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

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546 Chapter 11 Thermodynamic Relations

Equation 11.134, known as the Lewis–Randall rule, states that the fugacity of each compo-

nent 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 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:

 Introducing Eq. 11.134 into Eq. 11.132, the left side vanishes, giving or

(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

(11.136)

where V

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 compo-

nent at the same temperature and pressure. A similar result applies for enthalpy. In

symbols

(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

(11.138)

where and 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 mix-

ing is considered further for the special case of ideal gas mixtures in Sec. 12.4.

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 tem-

peratures and pressures of interest a component of a gaseous mixture may, as a pure sub-

stance, be a liquid or solid. An example is an air–water vapor mixture at 20C (68F) 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.

U 

i1

and

H 

i1

1ideal solution2

 u

 h

V 

i1



i1



i1

1ideal solution2

 v

 v

 0,

11.9 Analyzing Multicomponent Systems 547

 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 subsequent sections of the book.

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 chem-

ical potential of i between a specified state of the multicomponent system and the reference

state is obtained with Eq. 11.125 as

(11.139)

where the superscript 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

(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 and in Eq. 11.140 are, respectively, the chem-

ical 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

(11.141)

where is the Gibbs function per mole of pure component i evaluated at temperature T and

1 atm: (T, 1 atm).

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

(11.142)

where f

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

Eq. 11.142 into Eq. 11.141

(11.143)

In principle, the ratios of fugacity to pressure shown underlined in this equation can be eval-

uated from Eq. 11.124 or the generalized fugacity chart, Fig. A-6, developed from it. If com-

ponent i behaves as an ideal gas at both T, p and and Eq. 11.143

reduces to

(11.144)m

 g °

 RT ln

ref

1ideal gas2

T, p

ref

, f



p  f °



ref

 1,

 g °

 RT ln ca

b a

ref

f °

ref

1ideal solution2

 g°

 RT ln

f °



f °

g°

 g

g°

 g°

 RT ln a

f °

m°



f °

 m°

 RT ln

f °

activity

548 Chapter 11 Thermodynamic Relations

Chapter Summary and Study Guide

In this chapter, we introduce thermodynamic relations that al-

low u, h, and s as well as other properties of simple com-

pressible systems to be evaluated using property data that are

more readily measured. The emphasis is on systems involv-

ing a single chemical species such as water or a mixture such

as air. An introduction to general property relations for mix-

tures and solutions is also included.

Equations of state relating p, v, and T are considered, in-

cluding the virial equation and examples of two-constant and

multiconstant equations. Several important property rela-

tions based on the mathematical characteristics of exact dif-

ferentials 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 involving the volume expansivity,

isothermal and isentropic compressibilities, velocity of

sound, specific heats and specific heat ratio, and the

Joule–Thomson coefficient.

Additionally, we describe how tables of thermodynamic

properties are constructed using the property relations and

methods developed in this chapter. Such procedures also pro-

vide the basis for data retrieval by computer software. Also

described are means for using the generalized enthalpy and

entropy departure charts and the generalized fugacity coeffi-

cient chart to evaluate enthalpy, entropy, and fugacity,

respectively.

We also consider p–v–T relations for gas mixtures of

known composition, including Kay’s rule. The chapter con-

cludes with a discussion of property relations for multi-

component systems, including partial molal properties,

chemical potential, fugacity, and activity. Ideal solutions

and the Lewis–Randall rule are introduced as a part of that

presentation.

The following checklist provides a study guide for this

chapter. When your study of the text and end-of-chapter ex-

ercises has been completed you should be able to write out

the meanings 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–v–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 s, u, and h, 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 expansiv-

ity, 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 h and s.

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

partial molal properties.

 evaluate partial molal volumes using the method of

intercepts.

 evaluate fugacity using data from the generalized fugac-

ity coefficient chart, Fig. A-6.

 apply the ideal solution model.

Key Engineering Concepts

equation of

state p. 487

exact differential p. 494

test for exactness p. 494

Helmholtz

function p. 498

Gibbs function p. 498

Maxwell relations p. 499

fundamental

function p. 503

Clapeyron

equation p. 505

Joule-Thomson

coefficient p. 518

enthalpy and entropy

departures p. 526, 528

Kay’s rule p. 532

method of

intercepts p. 537

chemical potential p. 539

fugacity p. 542

Lewis–Randall rule p. 545

Problems: Developing Engineering Skills 549

Exercises: Things Engineers Think About

1. What is an advantage of using the Redlich–Kwong equation

of state in the generalized form given by Eq. 11.9 instead of

Eq. 11.7? A disadvantage?

2. To determine the specific volume of superheated water vapor

at a known pressure and temperature, when would you use each

of the following: the steam tables, the generalized compressibility

chart, an equation of state, the ideal gas model?

3. If the function p  p(T, v) is an equation of state, is (  )

a property? What are the independent variables of (  )

4. In the expression (  )

, what does the subscript v signify?

5. Explain how a Mollier diagram provides a graphical repre-

sentation of the fundamental function h(s, p).

6. How is the Clapeyron equation used?

7. For a gas whose equation of state is are the specific

heats and necessarily functions of temperature alone?

8. Referring to the p–T diagram for water, explain why ice melts

under the blade of an ice skate.

pv  RT,

0T0u

0T0p

9. Can you devise a way to determine the specific heat of a

gas by direct measurement? Indirectly, using other measured

data?

10. For an ideal gas, what is the value of the Joule–Thomson

coefficient?

11. At what states is the entropy departure negligible? The

fugacity coefficient, fp, closely equal to unity?

12. In Eq. 11.107, what do the subscripts T, p, and n

signify?

What does i denote?

13. How does Eq. 11.108 reduce for a system consisting of a

pure substance? Repeat for an ideal gas mixture.

14. If two different liquids of known volumes are mixed, is the

final volume necessarily equal to the sum of the original

volumes?

15. For a binary solution at temperature T and pressure p,how

would you determine the specific heat ? Repeat for an ideal

solution and for an ideal gas mixture.

Problems: Developing Engineering Skills

Using Equations of State

11.1 The pressure within a 23.3-m

tank should not exceed 105

bar. Check the pressure within the tank if filled with 1000 kg

of water vapor maintained at 360C using the

(a) ideal gas equation of state.

(b) van der Waals equation.

(d) compressibility chart.

(e) steam tables.

11.2 Estimate the pressure of water vapor at a temperature of

500C and a density of 24 kg/m

using the

(a) steam tables.

(b) compressibility chart.

(d) van der Waals equation.

(e) ideal gas equation of state.

11.3 Determine the specific volume of water vapor at 20 MPa

and 400C, in m

/kg, using the

(a) steam tables.

(b) compressibility chart.

(d) van der Waal equation.

(e) ideal gas equation of state.

11.4 A vessel whose volume is 1 m

contains 4 kmol of methane

at 100C. Owing to safety requirements, the pressure of the

methane should not exceed 12 MPa. Check the pressure using

the

(a) ideal gas equation of state.

(b) Redlich–Kwong equation.

11.5 Methane gas at 100 atm and 18C is stored in a 10-m

tank. Determine the mass of methane contained in the tank, in

kg, using the

(a) ideal gas equation of state.

(b) van der Waals equation.

11.6 Using the Benedict–Webb–Rubin equation of state, deter-

mine the volume, in m

occupied by 165 kg of methane at a

pressure of 200 atm and temperature of 400 K. Compare with

the results obtained using the ideal gas equation of state and

the generalized compressibility chart.

11.7 A rigid tank contains 1 kg of oxygen (O

) at p

 40 bar,

 180 K. The gas is cooled until the temperature drops to

150 K. Determine the volume of the tank, in m

, and the final

pressure, in bar, using the

(a) ideal gas equation of state.

(b) Redlich–Kwong equation.

11.8 Water vapor initially at 240C, 1 MPa expands in a piston–

cylinder assembly isothermally and without internal irreversibil-

550 Chapter 11 Thermodynamic Relations

ities to a final pressure of 0.1 MPa. Evaluate the work done, in

kJ/kg. Use a truncated virial equation of state with the form

where B and C are evaluated from steam table data at 240C

and pressures ranging from 0 to 1 MPa.

11.9 Referring to the virial series, Eqs. 3.29 and 3.30, show

that (C  B

)

11.10 Express Eq. 11.5, the van der Waals equation in terms of

the compressibility factor Z

(a) as a virial series in v

. [Hint: Expand the (v

 18)

1

term of Eq. 11.5 in a series.]

(b) as a virial series in p

)

and higher in the virial

series of part (b), obtain the following approximate form:

(d) Compare the compressibility factors determined from the

equation of part (c) with tabulated compressibility factors

from the literature for 0  p

 0.6 and each of T

 1.0,

1.2, 1.4, 1.6, 1.8, 2.0. Comment on the realm of validity

of the approximate form.

11.11 The Berthelot equation of state has the form

(a) Using Eqs. 11.3, show that

(b) Express the equation in terms of the compressibility fac-

tor Z, the reduced temperature T

, and the pseudoreduced

specific volume, v

11.12 The Beattie–Bridgeman equation of state can be ex-

pressed as

where

and A

, B

, a, b, and c are constants. Express this equation of

state in terms of the reduced pressure, p

, reduced tempera-

e 

A  A

a1 

B  B

a1 

p 

11  e21v  B2



a 

b 

p 

v  b



T v

Z  1  a





 B



RT, C



Z  1 



ture, T

, pseudoreduced specific volume, v

, and appropriate

dimensionless constants.

11.13 The Dieterici equation of state is

(a) Using Eqs. 11.3, show that

(b) Show that the equation of state can be expressed in terms

of compressibility chart variables as

. (Hint:

Expand the (v

 1e

)

1

term in a series. Also expand

the exponential term in a series.)

11.14 The Peng–Robinson equation of state has the form

Using Eqs. 11.3, evaluate the constants a, b, c in terms of the

critical pressure p

, critical temperature T

, and critical com-

pressibility factor Z

11.15 The p–v–T relation for chlorofluorinated hydrocarbons

can be described by the Carnahan–Starling–DeSantis equation

of state

where a  a

exp (a

T  a

), and b  b



T  b

. For Refrigerants 12 and 13, the required coeffi-

cients for T in K, a in J L/(mol)

, and b in L/mol are given

in Table P11.15. Specify which of the two refrigerants would

allow the smaller amount of mass to be stored in a 10-m

vessel at 0.2 MPa, 80C.

Using Relations from Exact Differentials

11.16 The differential of pressure obtained from a certain equa-

tion of state is given by one of the following expressions. De-

termine the equation of state.

dp 

1v  b2

dv 

v  b

dp 

21v  b2

dv 

1v  b2

b  b



4v,



1  b  b

 b

11  b2



RT1v  b2

p 

v  b



 c

Z  a

v¿

 1



b exp a

4

v¿

a 

b 

p  a

v  b

b exp

a

RTv

 10

3

 10

R-12 3.52412 2.77230 0.67318 0.15376 1.84195 5.03644

R-13 2.29813 3.41828 1.52430 0.12814 1.84474 10.7951

 Table P11.15

Problems: Developing Engineering Skills 551

11.17 Introducing into Eq. 6.10 gives

Using this expression together with the test for exactness,

demonstrate that is not a property.

11.18 Using Eq. 11.35, check the consistency of

(a) the steam tables at 2 MPa, 400C.

(b) the Refrigerant 134a tables at 2 bar, 50C.

11.19 At a pressure of 1 atm, liquid water has a state of maxi-

mum density at about 4C. What can be concluded about

( s p)

(a) 3C?

(b) 4C?

11.20 A gas enters a compressor operating at steady state and

is compressed isentropically. Does the specific enthalpy in-

crease or decrease as the gas passes from the inlet to the exit?

11.21 Show that T, p, h, , and g can each be determined from a

fundamental thermodynamic function of the form u  u(s, v).

11.22 Evaluate p, s, u, h, c

, and c

for a substance for which

the Helmholtz function has the form

where v and T denote specific volume and temperature, re-

spectively, at a reference state, and c is a constant.

11.23 The Mollier diagram provides a graphical representation

of the fundamental thermodynamic function h  h(s, p). Show

that at any state fixed by s and p the properties T, v, u, , and

g can be evaluated using data obtained from the diagram.

11.24 Derive the relation c

T(

g

)

Evaluating ⌬s, ⌬u, and ⌬h

11.25 Using p–v–T data for saturated water from the steam

tables, calculate at 50C

(a) h

 h

(b) u

 u

 s

Compare with the values obtained using steam table data.

11.26 Using h

, v

, and p

sat

at 26C from the ammonia tables,

estimate the saturation pressure at 30C. Comment on the

accuracy of your estimate.

11.27 At 0C, the specific volumes of saturated solid water (ice)

and saturated liquid water are, respectively, v

 1.0911 

3

/kg and v

 1.0002  10

3

/kg, and the change in

specific enthalpy on melting is h

 333.4 kJ/kg. Calculate the

melting temperature of ice at (a) 250 bar, (b) 500 bar. Locate

your answers on a sketch of the p–T diagram for water.

11.28 The line representing the two-phase solid–liquid region

on the phase diagram slopes to the left for substances that ex-

pand on freezing and to the right for substances that contract

0T0

c RT

v¿

 cT ¿ c1 

T¿



T¿

int

rev

int

rev

 dU  p dV

int

rev

 T dS

on freezing (Sec. 3.2.2). Verify this for the cases of lead that

contracts on freezing and bismuth that expands on freezing.

11.29 Consider a four-legged chair at rest on an ice rink. The

total mass of the chair and a person sitting on it is 80 kg. If

the ice temperature is 2C, determine the minimum total area,

in cm

, the tips of the chair legs can have before the ice in con-

tact with the legs would melt. Use data from Problem 11.27

and let the local acceleration of gravity be 9.8 m/s

11.30 Over a certain temperature interval, the saturation pres-

sure–temperature curve of a substance is represented by an

equation of the form ln p

sat

 A  BT, where A and B are

empirically determined constants.

(a) Obtain expressions for h

 h

and s

 s

in terms of

p–v–T data and the constant B.

(b) Using the results of part (a), calculate h

 h

and s

 s

for water vapor at 25C and compare with steam table data.

11.31 Using data for water from Table A-2, determine the con-

stants A and B to give the best fit in a least-squares sense to

the saturation pressure in the interval from 20 to 30C by the

equation ln p

sat

 A  BT. Using this equation, determine

sat

dT at 25C. Calculate h

 h

at 25C and compare with

the steam table value.

11.32 Over limited intervals of temperature, the saturation pres-

sure–temperature curve for two-phase liquid–vapor states can

be represented by an equation of the form ln p

sat

 A  BT,

where A and B are constants. Derive the following expression

relating any three states on such a portion of the curve:

where   T

 T

)T

 T

11.33 Use the result of Problem 11.42 to determine

(a) the saturation pressure at 30C using saturation

pressure–temperature data at 20 and 40C from Table

A-2. Compare with the table value for saturation pressure

at 30C.

(b) the saturation temperature at 0.006 MPa using saturation

pressure–temperature data at 20 to 40C from Table A-2.

Compare with the saturation temperature at 0.006 MPa

given in Table A-3.

11.34 Complete the following exercises dealing with slopes:

(a) At the triple point of water, evaluate the ratio of the slope

of the vaporization line to the slope of the sublimation line.

Use steam table data to obtain a numerical value for the

ratio.

(b) Consider the superheated vapor region of a

temperature–entropy diagram. Show that the slope of a

constant specific volume line is greater than the slope of

a constant pressure line through the same state.

used in analyzing steam turbines. Obtain an expression for

the slope of a constant-pressure line on such a diagram in

terms of p–v–T data only.

sat, 3

sat, 1

 a

sat, 2

sat,1

552 Chapter 11 Thermodynamic Relations

(d) A pressure–enthalpy diagram is often used in the refrig-

eration industry. Obtain an expression for the slope of an

isentropic line on such a diagram in terms of p–v–T data

only.

11.35 One kmol of argon at 300 K is initially confined to one

side of a rigid, insulated container divided into equal volumes

of 0.2 m

by a partition. The other side is initially evacuated.

The partition is removed and the argon expands to fill the en-

tire container. Using the van der Waals equation of state, de-

termine the final temperature of the argon, in K. Repeat using

the ideal gas equation of state.

11.36 Obtain the relationship between c

and c

for a gas that

obeys the equation of state p(v  b)  RT.

11.37 The p–v–T relation for a certain gas is represented closely

by v  RTp  B  ART, where R is the gas constant and A

and B are constants. Determine expressions for the changes in

specific enthalpy, internal energy, and entropy, [h(p

, T ) 

h( p

, T)], [u(p

, T )  u(p

, T )], and [s(p

, T )  s( p

, T)],

respectively.

11.38 Develop expressions for the specific enthalpy, internal

energy, and entropy changes [h(v

, T)  h(v

, T)], [u(v

, T) 

u(v

, T)], [s(v

, T)  s(v

, T)], using the

(a) van der Waals equation of state.

(b) Redlich–Kwong equation of state.

11.39 At certain states, the p–v–T data of a gas can be expressed

as Z  1  ApT

, where Z is the compressibility factor and

A is a constant.

(a) Obtain an expression for (  )

in terms of p, T, A, and

the gas constant R.

(b) Obtain an expression for the change in specific entropy,

[s(p

, T)  s(p

, T)].

[h(p

, T)  h(p

, T)].

11.40 For a gas whose p–v–T behavior is described by Z  1 

BpRT, where B is a function of temperature, derive expres-

sions for the specific enthalpy, internal energy, and entropy

changes, [h( p

, T )  h(p

, T )], [u(p

, T )  u(p

, T )], and

[s(p

, T)  s(p

, T)].

11.41 For a gas whose p–v–T behavior is described by Z  1 

Bv  Cv

, where B and C are functions of temperature, de-

rive an expression for the specific entropy change, [s(v

, T ) 

s(v

, T)].

Using Other Thermodynamic Relations

11.42 The volume of a 1-kg copper sphere is not allowed to

vary by more than 0.1%. If the pressure exerted on the sphere

is increased from 10 bar while the temperature remains con-

stant at 300 K, determine the maximum allowed pressure, in

bar. Average values of , , and  are 8888 kg/m

, 49.2  10

6

(K)

1

, and 0.776  10

11

/N, respectively.

11.43 Develop expressions for the volume expansivity  and

the isothermal compressibility  for

(a) an ideal gas.

(b) a gas whose equation of state is p(v  b)  RT.

0T0p

11.44 Derive expressions for the volume expansivity  and the

isothermal compressibility  in terms of T, p, Z, and the first

partial derivatives of Z. For gas states with p

 3, T

 2,

determine the sign of . Discuss.

11.45 Show that the isothermal compressibility  is always

greater than or equal to the isentropic compressibility .

11.46 Prove that (  p)

(  T)

11.47 For aluminum at 0C,   2700 kg/m

,   71.4  10

8

(K)

1

,   1.34  10

13

/N, and c

 0.9211 kJ/kg K. De-

termine the percent error in c

that would result if it were as-

sumed that c

 c

11.48 Estimate the temperature rise, in C, of mercury, initially

at 0C and 1 bar if its pressure were raised to 1000 bar isen-

tropically. For mercury at 0C, c

 28.0 kJ/kmol K,

/kmol, and   17.8  10

5

(K)

1

11.49 At certain states, the p–v–T data for a particular gas can be

represented as Z  1  ApT

, where Z is the compressibility

factor and A is a constant. Obtain an expression for the specific

heat c

in terms of the gas constant R, specific heat ratio k, and

Z. Verify that your expression reduces to Eq. 3.47a when Z  1.

11.50 For a gas obeying the van der Waals equation of state,

(a) show that ( )

 0.

(b) develop an expression for c

 c

, v

)  u(T

, v

)] and

[s(T

, v

)  s(T

, v

)].

(d) complete the u and s evaluations if c

 a  bT, where

a and b are constants.

11.51 Show that the specific heat ratio k can be expressed as k 

(c

  Tv

). Using this expression together with data from

the steam tables, evaluate k for water vapor at 200 lbf/in.

, 500F.

11.52 For liquid water at 40C, 1 atm estimate

(a) c

, in kJ/kg K

(b) the velocity of sound, in m/s.

Use Data from Table 11.2, as required.

11.53 Using steam table data, estimate the velocity of sound in

liquid water at 20C, 50 bar.

11.54 For a gas obeying the equation of state p(v  b)  RT,

where b is a positive constant, can the temperature be reduced

in a Joule–Thomson expansion?

11.55 A gas is described by v  RTp  AT  B, where A

and B are constants. For the gas

(a) obtain an expression for the temperatures at the

Joule–Thomson inversion states.

(b) obtain an expression for c

 c

11.56 Determine the maximum Joule–Thomson inversion tem-

perature in terms of the critical temperature T

predicted by the

(a) van der Waals equation.

(b) Redlich–Kwong equation.

11.57 Derive an equation for the Joule–Thomson coefficient as

a function of T and v for a gas that obeys the van der Waals



 0.0147

0000

Problems: Developing Engineering Skills 553

equation of state and whose specific heat c

is given by c



A  BT  CT

, where A, B, C are constants. Evaluate the tem-

peratures at the inversion states in terms of R, v, and the van

der Waals constants a and b.

11.58 Show that Eq. 11.63 can be written as

(a) Using this result, obtain an expression for the Joule–Thomson

coefficient for a gas obeying the equation of state

where A is a constant.

(b) Using the result of part (a), determine c

, in kJ/kg K, for

at 400 K, 1 atm, where 

 0.57 K/atm. For CO

A  2.78  10

3

/kg N.

Developing Property Data

11.59 If the specific heat c

of a gas obeying the van der Waals

equation is given at a particular pressure, p, by c

 A  BT,

where A and B are constants, develop an expression for the

change in specific entropy between any two states 1 and 2:

[s(T

, p

)  s(T

, p

)].

11.60 For air, write a computer program that evaluates the

change in specific enthalpy from a state where the temperature

is 25C and the pressure is 1 atm to a state where the temper-

ature is T and the pressure is p. Use the van der Waals equa-

tion of state and account for the variation of the ideal gas spe-

cific heat as in Table A-21.

11.61 Using the Redlich–Kwong equation of state, determine

the changes in specific enthalpy, in kJ/kmol, and entropy, in

kJ/kmol K, for ethylene between 400 K, 1 bar and 400 K,

100 bar.

11.62 Using the Benedict–Webb–Rubin equation of state to-

gether with a specific heat relation from Table A-21, determine

the change in specific enthalpy, in kJ/kmol, for methane be-

tween 300 K, 1 atm and 400 K, 200 atm.

11.63 Using the Redlich–Kwong equation of state together with

an appropriate specific heat relation, determine the final tem-

perature for an isentropic expansion of nitrogen from 400 K,

250 atm to 5 atm.

Using Enthalpy and Entropy Departures

11.64 Beginning with Eq. 11.90, derive Eq. 11.91.

11.65 Derive an expression giving

(a) the internal energy of a substance relative to that of its

ideal gas model at the same temperature: [u(T, v)  u*(T )].

(b) the entropy of a substance relative to that of its ideal gas

model at the same temperature and specific volume:

[s(T, v)  s*(T, v)].

v 





0 1v



T 2

11.66 Derive expressions for the enthalpy and entropy depar-

tures using an equation of state with the form Z  1  Bp

where B is a function of the reduced temperature, T

11.67 The following expression for the enthalpy departure is

convenient for use with equations of state that are explicit in

pressure:

(a) Derive this expression.

(b) Using the given expression, evaluate the enthalpy depar-

ture for a gas obeying the Redlich–Kwong equation of

state.

cific enthalpy, in kJ/kmol, for CO

undergoing an isother-

mal process at 300 K from 50 to 20 bar.

11.68 Using the equation of state of Problem 11.10 (c), evalu-

ate v and c

for water vapor at 550C, 20 MPa and compare

with data from Table A-4 and Fig. 3.9, respectively. Discuss.

11.69 Ethylene at 67C, 10 bar enters a compressor operating

a steady state and is compressed isothermally without internal

irreversibilities to 100 bar. Kinetic and potential energy

changes are negligible. Evaluate in kJ per kg of ethylene flow-

ing through the compressor

(a) the work required.

(b) the heat transfer.

11.70 Methane at 27C, 10 MPa enters a turbine operating at

steady state, expands adiabatically through a 5 : 1 pressure ra-

tio, and exits at 48C. Kinetic and potential energy effects

are negligible. If

 35 kJ/kmol K, determine the work de-

veloped per kg of methane flowing through the turbine. Com-

pare with the value obtained using the ideal gas model.

11.71 Nitrogen (N

) enters a compressor operating at steady

state at 1.5 MPa, 300 K and exits at 8 MPa, 500 K. If the work

input is 240 kJ per kg of nitrogen flowing, determine the heat

transfer, in kJ per kg of nitrogen flowing. Ignore kinetic and

potential energy effects.

11.72 Oxygen (O

) enters a control volume operating at steady

state with a mass flow rate of 9 kg/min at 100 bar, 287 K and

is compressed adiabatically to 150 bar, 400 K. Determine the

power required, in kW, and the rate of entropy production, in

kW/K. Ignore kinetic and potential energy effects.

11.73 Argon gas enters a turbine operating at steady state at

100 bar, 325 K and expands adiabatically to 40 bar, 235 K with

no significant changes in kinetic or potential energy. Determine

(a) the work developed, in kJ per kg of argon flowing through

the turbine.

(b) the amount of entropy produced, in kJ/K per kg of argon

flowing.

11.74 Oxygen (O

) undergoes a throttling process from 100 bar,

300 K to 20 bar. Determine the temperature after throttling, in

h*1T 2 h1T, v2

 T

c1  Z 





cT a

 p dd v d

554 Chapter 11 Thermodynamic Relations

K, and compare with the value obtained using the ideal gas

model.

11.75 Water vapor enters a turbine operating at steady state at

30 MPa, 600C and expands adiabatically to 6 MPa with no sig-

nificant change in kinetic or potential energy. If the isentropic tur-

bine efficiency is 80%, determine the work developed, in kJ per

kg of steam flowing, using the generalized property charts. Com-

pare with the result obtained using steam table data. Discuss.

11.76 Oxygen (O

) enters a nozzle operating at steady state at

60 bar, 300 K, 1 m/s and expands isentropically to 30 bar. De-

termine the velocity at the nozzle exit, in m /s.

11.77 A quantity of nitrogen gas undergoes a process at a con-

stant pressure of 80 bar from 220 to 300 K. Determine the

work and heat transfer for the process, each in kJ per kmol of

nitrogen.

11.78 A closed, rigid, insulated vessel having a volume of

0.142 m

contains oxygen (O

) initially at 100 bar, 7C. The

oxygen is stirred by a paddle wheel until the pressure becomes

150 bar. Determine the

(a) final temperature, in C.

(b) work, in kJ.

Let T

 7C.

Evaluating p–v–T for Gas Mixtures

11.79 A preliminary design calls for a 1 kmol mixture of CO

and C

(ethane) to occupy a volume of 0.15 m

at a tem-

perature of 400 K. The mole fraction of CO

is 0.3. Owing to

safety requirements, the pressure should not exceed 180 bar.

Check the pressure using

(a) the ideal gas equation of state.

(b) Kay’s rule together with the generalized compressibility

chart.

compressibility chart.

Compare and discuss these results.

11.80 A 0.1-m

cylinder contains a gaseous mixture with a mo-

lar composition of 97% CO and 3% CO

initially at 138 bar. Due

to a leak, the pressure of the mixture drops to 129 bar while the

temperature remains constant at 30C. Using Kay’s rule, estimate

the amount of mixture, in kmol, that leaks from the cylinder.

for the constants a and b.

(d) the rule of additive pressure together with the generalized

compressibility chart.

11.82 Air having an approximate molar composition of 79%

and 21% O

fills a 0.36-m

vessel. The mass of mixture is

100 kg. The measured pressure and temperature are 101 bar

and 180 K, respectively. Compare the measured pressure with

the pressure predicted using

(a) the ideal gas equation of state.

(b) Kay’s rule.

(d) the additive volume rule with the Redlich–Kwong equation.

11.83 Using the Carnahan–Starling–DeSantis equation of

state introduced in Problem 11.15, together with the following

expressions for the mixture values of a and b:

where f

is an empirical interaction parameter, determine the

pressure, in kPa, at v  0.005 m

/kg, T  180C for a mix-

ture of Refrigerants 12 and 13, in which Refrigerant 12 is 40%

by mass. For a mixture of Refrigerants 12 and 13, f

 0.035.

11.84 A rigid vessel initially contains carbon dioxide gas at

32C and pressure p. Ethylene gas is allowed to flow into the

tank until a mixture consisting of 20% carbon dioxide and 80%

ethylene (molar basis) exists within the tank at a temperature

of 43C and a pressure of 110 bar. Determine the pressure p,

in bar, using Kay’s rule together with the generalized com-

pressibility chart.

Analyzing Multicomponent Systems

11.85 A binary solution at 25C consists of 59 kg of ethyl

alcohol (C

OH) and 41 kg of water. The respective partial

molal volumes are 0.0573 and 0.0172 m

/kmol. Determine the

total volume, in m

. Compare with the volume calculated us-

ing the molar specific volumes of the pure components, each

a liquid at 25C, in the place of the partial molal volumes.

11.86 The following data are for a binary solution of ethane

) and pentane (C

) at a certain temperature and

pressure:

b  y

 y

a  y

 2y

11  f

21a



 y

mole fraction of ethane 0.2 0.3 0.4 0.5 0.6 0.7 0.8

volume (in m

) per kmol of solution 0.119 0.116 0.112 0.109 0.107 0.107 0.11

11.81 A gaseous mixture consisting of 0.75 kmol of hydrogen

) and 0.25 kmol of nitrogen (N

) occupies 0.085 m

at 25C.

Estimate the pressure, in bar, using

(a) the ideal gas equation of state.

(b) Kay’s rule together with the generalized compressibility

chart.

Estimate

(a) the specific volumes of pure ethane and pure pentane, each

in m

/kmol.

(b) the partial molal volumes of ethane and pentane for an

equimolar solution, each in m

/kmol.

Problems: Developing Engineering Skills 555

11.87 Using p–v–T data from the steam tables, determine the fu-

gacity of water as a saturated vapor at 280C, Compare with the

value obtained from the generalized fugacity chart.

11.88 Using the equation of state of Problem 11.10 (c), eval-

uate the fugacity of ammonia at 750 K, 100 atm and compare

with the value obtained from Fig. A-6.

11.89 Using tabulated compressibility data from the literature,

evaluate fp at T

 1.40 and p

 2.0. Compare with the

value obtained from Fig. A-6.

11.90 Consider the truncated virial expansion

(a) Using tabulated compressibility data from the literature,

evaluate the coefficients and for 0  p

 1.0 and

each of T

 1.0, 1.2, 1.4, 1.6, 1.8, 2.0.

(b) Obtain an expression for ln ( fp) in terms of T

and p

. Us-

ing the coefficients of part (a), evaluate fp at selected states

and compare with tabulated values from the literature.

11.91 Derive the following approximation for the fugacity of

a liquid at temperature T and pressure p:

where (T) is the fugacity of the saturated liquid at temper-

ature T. For what range of pressures might the approximation

f(T, p)  (T) apply?

11.92 Beginning with Eq. 11.122,

(a) evaluate ln f for a gas obeying the Redlich–Kwong equa-

tion of state.

(b) Using the result of part (a), evaluate the fugacity, in

bar, for Refrigerant 134a at 90C, 10 bar. Compare with

the fugacity value obtained from the generalized fugacity

chart.

11.93 Consider a one-inlet, one-exit control volume at steady

state through which the flow is internally reversible and

isothermal. Show that the work per unit of mass flowing can

be expressed in terms of the fugacity f as

11.94 Propane (C

) enters a turbine operating at steady state

at 100 bar, 400 K and expands isothermally without irre-

versibilities to 10 bar. There are no significant changes in ki-

netic or potential energy. Using data from the generalized fu-

gacity chart, determine the power developed, in kW, for a mass

flow rate of 50 kg/min.

11.95 Ethane (C

) is compressed isothermally without ir-

reversibilities at a temperature of 320 K from 5 to 40 bar.

Using data from the generalized fugacity and enthalpy de-

parture charts, determine the work of compression and the

heat transfer, each in kJ per kg of ethane flowing. Assume

steady-state operation and neglect kinetic and potential

energy effects.

int

rev

RT ln a

b

 V

 g1z

 z

sat

f 1T, p2 f

sat

1T 2 exp e

1T 2

3p  p

sat

1T 24f

, C

Z  1  B

 C

 D

11.96 Methane enters a turbine operating at steady state at

100 bar, 275 K and expands isothermally without irre-

versibilities to 15 bar. There are no significant changes in

kinetic or potential energy. Using data from the generalized

fugacity and enthalpy departure charts, determine the power

developed and heat transfer, each in kW, for a mass flow rate

of 0.5 kg/s.

11.97 Methane flows isothermally and without irreversibilities

through a horizontal pipe operating at steady state, entering at

50 bar, 300 K, 10 m/s and exiting at 40 bar. Using data from

the generalized fugacity chart, determine the velocity at the

exit, in m/s.

11.98 Determine the fugacity, in atm, for pure ethane at 310

K, 20.4 atm and as a component with a mole fraction of 0.35

in an ideal solution at the same temperature and pressure.

11.99 Denoting the solvent and solute in a dilute binary liq-

uid solution at temperature T and pressure p by the subscripts

1 and 2, respectively, show that if the fugacity of the solute is

proportional to its mole fraction in the solution:

where  is a constant (Henry’s rule), then the fugacity of the

solvent is where y

is the solvent mole fraction and

is the fugacity of pure 1 at T, p.

11.100 A tank contains 310 kg of a gaseous mixture of 70%

ethane and 30% nitrogen (molar basis) at 311 K and 170 atm.

Determine the volume of the tank, in m

, using data from the

generalized compressibility chart together with (a) Kay’s rule,

(b) the ideal solution model. Compare with the measured tank

volume of 1 m

11.101 A tank contains a mixture of 70% ethane and 30%

nitrogen (N

) on a molar basis at 400 K, 200 atm. For 2130

kg of mixture, estimate the tank volume, in m

, using

(a) the ideal gas equation of state.

(b) Kay’s rule together with data from the generalized com-

pressibility chart.

eralized compressibility chart.

11.102 An equimolar mixture of O

and N

enters a compres-

sor operating at steady state at 10 bar, 220 K with a mass flow

rate of 1 kg/s. The mixture exits at 60 bar, 400 K with no

significant change in kinetic or potential energy. Stray heat

transfer from the compressor can be ignored. Determine for

the compressor

(a) the power required, in kW.

(b) the rate of entropy production, in kW/K.

Assume the mixture is modeled as an ideal solution. For the

pure components:

10 bar, 220 K 60 bar, 400 K

h (kJ/kg) s (kJ / kg K) h (kJ/kg) s (kJ/kg K)

Oxygen 195.6 5.521 358.2 5.601

Nitrogen 224.1 5.826 409.8 5.911

11.103 A gaseous mixture with a molar analysis of 70% CH

and 30% N

enters a compressor operating at steady state at

 y

 ky