704
Chapter 11 Thermodynamic Relations
11.9.1 Partial Molal Properties
In the present discussion we introduce the concept of a partial molal property and
illustrate its use. This concept plays an important role in subsequent discussions of
multicomponent systems.
DEFINING PARTIAL MOLAL PROPERTIES. Any extensive thermodynamic
property X of a single-phase, single-component system is a function of two independent intensive properties and the size of the system. Selecting temperature and pressure as the independent properties and the number of moles n as the measure of size,
we have X 5 X(T, p, n). For a single-phase, multicomponent system, the extensive
property X must then be a function of temperature, pressure, and the number of
moles of each component present, X 5 X(T, p, n1, n2, . . . , nj).
If each mole number is increased by a factor ␣, the size of the system increases
by the same factor, and so does the value of the extensive property X. That is,
aX1T, p, n1, n2, . . . , nj2 ⫽ X1T, p, an1, an2, . . . , anj2
Differentiating with respect to ␣ while holding temperature, pressure, and the mole
numbers fixed and using the chain rule on the right side gives
X⫽
0X
0X
0X
n1 ⫹
n2 ⫹ . . . ⫹
nj
01an12
01an22
01anj2
This equation holds for all values of ␣. In particular, it holds for ␣ 5 1. Setting ␣ 5 1
j
0X
X ⫽ a ni
b
0ni T, p, nl
i⫽1
partial molal property
(11.101)
where the subscript nl denotes that all n’s except ni are held fixed during differentiation.
The partial molal property Xi is by definition
Xi ⫽
0X
b
0ni T, p, nl
(11.102)
The partial molal property Xi is a property of the mixture and not simply a property
of component i, for Xi depends in general on temperature, pressure, and mixture
composition: Xi1T, p, n1, n2, . . . , nj2. Partial molal properties are intensive properties
of the mixture.
Introducing Eq. 11.102, Eq. 11.101 becomes
j
X ⫽ a ni X i
(11.103)
i⫽1
This equation shows that the extensive property X can be expressed as a weighted
sum of the partial molal properties Xi .
Selecting the extensive property X in Eq. 11.103 to be volume, internal energy,
enthalpy, and entropy, respectively, gives
j
j
j
V ⫽ a ni V i ,
U ⫽ a ni U i ,
H ⫽ a ni Hi,
i⫽1
i⫽1
i⫽1
j
S ⫽ a ni Si
(11.104)
i⫽1
where Vi, Ui, Hi, Si denote the partial molal volume, internal energy, enthalpy, and
entropy. Similar expressions can be written for the Gibbs function G and the Helmholtz
function C. Moreover, the relations between these extensive properties: H 5 U 1 pV,
G 5 H 2 TS, C 5 U 2 TS can be differentiated with respect to ni while holding temperature, pressure, and the remaining n’s constant to produce corresponding relations
11.9 Analyzing Multicomponent Systems
among partial molal properties: Hi ⫽ Ui ⫹ pVi, Gi ⫽ Hi ⫺ TSi, °i ⫽ Ui ⫺ TSi , where
Gi and °i are the partial molal Gibbs function and Helmholtz function, respectively.
Several additional relations involving partial molal properties are developed later in
this section.
EVALUATING PARTIAL MOLAL PROPERTIES. Partial molal properties can
be evaluated by several methods, including the following:
c
If the property X can be measured, Xi can be found by extrapolating a plot giving
1¢X /¢ni2T, p, nl versus Dni. That is,
Xi ⫽ a
c
c
0X
¢X
b
⫽ lim a
b
0ni T, p, nl ¢ni S0 ¢ni T, p, nl
If an expression for X in terms of its independent variables is known, Xi can be
evaluated by differentiation. The derivative can be determined analytically if the function is expressed analytically or found numerically if the function is in tabular form.
When suitable data are available, a simple graphical procedure known as the
method of intercepts can be used to evaluate partial molal properties. In principle,
the method can be applied for any extensive property. To introduce this method,
let us consider the volume of a system consisting of two components, A and B. For
this system, Eq. 11.103 takes the form
V ⫽ nAVA ⫹ nBVB
where VA and VB are the partial molal volumes of A and B, respectively. Dividing
by the number of moles of mixture n
V
⫽ yAVA ⫹ yBVB
n
where yA and yB denote the mole fractions of A and B, respectively. Since yA 1
yB 5 1, this becomes
V
⫽ 11 ⫺ yB2VA ⫹ yBVB ⫽ VA ⫹ yB1VB ⫺ VA2
n
This equation provides the basis for the method of intercepts. For example, refer
to Fig. 11.5, in which V/n is plotted as a function of yB at constant T and p. At a
specified value for yB, a tangent to the curve is shown on the figure. When extrapolated, the tangent line intersects the axis on the left at VA and the axis on the right
at VB . These values for the partial molal volumes correspond to the particular
specifications for T, p, and yB. At fixed temperature and pressure, VA and VB vary
with yB and are not equal to the molar specific volumes of pure A and pure B,
denoted on the figure as yA and yB , respectively. The values of yA and yB are
fixed by temperature and pressure only.
V T and p constant
––
n
V as a function of y
––
B
n
vB(T, p)
VB(T, p, yB)
vA(T, p)
Tangent
line
VA(T, p, yB)
Fig. 11.5 Illustration of the evaluation of
0
(pure A)
yB
Mole fraction of B
1.0
(pure B)
partial molal volumes by the method of
intercepts.
method of intercepts
705
706
Chapter 11 Thermodynamic Relations
EXTENSIVE PROPERTY CHANGES ON MIXING. Let us conclude the present discussion by evaluating the change in volume on mixing of pure components at
the same temperature and pressure, a result for which an application is given in the
discussion of Eq. 11.135. The total volume of the pure components before mixing is
j
Vcomponents ⫽ a ni yi
i⫽1
where yi is the molar specific volume of pure component i. The volume of the mixture is
j
Vmixture ⫽ a ni Vi
i⫽1
where Vi is the partial molal volume of component i in the mixture. The volume
change on mixing is
j
j
¢Vmixing ⫽ Vmixture ⫺ Vcomponents ⫽ a ni Vi ⫺ a ni yi
i⫽1
i⫽1
or
j
¢Vmixing ⫽ a ni1Vi ⫺ yi2
(11.105)
i⫽1
Similar results can be obtained for other extensive properties, for example,
j
¢Umixing ⫽ a ni1Ui ⫺ ui2
i⫽1
j
¢Hmixing ⫽ a ni 1Hi ⫺ hi2
(11.106)
i⫽1
j
¢Smixing ⫽ a ni 1Si ⫺ si2
i⫽1
In Eqs. 11.106, ui, hi , and si denote the molar internal energy, enthalpy, and entropy
of pure component i, respectively. The symbols Ui, Hi , and Si denote the respective
partial molal properties.
11.9.2 Chemical Potential
chemical potential
Of the partial molal properties, the partial molal Gibbs function is particularly useful
in describing the behavior of mixtures and solutions. This quantity plays a central role
in the criteria for both chemical and phase equilibrium (Chap. 14). Because of its
importance in the study of multicomponent systems, the partial molal Gibbs function
of component i is given a special name and symbol. It is called the chemical potential
of component i and symbolized by ␮i:
0G
b
0ni T, p, nl
mi ⫽ Gi ⫽
(11.107)
Like temperature and pressure, the chemical potential ␮i is an intensive property.
Applying Eq. 11.103 together with Eq. 11.107, the following expression can be
written:
j
G ⫽ a ni mi
i⫽1
(11.108)
11.9 Analyzing Multicomponent Systems
Expressions for the internal energy, enthalpy, and Helmholtz function can be obtained from
Eq. 11.108, using the definitions H 5 U 1 pV, G 5 H 2 TS, and C 5 U 2 TS. They are
j
U ⫽ TS ⫺ pV ⫹ a ni mi
i⫽1
j
H ⫽ TS ⫹ a ni mi
(11.109)
i⫽1
j
° ⫽ ⫺pV ⫹ a ni mi
i⫽1
Other useful relations can be obtained as well. Forming the differential of
G(T, p, n1, n2, . . . , nj)
j
0G
0G
0G
dG ⫽
b dp ⫹
b dT ⫹ a a
b
dni
(11.110)
0p T, n
0T p, n
i⫽1 0ni T, p, nl
The subscripts n in the first two terms indicate that all n’s are held fixed during differentiation. Since this implies fixed composition, it follows from Eqs. 11.30 and 11.31
(Sec. 11.3.2) that
V⫽a
0G
b
0p T, n
⫺S ⫽ a
and
0G
b
0T p, n
(11.111)
With Eqs. 11.107 and 11.111, Eq. 11.110 becomes
j
dG ⫽ V dp ⫺ S dT ⫹ a mi dni
(11.112)
i⫽1
which for a multicomponent system is the counterpart of Eq. 11.23.
Another expression for dG is obtained by forming the differential of Eq. 11.108.
That is,
j
j
dG ⫽ a ni dmi ⫹ a mi dni
i⫽1
i⫽1
Combining this equation with Eq. 11.112 gives the Gibbs–Duhem equation
Gibbs–Duhem equation
j
a ni dmi ⫽ V dp ⫺ S dT
(11.113)
i⫽1
11.9.3 Fundamental Thermodynamic Functions
for Multicomponent Systems
A fundamental thermodynamic function provides a complete description of the thermodynamic state of a system. In principle, all properties of interest can be determined
from such a function by differentiation and/or combination. Reviewing the developments of Sec. 11.9.2, we see that a function G(T, p, n1, n2, . . . , nj) is a fundamental
thermodynamic function for a multicomponent system.
Functions of the form U(S, V, n1, n2, . . . , nj), H(S, p, n1, n2, . . . , nj), and C(T, V, n1,
n2, . . . , nj) also can serve as fundamental thermodynamic functions for multicomponent
systems. To demonstrate this, first form the differential of each of Eqs. 11.109 and use
the Gibbs–Duhem equation, Eq. 11.113, to reduce the resultant expressions to obtain
j
dU ⫽ T dS ⫺ p dV ⫹ a mi dni
(11.114a)
dH ⫽ T dS ⫹ V dp ⫹ a mi dni
(11.114b)
i⫽1
j
i⫽1
j
d° ⫽ ⫺p dV ⫺ S dT ⫹ a mi dni
i⫽1
(11.114c)
707
708
Chapter 11 Thermodynamic Relations
For multicomponent systems, these are the counterparts of Eqs. 11.18, 11.19, and 11.22,
respectively.
The differential of U(S, V, n1, n2, . . . , nj) is
dU ⫽
j
0U
0U
0U
b
dS ⫹
b dV ⫹ a a b
dni
0S V, n
0V S, n
i⫽1 0ni S,V, nl
Comparing this expression term by term with Eq. 11.114a, we have
T⫽
0U
b ,
0S V, n
⫺p ⫽
0U
b ,
0V S, n
mi ⫽
0U
b
0ni S,V, nl
(11.115a)
That is, the temperature, pressure, and chemical potentials can be obtained by differentiation of U(S, V, n1, n2, . . . , nj). The first two of Eqs. 11.115a are the counterparts
of Eqs. 11.24 and 11.25.
A similar procedure using a function of the form H(S, p, n1, n2, . . . , nj) together
with Eq. 11.114b gives
T⫽
0H
b ,
0S p, n
V⫽
0H
b ,
0p S, n
mi ⫽
0H
b
0ni S, p, nl
(11.115b)
where the first two of these are the counterparts of Eqs. 11.26 and 11.27. Finally, with
C(S, V, n1, n2, . . . , nj) and Eq. 11.114c
⫺p ⫽
0°
b ,
0V T, n
⫺S ⫽
0°
b ,
0T V, n
mi ⫽
0°
b
0ni T,V, nl
(11.115c)
The first two of these are the counterparts of Eqs. 11.28 and 11.29. With each choice
of fundamental function, the remaining extensive properties can be found by combination using the definitions H 5 U 1 pV, G 5 H 2 TS, C 5 U 2 TS.
The foregoing discussion of fundamental thermodynamic functions has led to
several property relations for multicomponent systems that correspond to relations
obtained previously. In addition, counterparts of the Maxwell relations can be obtained
by equating mixed second partial derivatives. For example, the first two terms on the
right of Eq. 11.112 give
0V
0S
b
⫽⫺ b
0T p, n
0p T, n
(11.116)
which corresponds to Eq. 11.35. Numerous relationships involving chemical potentials
can be derived similarly by equating mixed second partial derivatives. An important
example from Eq. 11.112 is
0mi
0V
b
⫽
b
0p T, n
0ni T, p, nl
Recognizing the right side of this equation as the partial molal volume, we have
0mi
b ⫽ Vi
0p T, n
(11.117)
This relationship is applied in the development of Eqs. 11.126.
The present discussion concludes by listing four different expressions derived
above for the chemical potential in terms of other properties. In the order obtained,
they are
mi ⫽
0G
0U
0H
0°
b
⫽
b
⫽
b
⫽
b
0ni T, p, nl
0ni S, V, nl
0ni S, p, nl
0ni T, V, nl
(11.118)
Only the first of these partial derivatives is a partial molal property, however, for
the term partial molal applies only to partial derivatives where the independent
11.9 Analyzing Multicomponent Systems
variables are temperature, pressure, and the number of moles of each component
present.
11.9.4 Fugacity
The chemical potential plays an important role in describing multicomponent systems.
In some instances, however, it is more convenient to work in terms of a related property, the fugacity. The fugacity is introduced in the present discussion.
Single-Component Systems
Let us begin by taking up the case of a system consisting of a single component. For
this case, Eq. 11.108 reduces to give
G ⫽ nm
m⫽
or
G
⫽g
n
That is, for a pure component the chemical potential equals the Gibbs function per
mole. With this equation, Eq. 11.30 written on a per mole basis becomes
0m
b ⫽y
0p T
(11.119)
For the special case of an ideal gas, y ⫽ RT/ p , and Eq. 11.119 assumes the form
0m*
RT
b ⫽
p
0p T
where the asterisk denotes ideal gas. Integrating at constant temperature
m* ⫽ R T ln p ⫹ C1T2
(11.120)
where C(T) is a function of integration. Since the pressure p can take on values from
zero to plus infinity, the ln p term of this expression, and thus the chemical potential,
has an inconvenient range of values from minus infinity to plus infinity. Equation 11.120
also shows that the chemical potential can be determined only to within an arbitrary
constant.
INTRODUCING FUGACITY. Because of the above considerations, it is advantageous for many types of thermodynamic analyses to use fugacity in place of the
chemical potential, for it is a well-behaved function that can be more conveniently
evaluated. We introduce the fugacity f by the expression
m ⫽ RT ln f ⫹ C1T2
(11.121)
Comparing Eq. 11.121 with Eq. 11.120, the fugacity is seen to play the same role in
the general case as pressure plays in the ideal gas case. Fugacity has the same units
as pressure.
Substituting Eq. 11.121 into Eq. 11.119 gives
RT a
0 ln f
b ⫽y
0p T
(11.122)
Integration of Eq. 11.122 while holding temperature constant can determine the fugacity only to within a constant term. However, since ideal gas behavior is approached as
pressure tends to zero, the constant term can be fixed by requiring that the fugacity of
a pure component equals the pressure in the limit of zero pressure. That is,
lim
f
pS 0 p
⫽1
(11.123)
Equations 11.122 and 11.123 then completely determine the fugacity function.
fugacity
709
710
Chapter 11 Thermodynamic Relations
EVALUATING FUGACITY. Let us consider next how the fugacity can be evaluated. With Z ⫽ py / RT , Eq. 11.122 becomes
RT a
0 ln f
RT Z
b ⫽
p
0p T
or
a
0 ln f
Z
b ⫽
p
0p T
Subtracting 1/p from both sides and integrating from pressure p9 to pressure p at fixed
temperature T
p
3ln f ⫺ ln p4pp¿ ⫽
冮 1Z ⫺ 12d ln p
p¿
or
cln
f p
d ⫽
p p¿
p
冮 1Z ⫺ 12d ln p
p¿
Taking the limit as p9 tends to zero and applying Eq. 11.123 results in
ln
f
⫽
p
p
冮 1Z ⫺ 12d ln p
0
When expressed in terms of the reduced pressure, pR 5 p/pc, the above equation is
ln
f
⫽
p
冮
pR
1Z ⫺ 12d ln pR
(11.124)
0
Since the compressibility factor Z depends on the reduced temperature TR and reduced
pressure pR, it follows that the right side of Eq. 11.124 depends on these properties only.
Accordingly, the quantity ln f /p is a function only of these two reduced properties. Using
a generalized equation of state giving Z as a function of TR and pR, ln f /p can readily
be evaluated with a computer. Tabular representations are also found in the literature.
Alternatively, the graphical representation presented in Fig. A-6 can be employed.
to illustrate the use of Fig. A-6, consider two states of water
vapor at the same temperature, 4008C. At state 1 the pressure is 200 bar, and at state
2 the pressure is 240 bar. The change in the chemical potential between these states
can be determined using Eq. 11.121 as
m2 ⫺ m1 ⫽ RT ln
f2
f2 p2 p1
⫽ R T ln a
b
p2 p1 f1
f1
Using the critical temperature and pressure of water from Table A-1, at state 1
pR1 5 0.91, TR1 5 1.04, and at state 2 pR2 5 1.09, TR2 5 1.04. By inspection of
Fig. A-6, f1/p1 5 0.755 and f2/p2 5 0.7. Inserting values in the above equation
m2 ⫺ m1 ⫽ 18.31421673.152 ln c 10.72a
240
1
ba
b d ⫽ 597 kJ/ kmol
200 0.755
For a pure component, the chemical potential equals the Gibbs function per mole,
g ⫽ h ⫺ T s . Since the temperature is the same at states 1 and 2, the change in the
chemical potential can be expressed as m2 ⫺ m1 ⫽ h2 ⫺ h1 ⫺ T1s2 ⫺ s12. Using steam
table data, the value obtained with this expression is 597 kJ/kmol, which agrees with
the value determined from the generalized fugacity coefficient chart. b b b b b
Multicomponent Systems
The fugacity of a component i in a mixture can be defined by a procedure that parallels the definition for a pure component. For a pure component, the development
11.9 Analyzing Multicomponent Systems
begins with Eq. 11.119, and the fugacity is introduced by Eq. 11.121. These are then
used to write the pair of equations, Eqs. 11.122 and 11.123, from which the fugacity
can be evaluated. For a mixture, the development begins with Eq. 11.117, the counterpart of Eq. 11.119, and the fugacity f i of component i is introduced by
mi ⫽ RT ln f i ⫹ Ci 1T2
(11.125)
which parallels Eq. 11.121. The pair of equations that allow the fugacity of a mixture
component, f i , to be evaluated is
0 ln f i
b ⫽ Vi
0p T, n
fi
lim a
b ⫽1
pS 0 yi p
RT a
(11.126a)
(11.126b)
The symbol f i denotes the fugacity of component i in the mixture and should be carefully distinguished in the presentation to follow from fi, which denotes the fugacity of
pure i.
DISCUSSION. Referring to Eq. 11.126b, note that in the ideal gas limit the fugacity f i is not required to equal the pressure p, as for the case of a pure component,
but to equal the quantity yip. To see that this is the appropriate limiting quantity,
consider a system consisting of a mixture of gases occupying a volume V at pressure
p and temperature T. If the overall mixture behaves as an ideal gas, we can write
p⫽
nRT
V
(11.127)
where n is the total number of moles of mixture. Recalling from Sec. 3.12.3 that an
ideal gas can be regarded as composed of molecules that exert negligible forces on
one another and whose volume is negligible relative to the total volume, we can think
of each component i as behaving as if it were an ideal gas alone at the temperature
T and volume V. Thus, the pressure exerted by component i would not be the mixture
pressure p but the pressure pi given by
pi ⫽
ni RT
V
(11.128)
where ni is the number of moles of component i. Dividing Eq. 11.128 by Eq. 11.127
pi
ni
ni RT/ V
⫽ ⫽ yi
⫽
p
n
nRT/ V
On rearrangement
pi ⫽ yi p
(11.129)
Accordingly, the quantity yip appearing in Eq. 11.126b corresponds to the pressure pi.
Summing both sides of Eq. 11.129, we obtain
j
j
j
a pi ⫽ a yi p ⫽ p a yi
i⫽1
i⫽1
i⫽1
Or, since the sum of the mole fractions equals unity
j
p ⫽ a pi
(11.130)
i⫽1
In words, Eq. 11.130 states that the sum of the pressures pi equals the mixture pressure. This gives rise to the designation partial pressure for pi. With this background,
we now see that Eq. 11.126b requires the fugacity of component i to approach the
partial pressure of component i as pressure p tends to zero. Comparing Eqs. 11.130
fugacity of a mixture
component
711
712
Chapter 11 Thermodynamic Relations
and 11.99a, we also see that the additive pressure rule is exact for ideal gas mixtures.
This special case is considered further in Sec. 12.2 under the heading Dalton model.
EVALUATING FUGACITY IN A MIXTURE. Let us consider next how the
fugacity of component i in a mixture can be expressed in terms of quantities that can
be evaluated. For a pure component i, Eq. 11.122 gives
RT a
0 ln fi
b ⫽ yi
0p T
(11.131)
where yi is the molar specific volume of pure i. Subtracting Eq. 11.131 from
Eq. 11.126a
RT c
0 ln 1 f i / fi2
0p
d
⫽ Vi ⫺ yi
(11.132)
T, n
Integrating from pressure p9 to pressure p at fixed temperature and mixture composition
fi p
RT c ln a b d ⫽
fi p¿
p
冮 1V ⫺ y 2 dp
i
i
p¿
In the limit as p9 tends to zero, this becomes
fi
fi
RT c ln a b ⫺ lim ln a b d ⫽
fi
p¿S0
fi
p
冮 1V ⫺ y 2 dp
i
i
0
Since fi S p¿ and f i S yi p¿ as pressure p9 tends to zero
lim ln a
p¿S0
fi
yi p¿
b S ln a
b ⫽ ln yi
fi
p¿
Therefore, we can write
fi
RT cln a b ⫺ ln yi d ⫽
fi
p
冮 1V ⫺ y 2 dp
i
i
0
or
RT ln a
fi
b⫽
yi fi
p
冮 1V ⫺ y 2 dp
i
i
(11.133)
0
in which f i is the fugacity of component i at pressure p in a mixture of given composition at a given temperature, and fi is the fugacity of pure i at the same temperature and pressure. Equation 11.133 expresses the relation between f i and fi in terms
of the difference between Vi and yi , a measurable quantity.
11.9.5 Ideal Solution
ideal solution
The task of evaluating the fugacities of the components in a mixture is considerably
simplified when the mixture can be modeled as an ideal solution. An ideal solution
is a mixture for which
f i ⫽ yi fi
Lewis–Randall rule
1ideal solution2
(11.134)
Equation 11.134, known as the Lewis–Randall rule, states that the fugacity of each
component in an ideal solution is equal to the product of its mole fraction and the
fugacity of the pure component at the same temperature, pressure, and state of aggregation (gas, liquid, or solid) as the mixture. Many gaseous mixtures at low to moderate
pressures are adequately modeled by the Lewis–Randall rule. The ideal gas mixtures
11.9 Analyzing Multicomponent Systems
considered in Chap. 12 are an important special class of such mixtures. Some liquid
solutions also can be modeled with the Lewis–Randall rule.
As consequences of the definition of an ideal solution, the following characteristics
are exhibited:
c
Introducing Eq. 11.134 into Eq. 11.132, the left side vanishes, giving Vi ⫺ yi ⫽ 0, or
Vi ⫽ yi
(11.135)
Thus, the partial molal volume of each component in an ideal solution is equal to the
molar specific volume of the corresponding pure component at the same temperature
and pressure. When Eq. 11.135 is introduced in Eq. 11.105, it can be concluded that
there is no volume change on mixing pure components to form an ideal solution.
With Eq. 11.135, the volume of an ideal solution is
j
j
j
V ⫽ a ni V i ⫽ a ni y i ⫽ a V i
i⫽1
c
i⫽1
1ideal solution2
(11.136)
i⫽1
where Vi is the volume that pure component i would occupy when at the temperature and pressure of the mixture. Comparing Eqs. 11.136 and 11.100a, the additive volume rule is seen to be exact for ideal solutions.
It also can be shown that the partial molal internal energy of each component in
an ideal solution is equal to the molar internal energy of the corresponding pure
component at the same temperature and pressure. A similar result applies for
enthalpy. In symbols
U i ⫽ ui ,
Hi ⫽ hi
(11.137)
With these expressions, it can be concluded from Eqs. 11.106 that there is no change
in internal energy or enthalpy on mixing pure components to form an ideal solution.
With Eqs. 11.137, the internal energy and enthalpy of an ideal solution are
j
U ⫽ a ni ui
i⫽1
j
and
H ⫽ a ni hi
1ideal solution2
(11.138)
i⫽1
where ui and hi denote, respectively, the molar internal energy and enthalpy of pure
component i at the temperature and pressure of the mixture.
Although there is no change in V, U, or H on mixing pure components to form
an ideal solution, we expect an entropy increase to result from the adiabatic mixing
of different pure components because such a process is irreversible: The separation
of the mixture into the pure components would never occur spontaneously. The
entropy change on adiabatic mixing is considered further for the special case of ideal
gas mixtures in Sec. 12.4.2.
The Lewis–Randall rule requires that the fugacity of mixture component i be
evaluated in terms of the fugacity of pure component i at the same temperature and
pressure as the mixture and in the same state of aggregation. For example, if the
mixture were a gas at T, p, then fi would be determined for pure i at T, p and as a
gas. However, at certain temperatures and pressures of interest a component of a
gaseous mixture may, as a pure substance, be a liquid or solid. An example is an
air–water vapor mixture at 208C (688F) and 1 atm. At this temperature and pressure,
water exists not as a vapor but as a liquid. Although not considered here, means have
been developed that allow the ideal solution model to be useful in such cases.
11.9.6 Chemical Potential for Ideal Solutions
The discussion of multicomponent systems concludes with the introduction of expressions for evaluating the chemical potential for ideal solutions used in Sec. 14.3.3.
713
714
Chapter 11 Thermodynamic Relations
Consider a reference state where component i of a multicomponent system is pure
at the temperature T of the system and a reference-state pressure pref. The difference
in the chemical potential of i between a specified state of the multicomponent system
and the reference state is obtained with Eq. 11.125 as
mi ⫺ m⬚i ⫽ R T ln
activity
fi
f ⬚i
(11.139)
where the superscript 8 denotes property values at the reference state. The fugacity
ratio appearing in the logarithmic term is known as the activity, ai, of component i in
the mixture. That is,
ai ⫽
fi
f ⬚i
(11.140)
For subsequent applications, it suffices to consider the case of gaseous mixtures. For
gaseous mixtures, pref is specified as 1 atm, so m⬚i and f ⬚i in Eq. 11.140 are, respectively,
the chemical potential and fugacity of pure i at temperature T and 1 atm.
Since the chemical potential of a pure component equals the Gibbs function per
mole, Eq. 11.139 can be written as
mi ⫽ g i⬚ ⫹ RT ln ai
(11.141)
where g⬚i is the Gibbs function per mole of pure component i evaluated at temperature T and 1 atm: g⬚i ⫽ gi (T, 1 atm).
For an ideal solution, the Lewis–Randall rule applies and the activity is
yi fi
(11.142)
f ⬚i
where fi is the fugacity of pure component i at temperature T and pressure p. Introducing Eq. 11.142 into Eq. 11.141
ai ⫽
mi ⫽ g⬚i ⫹ R T ln
yi fi
f ⬚i
or
fi pref yi p
mi ⫽ g⬚i ⫹ RT ln c a b a
b
d
p
f ⬚i pref
1ideal solution2
(11.143)
In principle, the ratios of fugacity to pressure shown underlined in this equation can
be evaluated from Eq. 11.124 or the generalized fugacity chart, Fig. A-6, developed
from it. If component i behaves as an ideal gas at both T, p and T, pref, we have
fi / p ⫽ f ⬚i / pref ⫽ 1; Eq. 11.143 then reduces to
mi ⫽ g⬚i ⫹ RT ln
yi p
pref
1ideal gas2
(11.144)
c CHAPTER SUMMARY AND STUDY GUIDE
In this chapter, we introduce thermodynamic relations that allow
u, h, and s as well as other properties of simple compressible
systems to be evaluated using property data that are more
readily measured. The emphasis is on systems involving a
single chemical species such as water or a mixture such as
air. An introduction to general property relations for mixtures
and solutions is also included.
Equations of state relating p, ␷, and T are considered, including the virial equation and examples of two-constant and multiconstant equations. Several important property relations based
on the mathematical characteristics of exact differentials are
developed, including the Maxwell relations. The concept of a
fundamental thermodynamic function is discussed. Means for
evaluating changes in specific internal energy, enthalpy, and