Moran M.J., Shapiro H.N. Fundamentals of Engineering Thermodynamics

Подождите немного. Документ загружается.

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

U 5 TS 2 pV 1

i51

H 5 TS 1

i51

(11.109)

° 52pV 1

i51

Other useful relations can be obtained as well. Forming the differential of

G(T, p, n

, n

, . . . , n

)

dG 5

T, n

dp 1

p, n

dT 1

i51

T, p, n

(11.110)

The subscripts n in the first two terms indicate that all n’s are held fixed during dif-

ferentiation. Since this implies fixed composition, it follows from Eqs. 11.30 and 11.31

(Sec. 11.3.2) that

V 5 a

T, n



and



2S 5 a

p, n

(11.111)

With Eqs. 11.107 and 11.111, Eq. 11.110 becomes

dG 5 V dp 2 S dT 1

(11.112)

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

dG 5

i51

Combining this equation with Eq. 11.112 gives the Gibbs–Duhem equation

i51

5 V dp 2 S dT

(11.113)

11.9.3

Fundamental Thermodynamic Functions for

Multicomponent Systems

A fundamental thermodynamic function provides a complete description of the ther-

modynamic state of a system. In principle, all properties of interest can be determined

from such a function by differentiation and/or combination. Reviewing the develop-

ments of Sec. 11.9.2, we see that a function G(T, p, n

, n

, . . . , n

) is a fundamental

thermodynamic function for a multicomponent system.

Functions of the form U(S, V, n

, n

, . . . , n

), H(S, p, n

, n

, . . . , n

), and C(T, V,

, n

, . . . , n

) 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

dU 5 T dS 2 p dV 1

i51

(11.114a)

dH 5 T dS 1 V dp 1

i51

(11.114b)

d° 52p dV 2 S dT 1

i51

(11.114c)

Gibbs–Duhem equation

11.9 Analyzing Multicomponent Systems 683

c11ThermodynamicRelations.indd Page 683 7/23/10 6:25:35 PM user-s146 c11ThermodynamicRelations.indd Page 683 7/23/10 6:25:35 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

684 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, n

, n

, . . . , n

) is

dU 5

V, n

dS 1

S, n

dV 1

i51

S,V, n

Comparing this expression term by term with Eq. 11.114a, we have

T 5

V, n



2p 5

S, n



S,V, n

(11.115a)

That is, the temperature, pressure, and chemical potentials can be obtained by dif-

ferentiation of U(S, V, n

, n

, . . . , n

). 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, n

, n

, . . . , n

) together

with Eq. 11.114b gives

T 5

p, n



V 5

S, n



S, p, n

(11.115b)

where the first two of these are the counterparts of Eqs. 11.26 and 11.27. Finally, with

C(S, V, n

, n

, . . . , n

) and Eq. 11.114c

2p 5

0°

T, n



2S 5

0°

V, n



0°

T,V, n

(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 combi-

nation using the definitions H 5 U 1 pV, G 5 H 2 TS, C = U 2 TS.

The foregoing discussion of fundamental thermodynamic functions has led to sev-

eral 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

p, n

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

T, n

T, p, n

Recognizing the right side of this equation as the partial molal volume, we have

T, n

5 V

(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

T, p, n

S, V, n

S, p, n

T, V, n

(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

c11ThermodynamicRelations.indd Page 684 6/21/10 9:36:32 PM user-s146 c11ThermodynamicRelations.indd Page 684 6/21/10 9:36:32 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

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 prop-

erty, 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 5 nm



m 5

5 g

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

5 y

(11.119)

For the special case of an ideal gas, y 5 RT

p, and Eq. 11.119 assumes the form

where the asterisk denotes ideal gas. Integrating at constant temperature

5 RT ln p 1 C 1T2 (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 advanta-

geous 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 5 RT ln f 1 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

5 y

(11.122)

Integration of Eq. 11.122 while holding temperature constant can determine the fugac-

ity 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

pS 0

5 1

(11.123)

Equations 11.122 and 11.123 then completely determine the fugacity function.

fugacity

11.9 Analyzing Multicomponent Systems 685

c11ThermodynamicRelations.indd Page 685 6/21/10 9:36:34 PM user-s146 c11ThermodynamicRelations.indd Page 685 6/21/10 9:36:34 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

686 Chapter 11

Thermodynamic Relations

EVALUATING FUGACITY. Let us consider next how the fugacity can be evalu-

ated. With Z 5 py

RT, Eq. 11.122 becomes

RT a

0 ln

RTZ

ln f

Subtracting 1/p from both sides and integrating from pressure p9 to pressure p at fixed

temperature T

3ln f 2 ln p4

p¿

1Z 2 12d ln p

cln

p¿

1Z 2 12d ln p

Taking the limit as p9 tends to zero and applying Eq. 11.123 results in

1Z 2 12d ln p

When expressed in terms of the reduced pressure, p

5 p/p

, the above equation is

1Z 2 12d ln p

(11.124)

Since the compressibility factor Z depends on the reduced temperature T

and reduced

pressure p

, 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 T

and p

, 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

2 m

5 RT ln

5 RT ln a

Using the critical temperature and pressure of water from Table A-1, at state 1 p

5 0.91,

5 1.04, and at state 2 p

5 1.09, T

5 1.04. By inspection of Fig. A-6, f

5 0.755

and f

5 0.7. Inserting values in the above equation

2 m

5 18.31421673.152 ln c10.72a

240

200

0.755

bd5 597 kJ

kmol

For a pure component, the chemical potential equals the Gibbs function per mole,

g 5 h 2 T s. Since the temperature is the same at states 1 and 2, the change in the

chemical potential can be expressed as m

2 m

5 h

2 h

2 T1

2 s

2. 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 par-

allels the definition for a pure component. For a pure component, the development

c11ThermodynamicRelations.indd686 Page 686 6/21/10 9:37:12 PM user-s146 c11ThermodynamicRelations.indd686 Page 686 6/21/10 9:37:12 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

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 coun-

terpart of Eq. 11.119, and the fugacity f

of component i is introduced by

5 RT ln f

1 C

1T2 (11.125)

which parallels Eq. 11.121. The pair of equations that allow the fugacity of a mixture

component

, f

, to be evaluated are

RT a

0 ln

T, n

5 V

(11.126a)

lim

pS 0

b5 1

(11.126b)

The symbol

denotes the fugacity of component i in the mixture and should be care-

fully distinguished in the presentation to follow from f

, which denotes the fugacity of

pure i.

DISCUSSION. Referring to Eq. 11.126b, note that in the ideal gas limit the fugac-

ity f

is not required to equal the pressure p as for the case of a pure component, but

to equal the quantity y

p. 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 5

nRT

(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 p

given by

(11.128)

where n

is the number of moles of component i. Dividing Eq. 11.128 by Eq. 11.127

5 y

On rearrangement

5 y

p (11.129)

Accordingly, the quantity y

p appearing in Eq. 11.126b corresponds to the pressure p

Summing both sides of Eq. 11.129, we obtain

i51

p 5 p

i51

Or, since the sum of the mole fractions equals unity

p 5

i51

(11.130)

In words, Eq. 11.130 states that the sum of the pressures p

equals the mixture pres-

sure. This gives rise to the designation partial pressure for p

. 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

11.9 Analyzing Multicomponent Systems 687

c11ThermodynamicRelations.indd687 Page 687 6/21/10 9:37:15 PM user-s146 c11ThermodynamicRelations.indd687 Page 687 6/21/10 9:37:15 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

688 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

5 y

(11.131)

where y

is the molar specific volume of pure i. Subtracting Eq. 11.131 from Eq.

11.126a

RT c

0 ln 1

T, n

5 V

2 y

(11.132)

Integrating from pressure p9 to pressure p at fixed temperature and mixture compo-

sition

RT clna

p¿

2 y

In the limit as p9 tends to zero, this becomes

RT cln a

b2 lim

p¿S0

ln a

bd5

2 y

Since f

S p¿ and f

S y

p¿ as pressure p9 tends to zero

lim

p¿S0

ln a

bS ln a

p¿

b5 ln y

Therefore, we can write

RT cln a

b2 ln y

2 y

RT ln a

2 y

(11.133)

in which f

is the fugacity of component i at pressure p in a mixture of given com-

position at a given temperature, and f

is the fugacity of pure i at the same tempera-

ture and pressure. Equation 11.133 expresses the relation between f

and f

in terms

of the difference between V

and y

, a measurable quantity.

11.9.5

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

5 y





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 aggre-

gation (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

ideal solution

Lewis–Randall rule

c11ThermodynamicRelations.indd688 Page 688 6/21/10 9:37:16 PM user-s146 c11ThermodynamicRelations.indd688 Page 688 6/21/10 9:37:16 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

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 V

2 y

5 0, or

5 y

(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

V 5

i51



1ideal solution2

(11.136)

where V

is the volume that pure component i would occupy when at the tem-

perature and pressure of the mixture. Comparing Eqs. 11.136 and 11.100a, the

additive volume rule is seen to be exact for ideal solutions.

c 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

5 u



5 h

(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

U 5



and



H 5



1ideal solution2

(11.138)

where u

and h

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 f

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 pres-

sure, 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 expres-

sions for evaluating the chemical potential for ideal solutions used in Sec. 14.3.3.

11.9 Analyzing Multicomponent Systems 689

c11ThermodynamicRelations.indd689 Page 689 7/23/10 6:27:20 PM user-s146 c11ThermodynamicRelations.indd689 Page 689 7/23/10 6:27:20 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

690 Chapter 11 Thermodynamic Relations

activity

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 read-

ily measured. The emphasis is on systems involving a single

chemical species such as water or a mixture such as air. An intro-

duction to general property relations for mixtures and solutions

is also included.

Equations of state relating p, y, and T are considered, includ-

ing the virial equation and examples of two-constant and multi-

constant equations. Several important property relations based

c CHAPTER SUMMARY AND STUDY GUIDE

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 p

ref

. 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

2 m8

5 RT ln

f 8

(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, a

, of component i in

the mixture. That is

f 8

(11.140)

For subsequent applications, it suffices to consider the case of gaseous mixtures. For

gaseous mixtures, p

ref

is specified as 1 atm, so m8

and f 8

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

81RT ln a

(11.141)

where g8

is the Gibbs function per mole of pure component i evaluated at tempera-

ture T and 1 atm: g8

5 g

(T, 1 atm).

For an ideal solution, the Lewis–Randall rule applies and the activity is

f 8

(11.142)

where f

is the fugacity of pure component i at temperature T and pressure p. Intro-

ducing Eq. 11.142 into Eq. 11.141

5 g8

1 RT ln

f 8

5 g8

1 RT ln ca

ref



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, p

ref

, we have

p 5 f 8

ref

5 1; Eq. 11.143 then reduces to

5 g8

1 RT ln

ref



1ideal gas2

(11.144)

c11ThermodynamicRelations.indd690 Page 690 6/21/10 9:37:19 PM user-s146 c11ThermodynamicRelations.indd690 Page 690 6/21/10 9:37:19 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

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

entropy are developed and applied to phase change and to

single-phase processes. Property relations are introduced involv-

ing the volume expansivity, isothermal and isentropic compress-

ibilities, velocity of sound, specific heats and specific heat ratio,

and the Joule–Thomson coefficient.

Additionally, we describe how tables of thermodynamic prop-

erties are constructed using the property relations and methods

developed in this chapter. Such procedures also provide the

basis for data retrieval by computer software. Also described

are means for using the generalized enthalpy and entropy depar-

ture charts and the generalized fugacity coefficient chart to eval-

uate enthalpy, entropy, and fugacity, respectively.

We also consider p–y–T relations for gas mixtures of known

composition, including Kay’s rule. The chapter concludes with a

discussion of property relations for multicomponent systems,

including partial molal properties, chemical potential, fugacity,

and activity. Ideal solutions and the Lewis–Randall rule are intro-

duced as a part of that presentation.

The following checklist provides a study guide for this chap-

ter. When your study of the text and end-of-chapter exercises

has been completed you should be able to write out the mean-

ings of the terms listed in the margins throughout the chapter

and understand each of the related concepts. The subset of key

concepts listed below is particularly important. Additionally, for

systems involving a single species you should be able to

calculate p–y–T data using equations of state such as the

Redlich–Kwong and Benedict–Webb–Rubin equations.

use the 16 property relations summarized in Table 11.1 and

explain how the relations are obtained.

evaluate Ds, Du, and Dh, using the Clapeyron equation when

considering phase change, and using equations of state and

specific heat relations when considering single phases.

use the property relations introduced in Sec. 11.5, such as

those involving the specific heats, the volume expansivity,

and the Joule–Thomson coefficient.

explain how tables of thermodynamic properties, such as

Tables A-2 through A-18, are constructed.

use the generalized enthalpy and entropy departure charts,

Figs. A-4 and A-5, to evaluate Dh and Ds.

For a gas mixture of known composition, you should be able to

apply the methods introduced in Sec. 11.8 for relating pressure,

specific volume, and temperature—Kay’s rule, for example.

For multicomponent systems, you should be able to

evaluate extensive properties in terms of the respective par-

tial molal properties.

evaluate partial molal volumes using the method of intercepts.

evaluate fugacity using data from the generalized fugacity

coefficient chart, Fig. A-6.

apply the ideal solution model.

c KEY ENGINEERING CONCEPTS

equation of state, p. 632

exact differential, p. 638

test for exactness, p. 638

Helmholtz function, p. 642

Gibbs function, p. 642

Maxwell relations, p. 644

fundamental thermodynamic

function, p. 647

Clapeyron equation, p. 649

Joule-Thomson coefficient, p. 661

enthalpy departure, p. 669

entropy departure, p. 672

Kay’s rule, p. 675

method of intercepts, p. 681

chemical potential, p. 682

fugacity, p. 685

Lewis–Randall rule, p. 688

c KEY EQUATIONS

Equations of State

Z 5 1 1

B1T

C1T2

1 . . .

(11.1) p. 632 Virial equation of state

p 5

y 2 b

(11.2) p. 633 van der Waals equation of state

p 5

y 2 b

y1y 1 b2T

(11.7) p. 635 Redlich–Kwong equation of state

Key Equations 691

c11ThermodynamicRelations.indd691 Page 691 6/22/10 7:20:52 PM user-s146 c11ThermodynamicRelations.indd691 Page 691 6/22/10 7:20:52 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New

692 Chapter 11 Thermodynamic Relations

Mathematical Relations for Properties

(11.14a) p. 638

Test for exactness

(11.14b) p. 638



5 1

(11.15) p. 639



521

(11.16) p. 639

Important relations among partial derivatives

of properties

Additional Thermodynamic Relations

c 5 u 2 Ts (11.20) p. 642 Helmholtz function

5 h 2 Ts (11.21) p. 642 Gibbs function

c 5

2ky

(9.36b) p. 658

Expressions for velocity of sound

(11.74) p. 660

Table 11.1 (11.24– Summary of property relations from exact

11.36) p. 644 differentials

Expressions for Du, Dh, and Ds

sat

2 h

T1y

2 y

(11.40) p. 649 Clapeyron equation

2 s

dT 1

(11.50) p. 653

Expressions for changes in s and u with T

and y as independent variables

2 u

dT 1

cT a

2 p ddy

(11.51) p. 653

2 s

dT 2

(11.59) p. 654

Expressions for changes in s and h with T

and p as independent variables

2 h

dT 1

cy 2 T a

ddp

(11.60) p. 655

2 h

5 h

2 h

2 RT

2 h

2 a

2 h

(11.85) p. 670

Evaluating enthalpy and entropy changes in

terms of generalized enthalpy and entropy

departures and data from Figs. A-4 and A-5,

respectively

2 s

5 s

2 s

2 R ca

2 s

2 a

2 s

(11.92) p. 672

(11.75) p. 661 Joule–Thomson coefficient

c11ThermodynamicRelations.indd692 Page 692 7/23/10 6:27:30 PM user-s146 c11ThermodynamicRelations.indd692 Page 692 7/23/10 6:27:30 PM user-s146 /Users/user-s146/Desktop/Merry_X-Mas/New/Users/user-s146/Desktop/Merry_X-Mas/New