the process and
  to calculate phase and chemical reaction
  equilibria.

9
Chapter 11Solution Thermodynamics Theory

• What is the most important property ?
• G.
• For pure component
• G G (T, P)
• For a homogeneous mixture e.g. containing i
  components mixture
• G G (T, P, n1, n2, , ni)

10
11.1 Fundamental Property Relation

• Equation (6.6) expresses the basic relation
  connecting the total Gibbs energy to the
  temperature and pressure in any closed system
• where n is the total number of moles of the
  system.

11
11.1 Fundamental Property Relation

• An appropriate application is to a single phase
  fluid in a closed system wherein no chemical
  reactions occur. For such a system the
  composition is necessarily constant, and
  therefore
• The subscript n indicates that the numbers of
  moles of all chemical species are held constant.

12
11.1 Fundamental Property Relation

• For the more general case of a single phase, open
  system, material may pass into and out of the
  system, and nG becomes a function of the numbers
  of moles of the chemical species present.
  Presumably, it is still a function of T and P,
  and we rationalize the functional relation
• where ni is the number of moles of species i.

13
11.1 Fundamental Property Relation

• The total differential of nG is then
• The summation is over all species present, and
  subscript nj indicates that all mole numbers
  except the i th are held constant.

14
11.1 Fundamental Property Relation

• The derivative in the final term is given its own
  symbol and name. Thus, by definition the chemical
  potential of species i in the mixture is

15
11.1 Fundamental Property Relation

• With this definition and with the first two
  partial derivatives replaced by (nV) and
• (nS), the preceding equation becomes
• Equation (11.2) is the fundamental property
  relation for single phase fluid systems of
  variable mass and composition.

16
11.1 Fundamental Property Relation

• It is the foundation upon which the structure of
  solution thermodynamics is built. For the special
  case of one mole of solution, n1 and nixi
• Implicit in this equation is the functional
  relationship of the molar Gibbs energy to its
  canonical variable, here T, P, and xi

17
11.1 Fundamental Property Relation

• Although the mole number ni of Eq. (11.2) are
  independent variables, the mole fraction xi in
  Eq. (11.3) are not, because
• ?i xi 1. This precludes certain mathematical
  operations which depend upon independence of the
  variables. Nevertheless, Eq. (11.3) does imply

18
11.1 Fundamental Property Relation

• Other solution properties come from definitions.
• When the Gibbs energy is expressed as a function
  of its canonical variables, it plays a role of a
  generating function, providing the means of
  calculation of other thermodynamics properties by
  simple mathematical calculations and implicitly
  represents complete property information.

19
11.2 The Chemical Potential

• Consider a closed system containing of two phases
  in equilibrium. Within this closed system, each
  individual phase is an open system, free to
  transfer mass to the other. Equation (11.2) may
  be written for each phase

20
11.2 The Chemical Potential

• Phases ? and ?.
• Free mass transfer between
• phases ? and ?

21
11.2 The Chemical Potential

• Two phases are closed, and at equilibrium.

22
11.2 The Chemical Potential

• Chemical Potential (?)
• is an extensive property,
• provides a measure of the work of a system is
  capable when a change in mole numbers occurs e.g.
  chemical reaction or a transfer of mass.

23
11.2 The Chemical Potential
For ? phases at equilibrium, and N is the number
of species.
Thus, multiple phases at the same T and P are in
equilibrium when the chemical potential of each
species is the same in all phases. What will be
happened?
24
11.3 Partial Properties

• The partial molar property any extensive
  property of a solution changes with respect to
  the number of moles of any component i in the
  solution at constant T, P and composition of the
  others.
• Partial property and molar property (except for
  ideal solution).

25
11.3 Partial Properties
M solution properties For example V, U, H, S,
G partial properties For example Vi , Ui
, Hi , Si , Gi Mi pure species properties For
example Vi , Ui , Hi , Si , Gi
26
11.3 Partial Properties
Partial molar property of species i in
solution is defined as It is a response
function, i.e., a measure of the response of
total property nM to the addition at constant T
and P of a differential amount of species I to a
finite amount of solution.
27
11.3 Partial Properties
Comparison of Eq. (11.1) with Eq. (11.7) written
for the Gibbs energy shows that the chemical
potential and the partial molar Gibbs energy are
identical i.e., Thus, the partial molar Gibbs
energy is the chemical potential.
28
11.3 Partial Properties
The definition of a partial molar property, Eq.
(11.7), provides the means for calculation of
partial properties from solution property data.
Implicit in this definition is another, equally
important, equation that allows the reverse,
i.e., calculation of solution properties from
knowledge of the partial properties.

29
11.3 Partial Properties
The total differential of nM is

30
11.3 Partial Properties
Because the first two partial derivatives on the
right are evaluated at constant n and because the
partial derivative of the last term is given by
Eq. (11.7), this equation has the simpler form

31
11.3 Partial Properties
Since ni xin And Substitute these terms
to Eq. (11.9), and then rearrange

32
11.3 Partial Properties
Summability relations
33
11.3 Partial Properties
Differentiate Eq. (11.11), Comparison of
this equation with Eq. (11.10) yields Gibbs /
Duhem equation

34
11.3 Partial Properties
This equation must be satisfied for all changes
in P, T, and the Mi caused by changes of state in
a homogeneous phase. For the important special
case of changes at constant T and P, it
simplifies to

35
11.3 Partial Properties
Gibbs/Duhem equation If T and P constant
36
11.3 Partial Properties
Solution property M Partial property
Pure-species property Mi
37
11.3 Partial Properties
Binary system
38
11.3 Partial Properties

• For two components

39
Example 11.3
The need arise in a laboratory for 2000 cm3 of an
antifreeze solution consisting of 30 mol
methanol in water. What volumes of pure methanol
and of pure water at 25 ?C must be mixed to form
the of antifreeze, also at 25 ?C ? Partial molar
volumes for methanol and water in a 30 mol
methanol solution and their pure-species molar
volume, both at 25 ?C , are Methanol (1) and
water (2)
40
Example 11.3
Solution
41
Example 11.3
Solution
42
Example 11.3
Solution The line drawn tangent to the V-x1 curve
at x10.30, illustrates the values of V140.272
cm3 mol-1 and V218.068 cm3 mol-1.
43
Relations among Partial Properties
We show now how partial properties are related to
one another. By Eq. (11.8), µi Gi, and Eq.
(11.20 may be written Application of the
criterion of exactness, Eq. (6.12) , yields the
Maxwell relation,
44
Relations among Partial Properties
We have two additional equations One can
write the RHS in the form of partial molar, and
change the composition from n to x.

45
Relations among Partial Properties

• Every equation that provides a linear relation
  among thermodynamic properties of a
  constant-composition solution has as its
  counterpart an equation connecting the
  corresponding partial properties of each species
  in the solution.

46
Relations among Partial Properties
47
Relations among Partial Properties

• This may be compared with Eq. (6.10). These
  examples illustrate the parallelism that exists
  between equations for a constant composition
  solution and the corresponding equations for the
  partial properties of the species in solution.
  We can therefore write simply by analogy many
  equations that related partial properties.

48
11.4 The Ideal-Gas Mixture Model

• Dalton Law
• Every gas has the same V and T.

49
11.4 The Ideal-Gas Mixture Model

• Amagat Law ?????
• Every gas has the same P and T.

50
11.4 The Ideal-Gas Mixture Model

51
11.4 The Ideal-Gas Mixture Model

• Properties of each component species are
  independent of the presence of other species.
• A partial molar property (other than volume) of a
  constituent species in an ideal-gas mixture is
  equal to the corresponding molar property of the
  species as a pure ideal gas at the mixture
  temperature but at a pressure equal to its
  partial pressure in the mixture.

52
11.4 The Ideal-Gas Mixture Model
53
11.4 The Ideal-Gas Mixture Model
54
11.4 The Ideal-Gas Mixture Model
Problem 11.1What is the change in entropy when
0.7 m3 of CO2 and 0.3 m3 of N2 each at 1 bar and
25 ?C blend to form a gas mixture at the same
condition? Assume ideal gases.
55
11.4 The Ideal-Gas Mixture Model
Solution
56
11.5 Fugacity Fugacity Coefficient Pure Species
As evident from Eq. (11.6), the chemical
potential µi provides the fundamental criterion
for phase equilibria. This is true as well for
chemical reaction equilibria. However, it
exhibits characteristics which discourage its
use. The Gibbs energy, and hence µi , is defined
in relation to the internal energy and entropy.
Because absolute values of internal energy are
unknown, the same is true for µi .
57
11.5 Fugacity Fugacity Coefficient Pure Species
Moreover, Eq. (11.20) shows that µiig approaches
negative infinity when either P or yi approaches
zero. This is true not just for an ideal gas but
for any gas. Although these characteristics do
not preclude the use of chemical potentials, the
application of equilibrium criteria is
facilitated by introduction of the fugacity, a
property that takes the place of µi but which
does not exhibit its less desirable
characteristics.
58
11.5 Fugacity Fugacity Coefficient Pure Species
The origin of the fugacity concept resides in Eq.
(11.28), valid only for pure species i in the
ideal gas state. For a real fluid, we write an
analogous equation that defines fi, the fugacity
of pure species i
59
11.5 Fugacity Fugacity Coefficient Pure Species
The origin of the fugacity concept resides in Eq.
(11.28), valid only for pure species i in the
ideal gas state. For a real fluid, we write an
analogous equation that defines fi, the fugacity
of pure species i
(11.31) This new property fi ,
with units of pressure, replaces P in Eq.(11.28).
Clearly, if (11.28) is a special case of Eq.
(11.31), then
60
11.5 Fugacity Fugacity Coefficient Pure Species

(11.32) and the fugacity of pure species i as
an ideal gas is necessarily equal to its
pressure. Subtraction of Eq. (11.28) from Eq.
(11.31), both written for the same T and P,
gives
By the definition of Eq. (6.41), Gi Giig is
the residual Gibbs Energy, GiRthus,
61
11.5 Fugacity Fugacity Coefficient Pure Species

(11.33) where the dimensionless ratio fi /P has
been defined as another new property, the
fugacity coefficient, given by symbolFi
(11.34)
These
equations apply to pure species i in any phase
at any condition. However, as a special case they
must be valid for ideal gases, for which GiR 0,
Fi 1, and Eq. (11.28) is recovered from Eq.
(11.31).
62
11.5 Fugacity Fugacity Coefficient Pure Species
Moreover, we may write Eq. (11.33) for P 0,
and combine it with Eq. (6.45) As explained
in connection with Eq. (6.48), the value of J is
immaterial, and is set equal to zero. Whence,

63
11.5 Fugacity Fugacity Coefficient Pure Species
And The identification of lnFi with GiR / RT
by Eq. (11.33) permits its evaluation by the
integral of Eq. (6.49)

64
11.5 Fugacity Fugacity Coefficient Pure Species
Fugacity coefficients (and therefore fugacities)
for pure gases are evaluated by this equation
from P V T data or from a volume-explicit
equation of state. For example, when the
compressibility factor is given by Eq.
(3.38),

65
11.5 Fugacity Fugacity Coefficient Pure Species
Because the second virial coefficient Bii is a
function of temperature only for a pure speciues,
substitution into Eq. (11.35) gives

66
11.5 Fugacity Fugacity Coefficient Pure Species

• Fugacity Coefficients from the Generic Cubic
  Equation of State

67
11.5 Fugacity Fugacity Coefficient Pure Species

• Vapor/Liquid Equilibrium for Pure Species

68
11.5 Fugacity Fugacity Coefficient Pure Species

• Vapor/Liquid Equilibrium for Pure Species
• For a pure species coexisting liquid and vapor
  phases are in equilibrium when they have the same
  temperature, pressure, and fugacity.

69
11.5 Fugacity Fugacity Coefficient Pure Species

• Fugacity of a Pure Liquid

70
11.6 Fugacity Fugacity Coefficient Species in
Solution

• The definition of the fugacity of a species in
  solution is parallel to the definition of the
  pure species fugacity. For species I in a
  mixture of real gases or in a solution of
  liquids, the equilibrium analogous to Eq.
  (11.20), the ideal-gas expression, is

71
11.6 Fugacity Fugacity Coefficient Species in
Solution

• where is the fugacity of species i in
  solution, replacing the partial pressure yiP.
  This definition of does not make it a
  partial molar property, and it is therefore
  identified by a circumflex rather than by an
  overbar.
• A direct application of this definition indicates
  its potential utility. Equation (11.6) is the
  fundamental criterion for phase equilibrium.

72
11.6 Fugacity Fugacity Coefficient Species in
Solution

• At equilibrium
• Thus, multiple phases at the same T and P are in
  equilibrium when the fugacity of each constituent
  species is the same in all phases.
• This criterion of equilibrium is the one usually
  applied by chemical engineers in the solution of
  phase-equilibrium problems.

73
11.6 Fugacity Fugacity Coefficient Species in
Solution

• For the specific case of multicomponent
  vapor/liquid equilibrium, Eq. (11.47) becomes
• Equation (11.39) results as a special case when
  this relation is applied to the vapor/liquid
  equilibrium of pure species i.

74
11.6 Fugacity Fugacity Coefficient Species in
Solution

• The definition of a residual property is given in
  Sec. 6.2
• Where M is the molar (or unit mass) value of a
  thermodynamic property and M ig is the value that
  the property would have for an ideal gas of the
  same composition at same T and P. The defining
  equation for a partial residual property

75
11.6 Fugacity Fugacity Coefficient Species in
Solution

76
11.6 Fugacity Fugacity Coefficient Species in
Solution

77
11.6 Fugacity Fugacity Coefficient Species in
Solution

78
11.6 Fugacity Fugacity Coefficient Species in
Solution

79
11.6 Fugacity Fugacity Coefficient Species in
Solution

80
11.6 Fugacity Fugacity Coefficient Species in
Solution

81
11.6 Fugacity Fugacity Coefficient Species in
Solution

• Fugacity Coefficients from the Virial Equation of
  State

82
11.7 Generalized Correlations for the Fugacity
Coefficient

83
11.8 The Ideal Solution Model

84
11.8 The Ideal Solution Model

• The Lewis/Randall Rule

85
11.9 Excess Properties

86
11.9 Excess Properties

• The Excess Gibbs Energy and the Activity
  Coefficient

87
11.9 Excess Properties

• Excess Property Relations

88
11.9 Excess Properties

• The Nature of Excess Properties

89
11.5 Fugacity Fugacity Coefficient Pure Species
Consider the change in G to G at very low
pressure
90
Fugacity Fugacity Coefficient Pure Species
91
Fugacity Fugacity Coefficient Pure Species
92
Fugacity Fugacity Coefficient Pure Species
93
Generalized Correlations for the Fugacity
Coefficient
94
Generalized Correlations for the Fugacity
Coefficient
The average properties at the critical point and
the 2nd Virial coefficient can be determined from
Equation 11.66-11.71
95
Fugacity of a Pure Liquid

• fi of a compressed liquid is calculated in 2
  steps
• fi of saturated liquid and vapor
• Compress liquid from Psat to P

96
Fugacity and Fugacity Coefficient Species in
solution

• fi of a solution is parallel to the pure solution
• The ideal solution (analogous to the ideal gas)
• At equilibrium

Thus, multiple phases at the same T and P are in
equilibrium when the fugacity of each constituent
species is the same in all phases.
97
Fugacity and Fugacity Coefficient Species in
solution
A partial residual property,
98
The Fundamental Residual-Property Relation
99
Fugacity Coefficient from the Virial EOS

• For mixture
• e.g. binary mixture
• B y1y1B11 y1y2B12 y2y1B21
  y2y2B22
• B y12B11 2y1y2B12 y22B22
  (11.58)

100
The Ideal Solution

• Serves as a standard to which real-solution
  behavior can be compared.

101
The Ideal Solution The Lewis/Randall Rule

• Fugacity calculation of i in ideal solution.

102
Excess Properties
Fundamental of excess property relation
103
The Excess Gibbs Energy and the Activity
Coefficient
104
The Excess Gibbs Energy and the Activity
Coefficient
105
The Excess Gibbs Energy and the Activity
Coefficient
106
Gibbs-Duhem Equation
107
The Nature of the Excess Properties

• All MEs become 0 as either species approaches
  purity.
• Plot between GE vs. x1 is approximately
  parabolic in shape,
• Both HE and TSE exhibit individualistic
  composition
• dependencies
• When an excess property has a single sign (as
  does GE in
• all six cases, the extreme value of ME (maximum
  or minimum)
• Often occurs near the equimolar composition.

108
Review

• Mixtures
• What is the definition of partial molar
  property? Try saying it in words rather than
  equation.
• Why is the partial molar property not the same
  as the pure property? What can happen when we mix
  different species?
• How is excess property defined?
• Do ideal gases always form ideal mixtures when
  allowed to mixed?
• Pick a property, say V,. Review the ways we can
  calculate the partial molar volume. What about
  straight differentiation? What is the alternative
  way that only works for binary mixture? What
  about graphically?

109
Review

• Mixtures
• What is Gibbs-Duhem equation? In what ways is it
  useful?
• For ideal solution, what is the molar volume?
• For ideal solution, what is the molar enthalpy?
• For ideal solution, what is the molar entropy?
• For ideal solution, what is the molar Gibbs free
  energy?
• What is the definition of infinite dilution
  property?