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PEARSON ALWAn LEARNING

Introduction to Thermodynamics

APS104 Introduction to Materials and Chemistry (Chemistry Topics, 2012)

Third Custom Edition for University of Toronto

Taken from: Thermodynamics, Statistical Thermodynamics, & Kinetics,

Second Edition by Thomas Engel and Philip Reid

Student Solutions Manual to accompany Physical Chemistry, Second Edit ion

by Thomas Engel and Philip Reid

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Cover Art: Courtesy of Photodisc/Getty Images

Taken from:

Thermodynamics, Statistic.J1 Thermodynamics, & Kinetics, Second Edition by Thomas Engel and Philip Reid Copyright C 2010, 2006 by Pearson Education, Inc. Published by Prentke Hall Upper Saddle River, New Jersey 07458

Student Solutions Manual to accompany Physical Chemistry, Second Edition by Thomas Engel and Philip Reid Copyright C 2010 by Pearson Education, Inc. Published by Prentice Hall

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ISBN 10: 1-256-46323-X ISBN 13: 978125646323-8

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Brief Contents 'f',,~cn from: Th~""'oJ)'ndlni('s. $1(l/iSlical ~ntIOdJ"""'io, & KiMli",. Sood F.d;!;on. byThoouos " ";

Contents Tak~n rrom: Thr""",I,.~a",k&. SI/II;s,ir/l1 ThrrmooJrllJmic:J, & Kinrrk Second Edi!ion. by Thomas Enj>el and Philip Reid.

Text

1 Fundamcmal ConccpLS of Thermodynamics 1

1.1 Whul ls Thermodynamics and Why Is II Useful'! I

1.2 8 asic Definitions Needed to Dc$cribe TIlcnnodynamic Syslcm~ :2

1.3 ThcmlOOlclry-l 1.4 F..quuliQns of Slale urnlthc Ideal Gas Law 6 1.5 A Brief Introduction \0 Real Ga.'iCS 'J

2 Heat, Work , Intern al Energy, Enthalpy, and the Fi rst Law of Thermodynam ics 15

2. 1 The 100ernai Energy and the FirM Law ofThcrmooynumics 15

2.2 Work 16 2.3 Heal 19 2.4 Heal Capacity 21 2.5 State FUIlCIioos and P.~lh FUBClioflS 23 2.6 F..4Ju ilibrium. Change. and Revclliibi lity 2S 2.7 Comparing Worl: for RC"elliibk and irre"ersible

Processes 26 2.8 Determining J.U and Introducing Emhalpy, a New

State Function 30 2.9 Cakulalingq. IV. Il.U, and "'" for Processes

In"oh'i llg Idea l Gases 3 1 2.10 "The Rc\"crsiblc Adi~b~t ic EXp;ln~ion and

Compression of ~n Ideal Gas 35

3 The Im portance of State Functions: Internal Energy and Enthalpy 41

3. 1 TIle I- Iathematical Properties of State Function.~ 4 1

3.2 The Dependence of U on V amI T 46 3.3 Docs the Internal Energy Depend Mor~ Strongly

onVorT '! 48 3.4 The V~riation of Enth~lp>' WiTh TemperJlUre

at Coostant I'n:ssure 51 3.S How Arc CI' ~ Ild Cv Related? 53 3.6 "The Variation of Enth~lp>' with Pressure ~t Coost~nt

Tcmper:lIure S4

3.1 ThcJoule--TIlomson Experiment 51 3.8 Liquefying Gases Using an lsentll:llpic

Exp;lnsion S9

4 Thermoche mistry 63 4.1 F~rgy St~d in Chemical Bonds Is Released

or Tat en Up in Chemical Rc~etioos 63 4.2 lntemal EnelID' and Enthalpy Changes Associated

with Chemical Reactions 64 4.3 Hcsss Law Is Based on Enthalpy Being ~ St~te

FuncTion 67 4.4 The TemperJture D.:pendence of React ion

Enthalpies 69 4.5 The Ellpcri menwlDelenninat ion of tlU and ill!

for Chemical Reactioos 1 1 4.6 (Supplemental) Differential Scanning

Calorimetry 73

5 Entropy and the Second and Third Laws of T hermodynamics 79

5.1 Thc Uni\'cl"5e Has a NalUrJI Dircctioo ofChangc 79

5.2 Heat Engines and the Scoond Law of"Thermodynamic$ 80

5.3 Intruducing Entropy 85 5.4 CalculaTing Changes in Entropy 86 55 Using Entropy to Calculate the NalUrJI Dircctioo

of a Process in an Isolated System 89 5.6 "The Cl~osios Inequality 9 1 5.7 "The Change of Entropy in the Surroundings

and ilS_ 1 - tlS + tlS, .. ,,,,,,,,,,,~,. \12 5.8 AbsolUTe Entropies and the Third law

ofThemlodynamks 94 5.9 Standard STateS in Entropy CalCU lations ')8 5.10 Entropy Changes in Chemical React ions 98 5.1 I (Supplemental) Energy Efficiency: Heat Pump~,

Refrigerators. and Real Engines 100 5.12 (Supplemental) Using Ihe Fuel th~t S Is u StaTe

Functioolo Determine the Dependence of S on V und T 107

5.1 3 (Supplemental) The Dependence ofSooTandP lOS

5.1 4 (Supplemental) The TIlermodynamic Temperature Scale l{)l)

_., rio. iW) ....... . n.rnu.xn~ .... t..trMnify

6 C hemical Equi librium 115 6, I The Gibbs Ern:rgy and the Ilelmhohz Energy 11 6 6.2 The Diffen"'ntiaI FormsofU. II. A.andG 120 6.3 1ltc Dependence of the Gibbs and Helmholtz Energies

onP.V.andT 121 6.4 TIle Gibbs Energy of a Re~ction Mixtu re 124 6.5 TIle Gibbs Energy of a Gas in u Mixture 125 6.6 Calcul~ting the Gibbs Energy of " Iixing for Ideal

Ga.ws 126 6.7 Calculating t.G~ for a Chemical Reaction 127 6.8 Introducing the Equi librium Constant for ~ Mi~ture of

Ideal Gases 129 6.9 Calculating the Equilibrium P.~rtial Pressures

inaMi:nureofideal Gases 13 1 6.10 1ltcV:uiationof Kl'withTenlpcr~ture In 6. I I Equilibria Involving Ideal Gases and Solid

or Liquid Phases 134 6. 12 Express ing the Equilibrium COIl.,tam in Terms

of Mole Fract ion or " lobrity 135 6. 13 TheDependenceoff.qonTandP 136 6. 14 (Supplemental ) A Ca.'\e Study: 1ltc Synthesis

of Ammonia 137 6. 15 (Supplemental) Expressing U alld H alld Heat

Capacities Solely in Terms of Measur~bl e Quantities 142

6.16 (Supplemental) Measuring AG for the Unfoldi ng of Single RNA ~ Iol ccu le.~ 146

6. 17 (Supplememal) The Role of Mi.,ing in Detennining F..4ju ilibrium in a Chemical Reaction 147

11 Electrochemical Cells, Batteries, and Fuel Cells 249

11 .1 1ltc EffC(:t of an Electrical Potential on lite o.emi",1 Putential of Charged Species 250

I 1.2 Con\"entions alld Stand:ml States in EIC(:trocltemistry 25 I

CONTENTS V

11.3 Measorement of the Reversible Cell Potential 254 11.4 Chemical Reactions in Electrochemical Cells and the

Nemst Equation 2S4 11.5 Combining Standard Electrode Potentials

to Determirn: the Cell P(){ential 256 11 .6 Obtuining Reaction Gibbs Energies and Reactioo

Entropies from Cell Putentials 257 11.7 The Relat ionship between the Cell EMF and the

r:.qu il ibrium COrlstunt 25R I 1.8 l)elermirlatioll of E" and Activity Coefficients Usi ng

an Electrochemical CelJ 260 11.9 Cell Nomenclature alld Types of Electrochemical

Cells 260 11. 10 The Electrochemical Series 262 11.11 llIcrmodynamics of B~ueries alld Fuel Cells 263 ' I I ~ The EleCTrochemistry of ComlTlOfl l)' Used

Balleries 263 II 13 Fuel Cells 265 I I 14 (Supplel11entul ) Elect rochemistry at the Atomic

Scale 267 I L .15 (Supplemental) Using Elcctrochemistry for Nanoscale

Machining 273 I L. 16 (Supplcmcnla l)Absolutc Half-Cell PUlentials 274

Appendix A Dala Tables 539 Appe nd ix C Answe rs (0 Selccted

End-of-Chaplcr Problems 579

Ind e x 59 1

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vi CONTENlS

Taken r,om: S/u,l~m Solu/joot M""uu/lo occompany Physk(J/ Ch~mi~Ir'J, SeCI)M Edilion. by 111onm' Eng.:! ~nd Philip Reid.

Student Solutions 1 Fundamental Concepts of Thermodynamics 2 Heal. Work. internal Energy. Enthalpy. and

The First Law ofTIlermodynamics 7 3 The Importance of State Functions:

Internal Energy and Enthalpy 16 4 Thennochemistry 22

5 Entropy and The Second and Third Laws of 1llcrmodynamics 29

6 Chemical Equilibrium 41 11 Electrochemical Cells. Batteries.

and Fuel Cells 83

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Text

Taken from: 'n1l' mlllilYI/(llIIics. SIlIIisliCIII Thermodylll/mics, & Kill e/irs, Second Edition. by Thomas Engel und Philip Reid.

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CHAPTER OUTLINE 1 U WHAT IS THERMOOYNAMICS AND WHY IS IT USEFUL?

U

U

U

BASIC DHINITIONS NEEDED to DESCRIBE THERMODYNAMIC SYSTEMS THERMOMETRY EQUATIONS OF STATE

Fundamental Concepts of Thermodynamics

AND THE IDEAL GAS LAW ,., A BRIEf INTRODUCTION

TO REAL GASES

Thermodynamics provides a desr~"'T_t'l-.-EtIgoInlPIIIIIp_. ~t'I~l.....-.g~~02012t'1 __ ~1oo:.

2 C H A PT E R 1 Fundl"n~l'I1a1 Concept5 of Tllefmodyn~m0::5

Pial'll cell

Milochondrioll Plasma membrane An lmel eell

FIGURE 1.1 Animal and planl l.....-.g~~02012t'1 __ Eo1aIia\,Ioo:.

I ~ BAS1( DEf IN1TIONS NEEDED TO DESCR IBE THERMODYNAM1( SYSHMS 3

t"Qr\t irlCnt~ and the ocean noor and too watcr-air interlace at the o are observed. If rlCither pressure gauge changes. as in Figure 1.2b. we refer 10 thc wall s as

'Fut 110" .,amc>I< ..... ...no .......... of''''' bubble" _...,~ 10 be ......... tl thot ,I cat> be ... 1 ~tlall"'cro Thio ;. '" kp", .. II)"II01I"OC ....sih ... of .. ..-iJhtk pi""'" _ (rictionh. pulley ...

:: "" .. " "ooOO .: ~ .: . " .. ,

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. . . .

oo" ". ..oo.

IOj

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. . ."" . . " .. " . . .

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4 C H A PT E R 1 Fundl"n~l'I1a1 Concept5 of Tllefmodyn~m0::5

being adiabatic. Because PI l' Pl. the systems are IlOt in thennal elluiliboriulTl and. therefore. ho!\'e diffcrentlemper~1Ures. An example of a s)stem surrounded by adiaOOt ie walls;s c..,ffee in a Stymf ...... m cup wilh a Styrofoam lid .l Expericoce Shows that;t ;s not possible 10 bring two systems eoclused by adiaoo.tic w.t ll s inlo Ihcrnl:ll equ ilibrium by bringing them into COIltact. because adiab:llic walls insubte ugainstlhe trJnsfer of '"heal:' If )'00 push a Styrofoam cup COIltaining hoI coffee against one COIltaining ice waler. they willnol: reach me same temper~lUre. Rely on your c.~pcrieoce al 1his point regarding lhe meaning of heal; a thel"lllOdynamie definition will be givcn in Chapler 2.

llw.: second limiting case is ~'n in Figure I .k. [n boringing!he s)'iilCntS inlo intimate eootact. both presswes change and reach the same \';due aflCl"some time. We conclude that lhe s),stems have the same tempcr~ture. TI : Tl and sa)' Ihatthcy are in thenTkll equi libri-um. We refer to the walls as beins diathernJaI. 1\0'0 s)'!ilcms in eont:oc1 SCpal":lled by dialheml:ll walls reach therm.:lI equiliborium bccaU$C diuthcrmal W"J lls cOllduct hc;.,1. HOI coffee stored in a copper cup;s an example of u syslem surroonded by diathermal walls. BecallM' the w.tlls are diathcm13l. the coffee will quickl)' reach ruom temperJlure.

The l~roth h,w or th~rmodynamics gencrJIi7.es lhe experiment illuSlrJted in Figure 1.2 and as-;ens the existence of an objeclive temperJIUre that can be used todefinc thc condition of thennal equilibrium. The formal statement of th is law is as follows:

l'wo sys[ems [lI:It are scpal":l[ely in [hcnnal equi li brium with a th ird system arc also in therm~1 equilibrium with one another.

There are four laws of thermodynamics. all of which are gcncrJli 7.ations from expe-rience rJther th~n m~themmical theorems. They have becn rigorously lested in more th~n a ccntury of experimenwtion, and no violations of these laws have been found. The unfortunate name a.~~igned to the "urt)1h"law iE due to the f~ctthat it was formulated after the firs! law of thennodynamics. but logically precedes il. l1lC 7.Croth law tells us

th~t we can detenninc if two systems are in thernlal equi librium without bringing lhem into con[xt. lmaginc the thi rd system to be a themWJnICter. which is defilled more pre-cisely in the next scclioo. The th ird system can be used 10 compare the tcmperJturcs of lhe other IWO systems: if they ha\"\: the same temperJture. the)' wi ll be in them131 equilibrium.

1.3 THERMOMETRY llw.: discussion of lhennal equi librium required on ly that a therm (Hlirler cx ist Ihal call mcasure relative I>otncss or coldnes.~. For ally u-;efol thermometer. the measured tcm-pel":l[un:. I. must be a single-va lued. continuous. and monolonic fuoction of some ther-mometric system property such as the "olume of a liquid. tile electrical resistancc of a metal or semiconductor. and the eicctromofive force gcncrJ lcd at the jUllction of two dissimilar mctals. The simplest case that OIIC can imagine is if the empiricaltcmperJ-ture. /, is linearl)' related to the value of the thermometric property . . 1:

I(X) - II + hoi" (U ) Eqllation (1.3) defines a tempera turt.' scale in lerms of a spec ific thermometric proper-[y. once the conswnts II and b an: detcmlined. The constallt II determines the zero of the tempcrJlUre scale because 1(0) : (I, and the com;tant h determines Ihe si?e of a unit uf temperJlUre_ called a degree.

One of the first practical thermometers was the men:ury-in-glass thermometer. [t utili1.es tile lllennomelric property lhultlle volume of mercury increases monotunically Octween - 38.8" alld 356.1"C;n which Hg is a liquid. [n 17045. Carolus Linnacus gave this thermometer a sWrKlardi zed scale by arbitroril), assigning the w lues 0 and 100 to the freeling alld boiling points of water, m;pectivcl)'. Because there are 100 dcgrces betwecn the IWU calibr~tion poillts. il is called the centigrade sclol lt .

__ .. n. )""" .... TlWdo,..,,~t:>r~"'T_t'l~EtIgoInlPIIIIIp_. ~t'I"--'l..,.a-.g~~02012..,. __ ~1oo:.

The centigr.ldc sca le has been superseded by the Celsius sca le. lllc CelsiuE .>ca lc (dcnoIed in units of ~C) is simi lar to lhe centigrJde scale. However. Miler than be ing determined by two fi~ed points. the Celsius scale is delermi ned by one fi~ed reference point at which ice. liquid water. lind gasoous water are in equilibrium. This point i .~ called the tri ple poinl (see Scrtion 8.2) and is assigned the value O.oJ c. On the Celsius sca le . the boiling poinl of waler al a ~ure of I atmosphere is 99.91S"C. 11Ie si: of the degree is chosen to be the.same as on the centigr.>de sca le.

Althoogh the Celsius sca le is used wide ly througl>oul Ihe world 10000y. the numeri -cal \-J.Jucs for tbis lemperJIUre scale an: completely arbi trJl)' beca~ a liquid OIherthan water could ha\'e been chosen as a reference. It would be prl'fe ... ble to ha\"t' a lemper ... -ture scale deri\'ed directly frum pbysical principles. lllI're is such a scale. ca lled the thermodynamic t~ II1 Pl'l"lIlu rt SC'Jle or absolute tempel"llturt' ~lI le. For such a scale. the temperJ.ture is independent of the substance used in the thermometer. and the eOll-stanttl in Equation ( 1.3) is leru. ",.., gas thtrll10meltr is a practical themlon1eler with which the absolute temperJture can be measured. 11Ie thermometric property is the tem-perJture depend-ence of P for a dilute gas at cOlistant V. The gas thermometer pfln"ides the international standard for thcnnolT\Ctry at very low temper.uures. At inlermediate temperatures. the electrical resistance of pbtinum wire is Ihe standard. and at hi gher temperatures. the radiated energy emitted from glowing silver is the slandard.

How is the gas thennometcr used to measure the IhemlOdynamic tempcmturc'! r..-Icasuremc nt.~ carried out by Jacques Charles in the Ilhh century demunstrated that Ihe prcs.~ure exerted by a fi xed amou nt of gas al constant Vvaries linearly with temper .. ture on the Celsius Sl,:a le .. s shown in Figure 1.3. At the time of Boyle's experiments. lemper-aTUres below - 3OOC were not lIuainable in the laOOrJlOry. However. the P versus T data can be extrapolated to the limiting T va lue at which P --+ O.1t is found that these straight lines obtained fOf differcnt values of V intersect al a common point on the Taxis th .. t tics near - 213C.

The lbta show that at constant V. the thermometric propcny P varies with tempera-ture as

( L4)

where I is the tcmper.nure on the Celsius sc.,le. and c and dare experiment:llly obtained proponionality constants.

Figure . 3 shows tbat alllincs inte~t at a single poim. even for di fferent gast'$. This suggests a unique reference point for tempe ... ture .... ther than the two rc:ference points u.-.ed in constructing the centigr..ldc sc .. le. lbe \'alul' zen.> is given to the tempe ... . IUre .. t which P-O. Ilowever. this choice is nOi sufficient to define the temperJture scale. becaU$e the Sill' of the degree is undefined. By CQIl\"t'ntion. the si1.e of the degree on the ab.wlute temperJ.ture sc .. lc is set equal to the size of the degree on the Celsius sc .. le. With these IWO choices. the .. bsolute and Celsius tcmpeMure scll ics are rebted

1.3 THERMOMETRY 5

r' l. 1.0 ,

/" .. 1;:'G~:O--: .,'-.::;. ,',;);;-; - 200 - 100 0 100 200 300 .00

TemperalUteiCelalus

0. 1 l

0.2l

0.3l

O.H 0.5l 0.6l

FIGURE 1.3 Th(fl'\'~~u ...,(~cnedb}"5.00 x 10-1 mol of a dilut( sa~ is ~h"","n as a function of (h( tempe .... lu'" nlOasurcJ on the Cel,;u, !'Calc for differenl r.,eUr~"'T_t'l~EtIgoInlPIIIItp_. ~t'lPwD>~~~02012t'1 __ ~1oo:.

6 C H A PT E R 1 Fundl"n~l'I1a1 Concep15 of Tllefmodyn~m0::5

eo",

L---'~". rW \jilIn11W

....

fiGURE 1.5 n.e ~bsolu1~ Icmpenl1ure is lJIoJw~ un a logarithmic >eale 1og~lhcrw;lh lho

Ir iflhe data are extrJpolated to ~.cro pressure. 11 is in this limit that the gas thermom~ter provides a measure of the tllCm lQdyoamic lempcrnlUfe. Because I bar '" lif Pa. gasindependent T values are only obtaiocd below P - 0.01 bar. The aMolute temperJture is ,;hown in Figure 15 un a logarithmic scale together with assoc i ated physical p/lcroomcna.

14 EQUATIONS OF STATE AND THE IDEAL GAS LAW ~I",roscopic models in which the system is described by a liet of variables are based un C!lperience. It is particularly useful 10 formu lale an llIlI t;oo of statl'. which relales lIIe Siale variahles. Using tile abwlllle lemperJture scale. it is possible 10 obtain an equation of state for an ideal gas from e~perimellis. If lhe pressure of He is measured as a func lion of lhe volume for differelll va lues of te mperJlure. Ihe SCI of noointers.ecting hyper bolas shown in Figure 1.6 is obtained. TIle curves in th i ,~ figure can be quanti tatively fit by tile functional form

PV '" ttT (1.1) whcre T is thc absolute tcmpemture as defined by &!uation ( 1.6). allowing It 10 be deteml incd, which is fou lld 10 be directly prOpOftiOllal1O the mass of gas uscd . It is usc fulto ,;cparate out th is dependence by writing it '" tlR. where II is the number of moles of the gas. and R is a constant that is indepenlll' nt of the size of the system. TIlc resu lt is the ideal gas C

1 4 EQUATIONS QF STATE AND THE IDEAL GAS LAW 7

TABLE 1.1 UNITS OF PRESSURE AND CONVERSION FAaORS

Unilof l'~u..., Symbol l'I'umrrical V"I""

""'" '" I NmZ _ lkgm ls 1

""'-" .m I ~m ,. 101.3"..5 p~ r

8 C H A PT E R 1 Fundl"n~l'I1a1 Concep15 of Tllefmodyn~m0::5

TABLE 1 .2 THE IDEAL GAS CONSTANT, R, IN VARIOUS UNITS

R _ ~.3 1 4 J K'1 moL ' l R _ lL.3l~ Pa m ) K- I mol-I R _ 8.314 X 10-1 L bar K- I mol-I R _ 8.206 X 10-1 L aun K- I mol-I R _ 62.36 L T..,- K- I mol-I

c~lcul~tions forOlher units of pressure and \"l)lume. \'alucs of the oonslam II. with differ ent combin~tions of un its ~re ~i\"en in Table 1.2.

EXAMPLE PROBLEM 1.2 Consider the cQmpsite system. which is he ld ~t 2

1 S A BRIEf INTRODUCTION TO Rill GA)I;S 9

1be p:lni~1 pressures ~re given by

L PU, " XIf,P - 0.261 x 1.92 bar = 0.50 1 bar PN - XN.P = 0.653 X 1.92 bar = 1.25bar P", - xIC,P - 0.0860 x 1.92 bar = O.l65bar

1.5 A BRIEF INTRODUCTION TO REAL GASES J

Tlle ideal g;t.,~ law provides 3 firsl Iool; 31 the usefulness of describing a system in lerms of nllJero:scopic p:lt'~meters. HO'oIoe'er. we should ~Iso emphasize the downside of not mking the microscopic n~ture oftl>c system inlO accoonl. Fore~~mple. the ide~ 1 gas law only holds for gases:l1 low densil ies. E;o;pcrime111s show Ihal Equation (1.8) is ac,ut',Uc 10 higher values of pressure and lower values of!empet'JIure for lie Ihan for NH,l. Why is this lhe case? Real gases will be discussed in dewil in Chapler 7. However. ba::ause we need 10 lake nonidealilas behavior into account in Chapters 2 through 6. we inlro_ duce an equalion of Stale Ihal is valid 10 hiJ!her densit ie.s in th is s.ection.

An ideal gas is described by two assumplions: the aloms or molecules of an ideal ga~ do not interact with one another. and the atums or mulecules can be lreatcd a~ point masses. These a~sumptions have :l limited range of va lidity. which can be discussed usinilihe potential energy function typical for a reJ I g~s. as shown in Figure 1.7. This figure shows Ihe potential enugy I)f intet'~ction I)f (WI) gas moloxu le.s as ~ fu nction of the di stance between them. TIle intermolecular potential ClIO be divided into regions in which the potential energy is essentially zero Ir > ""'"';'''')' negalive lattrJctive inter-action) Ir"'" .. Jt .... > , > 'v .o). and positive (repulsi\'C imer.ICl;on) Ir < 'v_oj. Tlle

di ~tJnce r,,,,,,,,_ is 1101 uniquely de.finoo and depends 00 the energy of the molecule. It C:lJ1 be e.stintaled from Ihe relation IV(r,-.-" ..... )J "" kT.

As the density is increased from very low ' r,,,,.,,,~,,,. and the value of r,,,,.,i,,,,,, is sobstance dependent. Real gases will be discussed in much grealer dctail in ChJpter 7. However. at Ihis poi lll we introduce a real gas cquation of stute. because i( win be u'5Cd in (he ne~t few chaplers. The "an d e, \Vllals equullon (,r stllte IUkes both Ihe finite size of molecules and the allmctive poten-tial into account. It h~s the foml

"NT ,,1{/ p=----V - "b V2

0 .12)

This cqualion of Slate has two p:lt'~meters trot are subst:lnce &pendent and mUSI be cxperimentally delermined. Tlle p.:lrJlllClers b and u take the finite size of the molecu les aoo the strength of the al1r.lcti\'e internction imo ;[ccoum. respecti,ely. (Va lues of u and b for se lccled gases an: listed in Table 7.4.) The van & r W""ls eqll:llion of Slale is more

c.,..=,~,..=-__ ... ,,..

FIGURE 1.7 The poIen!i.] ~""'ly "f infcnoction of two "",Iocules ...... lonlS if Iohown u. function oflhdr~iun. r. The ydl ...... OIr'VC ~ the poI~nlial cllC1lY function f ..... #Jl ide;ti gas. Thc dlo!ohcd blue lim: indi

10 C H A PT E R 1 Fundl"n~l'I1a1 Concept5 of Tllefmodyn~m0::5

accur~te in calculating the rcl:!t ionship between P. V, and T for gases than the idcal gas law because" and b have been optimized usi nge~pcrimcntal re su lts. l lowe'cr. there are other more accurate equations of state that are valid o .... er:t wider range than the 'an der Waals equation. Such cr~"'T_t'l~EtIgoInlPIIII1p_. ~t'I~~~~02012t'1 __ ~1oo:.

(I mol)l x 1,370 oor dm6 moI-1 (250. Li

- 9.98 X 10-1 bar

QUESTIONS ON CONCEPTS 11

I mol X 8.3 14 X 1O- 1 Loormol- 1K- 1 X 300 K O.IOOL Imol xO.oJ87dml moi l

( I moI)l X L37000rdm6 mol- 1 (0. 100 L)2

- :!10. boIr 1l.. Noee Ih;u the result is idemicul with that for the ideal gas law for V .. - 250 1..

and thutt1le result calculated for V .. - 0.100 L deviates fmm t1le ideal g ... ~ law result. Because ",fill > p iJaJl. we coodude that the repulsive inter.IC-J tion is more importun! than the unruetivc intcr~ctioo for this specific value of molar volume and tentpcr~ture.

Vocabulary ab.>olutc te nlper~ture sca le adiabatic Boltzmann cons,ant boundary Celsius sca le

ide..l ga.~ law intensive variable

isolated sys'em kelvin

. ystCln

system variable.s ,e",perature

,empemture sca le

centigrudc !;Cale dosc:d sYStem diathennal

macroscopic scale

m:K'TOSCopic Il""ariables

mole fmction

themlal eljui librium thernlOdynamic equilibrium

,1lemuxlynamic ,emperature scale

thermometer equation of liIa,e equi librium

e.~tensive \":Iriable gas Ihermometer ideal gas COIlSlani

open sysrem

p;lrtial pressure

pl'Cssure

sUm.>Undings

Questions on Concepts Q 1.1 Real walls are never to'ally adiabatic. Order the fol -lowing wall.~ in increasing order with rcspcc, to their be ing diathermal : I-ern-th ick concrete. I-cm-thick vacuum. I-cm-thick copper. I -cm-Ih i~k cork. Q I.2 The parameter u in Ihe van der Waa ls eljuation is greater for II~ \lUln for He. What does this say about the fom' of the polemial function in Figure 1.7 for the tWo gases1 QI .J Give an example biJsed on molecule-molecule interaClions illu . trating how the total pre.~sure upon mixing tWO !"Cal ga'l's could be different from the sum of the panial pressures. QIA Can temperature be meusured din:ctl y'1 u:plain yoor answer.

QI .5 Explain how the ideal gas law can be deduced for the measuremc:n,s shown in Figures 1.3 and 1.6.

\hJn der Waals equa,ion of state wall

:a:roIh law of ,hennodynamics

Q I.6 The location of the boundar)' between the system and the ~uTT{)un din gs is a choice that mu~t be made by ,he thermo-dynamicist Consider a beaker of boiling water in an ainight room. Is lhe syslcm Ol)en or closed if you place the boundary just outside the liquid waler? [ ~the syslem ollen or dosed if you place the boundary JUSt inside lhe walls of the room? QI.1 Give an example of lWO systems that arc in equilibri. urn with respect,o only onc of two state variables. QUI At . ufficic!I\ly high tempcr~turcs.thc Van der Waals equation has the form P "" RT/ ( V", - b). Nme lhat the alfr.K:,ive part of the potentbl has no influcnce in lhis expres-sion. Justify this bch:Jvior using'he potenti~1 energy di~gmm of Figure 1.7. QUI Give an example of ''.1."0 sys,cms SCpalrdtoo by a wall thai are in thermul bill not chemkal equilibrium.

__ .. n. )""" ... . TlWdo.....~t:>r~"'T_t'l~EtIgoInlPIIIIp_. ~by~l..,.a-.g~~02012..,. __ ~1oo:.

12 C H A PT E R 1 Fundl"n~l'I1a1 Concept5 of Tllefmodyn~m0::5

Problems Problem numbers in red illdicate that the SOlution to thoc pr0b-lem is given in the SIUJtm 's SQlu,io-ls Matmul. 1'1 .1 Approximately how mally o.\ygen molecu les amve each :lCCond at the mitochondrion of an active peniOf1? The following data...-e ~\hJi lablc: Oxygen oonsumption is about -10. mL ofOl per minute per kilogr-Jm of body "reigJl1. meas-ured at T = 300. K and P - 1.0 atm. In an :.dull wi[h a body weighl of 64 kilogrJIIlS lhere are about 1.0 x 10[1 ce lls. Each cell cIMlIains abou[ 800. mitochondria. 1'1 .2 A comprcssed cylilldc r of gas contains 2.25 x 103 g of Nl gas at a l)ressuTC of 4.25 x 101 f>'J and a 1CmperalUre of 19AoC. What volume of gas has ~cn relcased in[o the at mos-phere if[hc fi nal pressure in thc cyli lldcr is 1.80 x I ():I P-J? Assume idea l behavior and that [hc gas [CmpcrJlUre is unchanged 1'1 .3 Calculatc the pres..ure exerted by Ar for a molar vol-ume of 1.04 l mol ';11 475 K using the va n der Waa ls equa-tion of statc. The vun dcr Wua ls p:trameters II alld b for Ar are 1.355 b;H dm6 11101- l and 0.0320 dm) 11101- 1 respectiv.-:ly. Is the al1 mctive or repu lsive portion Oflhc potentia l dominant under lhese conditions? 1'1.4 A sample ofprOp;Jne (CJHa) is placed in a elosed vesscllogether with an amounl ofOl lhal is 2.50 times lhe amoum rlCCded to completely ox idize the pnlp;Jne 10 COl and IllO at constam temperJture. Calculate the mole frnction of each component in the Te$ulting mixture after o~ idation assumi llg th;.tt the HJO is present :as a gas. P1.5 A gas sample is known to be a mixture of ethane and but:lJ1C. A bulb havi ng a 2 15.0'J at 19.2"C. [fthe weight of the gas in the bulb is 0.3554 g. what is the mole percent of butane in the mi .\lUre? 1'1 .6 One liter of full)' oxygen~ted blood can carry 0.20 litelli ofOl measured at T - 213 K and P ,. 1.00 atm. Cal-culate lhe number of molcs ofOl camed per li ter of blood. Hemoglobi ll. the o~ygcn trJnspon protein in blood. has four oxygen binding sites. 1101'0' many hemoglobin molecu les arc "''1uired to trJnspon the OJ in 1.0 l of fully oxygenated blood? 1'1.1 Yeast alld other organisms call com'cn glucose (C6H U0 6) to ethullol (CHlCH~H) by a pr"t: SCp;lrJting thoc two ves-sels is opened and bulh are cooled to a temper~TUre of 14.5OC. What is the fi n.:tl pressure in lhe >'esse ls? PL I3 A mi_uure of oJlygen alld hydrogen is analyzed by passing it O\'Cr hof copper o~ ide and through a dryi ng tube. Hydrogen reduces the CuO according to the reaction CuO(s) + HJ{g) - Cu(s) + HJ0(I).andoxygen TeQx idizes the copper formed accordi ng 10 Cu(s) + 1/ 2 0 !1g ) - Cu()(s). At 25C and 750. Torr. 125.0 cml of the mi.\lure yields 82.5 emJ of dry oxygen measured a[ 25"C and 750. TOfT after passage O~1!r CuO alld the dry ing agent. What is the OI'iginal composition ofThe

mi.~ture? 1'1.14 An athlete at high perfomlance inhales - 4.0 L of air a[ 1_00 aIm alld 298 K. The inha led and cxhaled air conta in 0.50% a nd 6.2% by >'olume of water. resllectivc1y. For a res-pirJtion rJte of 40. breaths per minute. how mally moles of water per mi nUTe are expe lled from the body through the lu ngs? 1' 1.15 Devise a te mperature .>cu le, abbreviated G. for whidl the mag niTUde of the idea l gas con.~ta n t is 3. 14 J G- 1 11101- 1. 1'1 . 16 Aerob ic cel ls metabolize glucose in the respirJtory system. This reaction proceeds according to the ovcrJ I1

rea~l ion

Calculate Ihc volume of o_tygen required at 511' to metabol ize 0.0 10 kg of glucose (~,A). STP refers to smndard tcmpcr-arurc and pressure. that is. T = 27J K and P '"' 1.00 atm. Assume o~)"gcn behavcs ideall y at 5TP.

__ IO n. )""" .... TlWdo.....~t:>r~"'T_t'l~EtIgoInlPIIIIIp_. ~t'I,...",l.....-.g~~02012t'1 __ ~1oo:.

" 1.17 An atlilete at high pcrfoonance inhales - 4.0 L of air at 1.0 atm and 298 K at a respiration r~te of 40. breaths pcr minute. If the exhaled and inhaled airconlJin 15.3% and 20.9% by \'olume of oxygen re5pccti\ely. how many moles of oxygen per minute are abw

14 C H APT[ R 1 Fundamen1a1 ConctP1~ 01 Thtrrt'lOdynamocs

is removed. 11le remaining gas is pure hyd~en and exens a pressure of 0.250 aIm when measured 31 !he wme \';lluI'S of T and Vas !he original mixture. What was the romposition of the original mixture in mole percent? 1'1.32 Suppose that you measured !he product PVof 1 mol of a dilu1e gas and fou nd th;n PV eo 27.54 l a1m a1 O.OO"C and 37.32 l aun~! IOO."C. Assume !h~! the ideal gas law i ~ \"~Iid. with T - (( OCl + ReicI. PubIioIlod by __ LN

CHAPTER OUTLINE

U THE I~TERNAl ENERGY 2 AND THE FIRST LAW Of THERMODY~AMICS ,~

U

>A

'"

WOO<

HEAT

HEAT (APACITY STAT fUNCTIONS AND PATH fUNCTIONS

Heat, Work, Internal Energy, Enthalpy, and the First Law

of Thermodynamics , .. EQUILIBRIUM. CHANGE,

AND REVERSIBilITY

1.7 COMPARING WORK fOR REVERSIBLE AND IRREVERStBlE PROCESSES

1.8 DETERMI~I~G.J.(J A~D INTRODUCING ENTHALPY, A NEW STAT fUNCTION

1.9 CALCULATI~G q. W, l(J, AND lH fOR PROCESSES INVOLVING IDEAL GASES

1.10 THE REVERSIBLE ADIABATIC E)(PANStON AND COMPRESSION Of AN IDEAL GAS

In this chapter, the internal energy. U, is introduced. The first law of thermodynamics relates a.U to the heat (q) and work (w) that flows across the boundary between the system and the surroundings. Other important concepts introduced include the heat capacity, the differ-ence between state and path functions, and reversible versus irre-versib le processes. The enthalpy, H, is introduced as a form of energy that can be directly measured by the heat flow in a constant pressure process. We show how a.U, :lH, q, and w can be calculated for processes involving ideal gases.

2 1 THE INTERNAL ENERGY AND THE FIRST LAW OF THERMODYNAMICS In this section. we focus on thoc chJngc in encrgy of the systcm and surroundings during ~ thermodyn~mic proceS$ $uch a$ an c.~""nsion or compression of a gas. In thermody-n~mics. we ~re intcrested in the inlem~1 energy of [he system. as opposed [0 [he energy ~ssoci~ted with the system rdative 10 J ""nicular frame of reference. For example. u spinning cont~incr of gus ha$ a kinetic energy rdative to ~ stationary observer. Howev-er. the intcrnal energy of the gas is defined rel~tive [0 ~ coordi n~le system fixed on the

con[~incr. Viewed ~t a microscopic level. the inlemal energy Can take on u number of forms such as

the kinctic ctlergy of the molecules: the potential cne'llY of the constituents of the system: for example. a crystal consist-ing of polarizable molccules will experience a change in its poIential ene'llY as an electric field is appl ied to thoc s~tcm: the internal energy ston:d in the form of molecular \"ibl1ltiOlls and rotations: and the internal energy stored in the fOflll of chemical bonds that can be released through a chemical reaction.

__ .. n. )""" .... TlWdo.....~t:>r~"'T_t'l~EtIgoInlPIIIItp_. ~by~l.....-.g~~02012..,. __ ~1oo:.

1S

16 C H A PT E R 2 Hut. WorK. Intll1'~1 Erot'O'I', {",I\aIP'!', and the fl~t law of Thelem i' ......... n in whictl wmrn-ssioo ... 'Ork i. IIc system and the surroundings are taken into ac-coum. Th is law can be formulatcd in a number of equ ivalent forms. Our initial formula-tioo of this law is stated as follows:

The internal energy. U. of an isolated i)stem is constant.

This (orm of the first law looI.::s unimeresting. because it suggests that nothing happe ns in an isolated system. How can the fi rs{ law te ll us an)'lhi ng about themlodynamic prOCCMeli such as chemical react ions? When changes in U occur in a system in conmct with its SUTTO\l nd ings. t..VIOItJI is given by

t..V,,,,,,, - t1V., ., .. + t1V ... ,,,_ .. , . - 0 Therefore, the fin;1 law becomes

(2 .1 )

(2.2)

Fur uny decrease of V,),,,,,., Uf"".)".J"'~f must increase b)' exact ly the same amount. For example. if a gas (t he system) is cOr~"'T_t'l~EngoInlPIIIIIp_. ~t'I~~~~02012t'1 __ ~1oIo:.

Work is trJllsilOry in tll~t it only appeaTIS during a change in st;lte of the system and surroundings. Only eroergy. and not woO::. is assoc iated witll tile in itial and fi nal states of the systeilis. The roe! effect ofworl; is 10 ellange U of the system and surroundings in acoord:lnce with the first law. If the on ly change in the surroundings is that a mass has been rolised or lowered. wo .... has flowed between the system and the surroundings. Tlte quantity of worl; can be calcu lated from the change in potential energy of!he mass. AE,.,..., ... , - mgll. when: g is the gr.lvitatioool accelcr .. llion and I. is the change in the heigh! of the mass. m. The sign COllvention for ","QrI( is as follows. Irthe height of the mass in the mrround ings is lowered. tv is positive: if the height is r.t.ised. tv is roegative. In shun. tv > 0 if AU > O. It is common usage to say that if w is positive. wurl;: is done OIl the sys-tem b)' the sUrToondings. If tv is negative. work is done by the system 00 the s01Tl)und ings. How mucll \VOrl: is done in the process shown in Figure 2. 1? Using a definition from

physics. work is done whe n an object subjcct 10 a force. fl. is moved through a distance. dl. given by

111 '" JF'dl (2.4) Us ing Ille dcflllilion of pressure as the force pe r onit area. tile work done in mov ing Ille mas.~ is given by

(loS)

The minus sign appears becaose of oor sign convemioo for work . Nore that the pn:,;sore that appears in this exJl'f"C$sion is the exte rnal Pf"$sure. P o.~""'" which need not ('(Iual the system pressure. P.

An e.tample of allOlhcr important ki nd ofworl;. namely. electrical ",uri;. is hO\lo'n in Figure 2.2 in which the content of the cyl inder is the system. Electrical current flows through a conducti"e aqueous solutioo and w.lIer undergoes elecno lysis H.> produce H l and O~ gas. The current is produced by a gcroerator. like that used to power a light on a bicycle through the ItlIXhanical worl; of pedaling. As current flows. the m:t5s that Urivcs the generJtor is lowered. In this casco lhe surroundings do the electrical 100'0 .... on the sys-lem. As a resu lt . some of the liqu id W'Jte r is trolnsformed to Hz and 02' From electrostat ics. Ihe worl; done in transporting a charge. Q. through an electrical potential difference. ,po is

It., ... " .... ., Q (2.6) For u conswnt CUrTent. I. that flows for a time. I. Q = II. Therefore.

(2 .7)

TIle system ulso docs work on the sorroundings through the incre~se in the vo lume of the ga~ phase ut cunMu nt extcrnu l pressore. n,e tutul work done is

II! " Wp _y + lV.I ... ,,,,,,I " l t - J p"""",/tiV .. II - P""nw/ ! dV = 11 - p.x""",/(V! - V. l (2.8)

Other forms of work include the work of expanding a surface. such us a wap bub-ble. ugainst the surface tension. Table 2. 1 shows It.c e~prcssioos for work for four dif-ferent ca~s. F..uch of these diffcrent types of ",urk poses a requirement on the wull s SCfl"rating the system and surroundings. To be able 10 curry out the first three types of work. the wa lls must be mowble. whereas for electrical work. they must be conduct i, .... Scver .. 1 e.~amples of wori; cakulalions are gi'"Cn in Ellample Problem 2. 1.

22 WORK 17

Initial statB

Final state

FIGURE 2.2 Cum:nt prodlk"Cd b)' a generalor i. u!Oed to ekctrulY1.e ""al~r and lhe",b,. do wort 0

18 C H A PT E R 2 HUI. WorK. Inlll1'~1 Erot'O'I', {",I\aIP'!', and the fl~t law of The

2.3 HEAT Ueat l is defined in thermodynamics as 11M: quantity of energy that flows across the boundary between the system ~nd sUJTOUndings because of a temper~ture differencc be-tween the system and the SlIrruunding.s. Heal always flows spontaneously from regions of high l('mper.uure 10 regions of low temperature. lust as for wort, sc\,cr:tl imporlam char.ICleristiC5 of he;u ure of imporlaocc:

Heat is tr.msitory. in that it (111)' appears during a change in state of the system and surroundings. Only energy. and IlOf Ileal. is associated with the initial alld final Slates of .he ~ySlCm and the su rroundings. The net effect of heal is to change tile internal ellCrgy of the system and surround-ings in accordaroc:e with the first law. If the only change in the surroundings is a change in temperJIIIn: of a reservoir, heat has flowed between system and surmuRd-;ngs. TIle quantity of heal that has flowed is directly proponionallO the ch:mge in tcmpcrJlUre of lhe rese rvoir. The sign eonvenlion for heal is as follows. If lhe lemperalure of lhe urroundings is lowered. q is posilive: if il is rJised. q is negalive. It is common usage 10 say lhul if q is l)Qsilive. heul is WilhdrJwn from lhe surrounding , and deposiled in lhe sys-lCI11. If II is negative. heal is wilhdrJwn from lhe system und deposiled in lhe surrounding.~. ))clining lhe .~u rroundings as the reSl of lhe universe is impractical. becuuse it is not

realislic to !.Cureh through the whole universe 10!.Ce if a mass has been raised or lowered and iflhe tcmperJture of a reservoi r has changed. Experience shows lhal in gencral only tho!.C part.~ of lhe uni,'cI"SC close to lhe systcm inter Jet with the system. Experiments can be constructed to ensure lhal this is lhe case. as shown in Figure 2.3. Imagine that you are intcreJ;ted in an cXOIherrn ic chemical reaction that is carried OUI in a rigid sealed con",iner with di athemlal walls. You define lhe system as consisling solely of the reac-lant and product mixture . The vessel contain ing lhe systcm is immersed in an inner waler oo.lh separJted from an OUter wale r oo.th by a comainer with rigid di:uhennal walls. During the reaction. heat nows OlU of the system (q < 0). and the temperJlUre of the inner Water oo.th increases to Tlo Using an electrical heater. llle lempcrJlure of tile outer water oo.th is ifl("rcascd so that at all times. T""", = T_ ... Because of this condi-lion. no heat flows across the boundary between the two waler oo.ths.. and because lhe con",iner enclosing the inncr water oo.th is rigid. no wort flows aclQSS [his boundary. Thcrcforc. ,),U - q + IV - 0 + 0 - 0 lor ll,e composite S.,slem made up of the inner water oo.th and e"erything within it. Therefore. this composile system is an isolated 5)'5-tem that does n-oc inter.ICt with the rest of the universe. Todetermine q and 10 fOf lhe re-actant and product mixluTC. we need to examine only the composite system and can disregard the rest of the uni'eI"SC.

To emphasize the distinction between q and 1V. and the relationship between q. w. and ~U, we disco.'s the IWO processes shown in Figure 2A. They are carried out in an isolated ystcm. divided illlo two suOsystems. [ and II. [n both cases. system J consists solcly of the liquid in the beaker. and eveT)'lhing else including the rigid adiaootic wall s is in system 11. We a$~ume that the tcmper~ture of the liquid is well below its boiling poinl so thaI its vapor pressure is IIcgligibly small. TI,is cnsures that no liquid is vapor_ ized in the process. and both systems can be viewed as closed. We also assume lhul lhe change in tempemture of system I is wI)' small. System 11 can be viewed as lhe sur-

rounding.~ for system ' and vice versa.

' It ... " pl.....-.g~~02012..,. __ Eo1aIia>.Ioc.

20 C H A PT E R 2 HUI. WorK. Inlll1'~1 Erot'O'I', {",I\aIP'!', and the fl~t law of The O. so we ,oncludc that qr > O. The iocreasc: in UI is due to heat flow from system" to system I calL'iCd by the diff,rence between t~ tempcr~ture ofthl: heating filame nt and the liquid and nOi by the elcctrical ",uri.: done on the fi lament. NOie that because .!lUt + .!lUn = O. the hCOlI now from system 11 to system (can be calculatL-d from the electrical wor\.:: done entire' Iy with in system IL qr = - wrr = ' ,pI. I.,aUrr < O,onsistcnt with

1,00 X 101 S 10 effL-.:lthe IrJnsfonnation. 11Je densities of liqu id and gaseOUE waler under tllcse conditions are 997 and 0.590 kg m J. respectively. II. It is often useful to repl:tee a real proc~s by a model that e~hibits the im-

po

22 C H A 1'1 E R 2 Hut. WorK. tnlll1'~1 Erot'O'I'. {",I\aIP'!', and the f,~t law of The< varialion ()f Cp' .. with t~",p.:r~luru is iOOwn forCi I'

where C is in the SI unil of J K I. It is un e~tensi\'e quantity thaI. for c:c.~mp1c. doubles ~s the mass of Ille system is doubled. Often. tile IllQlar heat c~pacity. Coo is u~d in cal culations. Il is an intensive quantity with the uni ts of J K - I mol I. E~pcrimcntall)'. the heat capacity of fluids is measured by immer.;ing a healing ooil in Ille fluid ~nd cqU~Iing Ille electrical wot1; done 0Il11le coi l with Ille Ileal flow inlO the sample. For solids. the heating coil is wrapped around the solid. In both ca~s. Ille upcrimcntal results most be corrected for heat losses 10 lhe sUlTOU ndings. llJc signirlCanee of the notalion 4q for an iocremenTal amount of heal is aplained in lhe ne.~1 .st ion.

The valoe of the heat capacily depends on the e~perimcntal conditions under which il is determined. The most common conditions are constant wliumc or constant pres-sure. for which the hellt capacity is denoted C v and C,.. respe

2 5 STATE fUNCTIONS AND PATH fUNCTIONS 23

Solution T .... ./

qp " - J C;:''''''' ..... '''( T)dT = - C';""" ..... '" il.T T_,., L -- I.SOkg X.U 8Jg I K I XI -'.2K= - 89.lkJ

How ;lfe C,. and Cy related for a gas? Consider the processes shown in Figure 2.6 in whith u Ii\ed amount of heat flows from the surroulldings imo a gas. In the tonst~m pressure procc"ss. the ga5 e~pands a5 ilS temper-ature increases. Therefore. the syStem does work on (he surroo lldings. As U consequence. nOi all the heat flow into the system tan be used to incre= il.U . No such work occurs for the tOfrl:sponding constant \"Qlume process. and all Ihe heal flow imo Ihe syslem can be used I" increa.ere-fore. ar,. < dr" for the same heat flow 4q. For this reason. C,. > Cv for gases.

The Same argulllcnt allplies 10 Iir....-y"'T_t'l"""""-EtIgoInlPMp_. ~t'lI'NrDl~~~02012..,. __ Eo1aIia\.Ioo:.

24 C H A PT E R 2 HUI. WorK. inlll1'~1 Erot'O'I'. {",I\aIP'!', and the fl~t law of TheO:"1-bly. The gas in Ihe iniHal .lalC V" T, i. compressed 10 an inlenn .. 'di.iC i-l~IC, "'Ilereby thr; tempr:('~turt i!!Creascslu lhe , ... 1"" T2. II is then brooShl inlo romillC1 ",ith l lhe"",,! ",scl'\'ui, I T). 1c:adins. 10 a funhe,risc in lemperature . The fillOl' state i. V2 T) . Th.;: rnhanical st. allow the s)'i-lcm volume to ~ ooly VI or V~.

nO! the path I~l:en 10 reach the ~tate. This property Can be expressed in a malhem~tieal form. Any state funetlOll. for example. U. must satisfy the equat ion

I

.1U=< jdU =< Ur-Ui (I. U ) ,

whe~; aBd/dellQle the initial aBd final Slales. This equation Mlltes that in ordu for tJ,U to dcpe Bd only on the initial and final SlaltS charJoCleri1.ed hen: by; aBdf. the \'alllC of tIM: intcgrJI must be indepeBdent oflhe path. If this is the case. U can be expressed as an infinitesilTkll quantity. dUo thai when integrJted. depeBds onl)' on the initial and final Slates. TlJc quantity dU is called an t:

26 EQUIlIlRIUM, CHANG. AND RMRSIBLITY 25

und p.,,,,rtti>J is different for e~ch value of the m'lSs. or for e~ch path. whereas.l V is con-SWnt. lherefon:. 111 is also different for exh path~ we c~n choose one path from VI. TI to V1 TJ und a different path front Vl. TJ bad.: to VI . TI . Because the wori< is dif-fen:nt along thc:sc paths. the c)'c lic inlcgr.d of wori< is not equal to rero. lhen:fon:. w is I'I1.II a SlJte function .

Using the firstl~w to calculate q for each ofthc paths. we obtain the result

(2. 18) BecallSI witll ~ ,",Iunl(' vi 4.(lI.;oru ~'n as blldcltf"C$Thlllli(inlheP V Tl ur_ f...., . The thin! Cur.e COlTl'~ tu paTh bcTwffn "" inili. 1 ~"e i and a til\lll ~(ef.hat il neilher a roOSlanllempefl' TUre nor Mcunil~n. wlu""" paTh

__ .. n. )'"" .... TlWdo.....~t:>r~"'T_t'l~EtIgoInlPIIIItp_. ~by~l....-.g~~02012..,. __ ~1oo:.

26 C H A PT E R 2 HUI. WorK. Inlll1'~1 Erot'O'I'. {",I\aIP'!', and the fl~t law of Thec~ or exactly I kg each an: ron-n.cc1cd by a win: ofuw mal$ running o'cr. friocti""less pUlley. ~ $)'~Iem;$ in mhanical equilibrium and the mOl$$Q *" SoI~tion:.ory.

Nellt. consider a pruces~ in which the ~yslem change . from an initial Stale char.lCter-iud by p;. V;. and T{ to a final ~tale ehamctcriud by Pf V,. and T, as shown in Figure 2.8. If the r.lle of change of the macroscopic variables is n.egligibly small. thoc system passel through a su=ion of states of internal equil ibrium as it goes from the initial to the final stale . Such a process is callcd a quasislallc P I'OCfS~. and internal equilibrium is maintained in a quasi-static process. If the r .. te of change is sufficient ly large. Ihe rates of diffusion and intemJQlC(:ular col lisions may IlOl be high enough to maintain the system in a Sl.:lte of imernal equilibrium. 111cnnodynamic calculations for such a process an: valid only if it is mean ingful to assign a single \h~lue of thoc maclmCopic \'llriables P. V. T. and coocemratioo to the systc:m undergoing change. The same con~id

cr~tions hold for the sUm.>llndings_ We only consillcr quasi-static processes in this le};t as SoIate variables ~uch as P and Tare only meaningfu l in Ihis limit.

We now \'i~uatize a proc~ in which the s)~tem unOergoes a major cholnge in terms of a directed path consisting of a sequence of quasi-Solatic processes and distinguish be tween two wry importam classes of quasi,slatic processes. namely. reversible and irre-versible processes. It is useful to consider the meehani~'al s)'steln shown in Figure 2.9 wilen discu~ing reversible and im:versible processes. Because the two masS(.s h~\'C the Same value, the net force acting on c~ch end of the wire is 1.cro, and the masscs will not move. If an additional mass is pl~ced on either of the two ma:ISCS. the system is no longe' in mechunical equilibrium and the masses will move. In the limit that tile incre-mental mass approochC5 1.ero, the velocity at which the initiul mu~se.~ move approo~hc~ zero. In this case, one refers to the process us be ing re\'ersl hlc, meaning thm the di rec-tion of the process CJn be reversed by placing the infinites imal mass on the other side of the pulley,

Reversibility in a chemical system can ~ illustr~tcd by a system consiMing of li4Uid water in e4uilibrium with gaseous water that is surrounded by a lherm~1 reservoir. The system and surroundings are boI:h at temper Jilin: T. An infinitC5imully small increase in Trcsults in a sma ll inerea."" of the amount of water in the gaseous phase, and a small de-crease in the liquid phase. An equally small decrease in the tcmpcr~ture has the oppo-... ite effect. Therefore, fluctuations in T gi\1: ri.l.e 10 corresponding fluctuat ions in the composition of tbe system. If an infinitesimal opJlOl>ing cholnge in tbe \':Iriablc 11t:11 drives the process (lI'mperJlUre in this case) causes a re''crSa1 in tIM: di rection of the process. the process is reversible .

If an infinite1limal change in tIM: driving \"riable does IlOl change the direction of the process, one says that the process is Irn.'nrslblt . Fur c.(amplt:. if a large stepwise tem-perature increase: is indueed in the system using a heat pulse. lhe ~mount of W'Jtcr in the gas phaM: increases abrupl ly. In this case:. the composition of the S)'Stem cannot be re-turned to it ... inilial \'lllue by an infinile1limal temper~tun: deerease. This relationship is characteristic of an im:versible process. Although any process that takes place at a rJpid

r~te in the real world is im:\'ersible, real processes ean appro.xh reversibility in the ar-propriale limit. For e.tample, a slow increase in the electrical potentia l in an electro-chemicaL cell Can convert reactant~ to products in a nearly re\1:rsibLe III'OCC5S.

2 7 COMPARING WORK FOR REVERSIBLE AND IRREVERSIBLE PROCESSES We concluded in Sect ion 2.5 that w is n"t a State function and thutthc work associated with a proccss is puth dependent. This .tatrment can be put on a qUamitalive footing by comparing the work associated with the reversible and irreversible c~punsion and COlll-

pres~ion of an ide:!l g'b. This process is di scussed neXt and illustr~ted in Figure 2. 10. Omsider the following im:versibLe process, meaning Ihat the inlernal and external

pressures are nOi equ~1. A quantity of an ide~l gas is confined in ~ cylinder wi th ~ weightless 1lIQ\'llble piston. 111c w.tlls of thoc system are diathemlal, allowing Ile~t to flow between the sy~tem ~nd surroundings. 111crefore, the process is iSOthocmlul at the temperature of the surroundings, T. The ystem is initiully defined by the \':Iriablcs T.

__ oI:> n. )""" .... TtWIIO'-~""~"'T_Io\I~EtrgoI"""ptJIIp-. ~1o\I"--'l...-.g~~ Cl 2012~ __ EdI.o::Mi

2 7 COMPARING WORK fOR REVERSIBLE AND IRREVERSIILE PROCESSES 27

PI. and VI' TIle position of tl>c piSlOn is Iktemlined by P~'''''''''I = PI. whicll can be cll~nged by :uiding Of removi ng weights frum the pist(Hl. Bec~use the weights are moved horironlally. no wor~ is done in adding or removi ng them. 'The gas is first u-panded at CQIIstant tcmper~lU re by decreasing P .. "' .... ~brupl:ly to the v.:tluc Pl ( ..... eights ure renlO\cd). where Pl < PI ' A sufficient amoum of heat nOWlii into the sySlem thfQl,lgh lhe dialhcrnl:l l wa lls to ~ecp the temper~ture at !he C(Hlstam \'alue T, The sys-tem is now in the Siale defined by T. P2. and V2. where V2 > VI' "The system is then re-turned to its original statc in an isotherm al p roass by iocreasing P D."ntI>/ ubrupll)' to its original value PI (weights are added), Heal nows out of the syslem into the sur-rwndings in this slep. The s),stem has been restOfl'd to its original stale and. because this is a C)'clic process. AU = O. Are c work for each individual step:

It1OOJ = 2 - Pu",~., AV, = 1t'~_.,_ + tV"""'P"m... ,

- - P.!(V2 - VI ) - PI(VI - Vl) " " , - -( p.! - P') )( (Vl - V.l > 0 because Pl < PI and V1 > VI (2.20) FIGURE 2.10

The re l mion~hip between I' and V for the process under considerJtion is shown gmphi-ca ll )' in Figure 2. 10. in what is call ed an Indicp tor diagra m. An indicatOf diagrJm is u>lCfu l becau.~ the wor ~ done in the e~JXI n s ion and cuntrJct ion steps can be e\'a lUaled from the appropriate area in the figure. which is equivalent to evaluat ing the integral 1V - - J p"",,,,,,. dV, NOle that the work dOlle in the e~pallsion is negative because AV > O. and that done in the compression is positi\"C because AV < O. Because P1 < PI. lile magnitude of the worl.: done in the eompression process is more Ihan Ihal done in Ihe e~JXInsion process and II',,,, > O. What can (Hle say aboot 'I-.I? TIle first

I~w st~tes that bec~use ilU = '1_ + Il'_. = 0, qlOUJl < O. The SlIme C)'clic~1 process is carried OIl! in a reve~ible C)'cle. A nccc.\S.l1)' condition

for reversibi lity is Ihat I' - Pt> .. ~ at e"ery Slep of the cycle. Th is means thai I' changes during the e~p'l1Ision ~nd compressioll steps. The work :&ssociated with the e~JXIllsion is

I I I 'V V, Il'r..,..... .... - - p.~"'-ttlV - - PtlV - -"RT If = - " RT InY; (2.2 1) This worl.: is sho\I.n schem:llicall), as the red area in the indicator diagr.un of Figure 2. 11 ,

If this process is re\'ersed and the compressioo worl.: is cukulated. the following \C-su lt is obtained:

(2.22)

It is seen that the magnitudes Qf the worl.: in the forward and reverse processes arc equal. TIle tota l work done in this cycl ical process is given by

V1 VI IV .. "IV,x..,,,,1too + we""",,,,,""" = - ,.RT IWi/;" - " RT InY; --IIRT ln~ + I/RTln ~=O

VI VI (2.23)

TIlercforc. the worl.: done in 11 re\"c~ i b le i'lQlhcrmal cycle is zero. Becau.~ AU - II + 111 i.. a . tate function. 'I - 111 - 0 for this reversible iSOl:llcrm~1 process, L.ooJ.inS at the heights of the weights in the surrou ndi ngs at tl>c end of the process. we find Ihal lhey are the sallie us ut the beginning of the proce..s. To compare Ihe won: for reversible ~nd irreversible processes, the state ,-ariablu need to be g.iwn speci fic \-aIU('s as is done in EJo:lImple Problem 2.4.

The work for each Mep and the 100ai wort< cun b.: OMailep. II' i, given by Ihe 100ul area in red and yellow: for the e~l"'nsi()n ~ICp. '" i~ gi,,,n by Ihe ",d a",a. The IIm>WS indkale the direclion of change in V in Ihe IWO sleps. The sign of", is OJlIl".'Sile for IheSl:' ",", processes. The IOI~J work in Ihe '}'ck i! the )'ellow

to 15 20 25 "'-

FIGURE 2.11 IndklMor di~g" .. n for" ...,'"rsible process Unlike .. ig....., 2.10. tl>(:.rcas under lhe P- VOl .... es;u-e lhe &;orne in the forwanl :uwJ I"Co'(fK directiuns.

__ .IOn.. ) .. """'. TIinIo.....~t:>r~"'T_t'l""""'-Et1goInlPMp_. ~t'lI'NrD1~~~02012""' __ Eo1aIia\,Iolo:.

28 C H A PT E R 2 Hut. WorK. tnlll1'~1 Erot'O'I'. {",I\aIP'!', and the f,~t law of The

2 7 COMPARING WORK fOR REVERSIBLE AND IRREVERSIILE PROCESSES 29

Icf Pa w",,,,,, ... ,.= - P., ........ AV = - 11.Obar Xt;;;;:""" x ( 10.2L - 4.S0L)

10 l ml Icf Pa 1O-1m1 X - L- - .. SO bar x --;;- x (25 OL - 10 .2 L) x - L-

U = - 12.') X loJ J 1l1c nmgnilUde orille woO.: is greater for tlIc two-step process tllan for the single-step process. but less than that for the n:\"CTSible procc:o;s. J Example Problem 2.4 shows that the maj!.nitude of w for the irreversible eJ(pansion

is less tllan that for the Te\ersiblc exp,msion_ but also sUj!.ge.st5 that the magnitude of til for a multistep eJ(pall.'lion process increases with the number ... f steps. This is ir>(]eed the

ca~. as shown in Figure 2. 12. Imagine that the number of steps" increases indefinitely. As " increases. the pressure difference p,~", ..... - P for each individual step decreases. In the limit that " ..... 00 . the pressure difference P,~",,,,,,, - P ----> O. and the total urea of the rectangles in the indicator diaj!.rJ.m approaches the area under the reversible curve. In this limit . too irre"ersible pn.x.:ess becomes re"ersible and the value of the wor~ equuls thul of Ihe revers ible process.

By con trust_ Ihe tlwgnitudc of the irreversible compression wor~ exceeds Ihat oflhe revers ible pn.x.:ess for finile vuluc.o; of /I und becomes equal to thai of the reversible proce.'s a.~ /I ..... 00. The difference belween Ihe expansion and comptes,ion pnJCC"~S

re>u l l.~ from Ihe requireille n! that P,~",,,,,,, < P ullhe heg inning or each expansion slep. whcrca.~ p, .......... > P allhe beginning of euch compression slep.

\ FIGURE 2.12 Thc wl)'cllow ar~~) in lhe lop p

30 C H A PT E R 2 Hut. WorK. tntll1'~1 Erot'O'I'. {",I\aIP'!', and the f,~t law of Ther~"'T_t'l~EtrgoInlPIII/tp_. ~t'I~~~~02012t'1 __ ~1oIo:.

(2.27)

29 CAlCULATlNG q, w, lU, AND lH FOR PROCESSES INVOLVING iOEAl GA,ES 31

~nd inlegrJlc Ihis e~prcssion between Ihe initial ~nd fin:.l st~leS: f !,w - v, - Vi = ! .(fqp - ! PdV = lip - P(Vf - V; ) ,

(2.28) In order to evaluate thoc integral in\'ol~'ing P. we must kllOW p(V). which in this case is

~ = PI = P where P is constant. n.e integral o,er.(f lip has a unique: ~'al ue: because the path ( P = constant) is defined . RcalT:lnging the lasl equat1Qn. "'e obtain

(2.29) Because P. V. and V are all statc fUBetions. U + PV is a swte fUBelion. This new swte function i .~ ca lled t nthalp)' and is givcn the symbol H. fA more rigorous demonStrJl ion that H is a St3tc funct ion C3n be givcn by innH;:ing the first law for U 3nd applyi ng the criterion of Equation 13.5) to the produci PV.l

H _ U+ PV (UO)

As is the ca.'iC for U, /I has Ihe unils of energy. and it is an extensive property. As shown in Equution (2.2UITOundings 31 conswnl pressure:

(2 .31)

This equal ion is lhe conSlanl pressure ana logue of F~lLllion (2.26). Because chemica l re ~Cli,)Il $ an: "Iuch mOl"C frequemly curried oot ~t Ct)nstant P than Ct)nStanl V. the energy change measured c.1perimcnmlly by monitoring the heat flow is fllf r..thcr th~n flU .

2 9 CALCULATING q, W , .l.U, AND .l.H fOR PROCESSES INVOLVING IDEAL GASES In this section. we discuss how flU and flH . as well as q and 1(!, can be caJcula1cd frum Ihe inilial und fi nal stme \'ariabh:s if the path between Ihe inilial and final state is kllOWn. n.e problems at lhe end oflhis c .... pler ask )'00 10 calculate q. 111. flU. and flff for s im-ple and multistep processes. Because an equatioo of state is often needed 10 carry OUI soch taltulations. the syslcm will generally be an ideal gas. Using an ideal gas as a sur-rogate for more comple.>: syslCnls has the significanf ad~'amage Ihal lhe mathematics is simplified. allowing one to corn.:cntr,ltc on the process rather Ihan Ihe manipu lalion of equations and the eva lual ion of intcgrJls.

What docs one nccd 10 know 10 cakulale flU? The following discussion is I1'striet-ed 10 processes Chat do not involve ,:hcmical Il'actions or c .... nes in phase. Because U

i .~ a state funclion. flU is independe nt of the P'Jth between the initial and final Statcs. To describe a fixed a rolount of an idea l gas (i.e .. " is con SlanT). the values of two oflh.., vari -ables P. V. and T muSI be known. Is Ihis also lrue for flU for processes involving ideal gases? To answer this question. consider Ihe expansion of an ideal gas from an iniliul stalc VI . TI toa fi nal sta le Vl Tl . We firsl assume Ihal U is a funclion of both V and T. [s Ihis assumpt ion correct? fl ecuuse U does nOI depend on V for an ideal a5. dU mUSI be a fu ncl ion of T only for an ideal gas. flU = flU(T}.

We alsu know that for a ICl11pcr~lurc rJngc over which Cv is conSI;ml.

flU - 'IV - Cv(Tf - 7; ) (2.32)

_ Y the ~emlstry " L place-

2.1 Heal Capac'ty Is Ihi s equut1Qn on ly valid for constunl Vl Because U is a function of Tooly for an ideal gas. Equ~lion (2.32 ) is also ''ll iid for processes involving idcal gases in which V is not conslan!. n.erefore. if one knows C", TI. and Tl . flU can be calculaled . regardless of lhe P'Jlh belween lhe inltb l and fi nal Slates.

__ .. n. ) """ .... TtWII o.-~""~"'T_b\l~EtIgoI""" ptJllpRtid. ~b\I "--' l....-.g~~02012~ __ Educali

32 C H A PT E R 2 Hut. WorK. Intll1'~1 ErotrO'l', {",I\aIP'!', and the fl~t law of The

29 CAlCULATlNG q, w, lU. AND lH FOR PROCESSES INVOLVING iOEAl GA,ES 33

Solution We begin by a.l,king whelher we can evaluate q. U:. tlU. or tlH for any of tile segments without ~ny culculations. Because the p;llh between SUItes 1 and J is isothennal. tlU and tlH ure 1.CO) for Ihis segment. lherefore. from the firs! law. lIl-t .. - 1(1)- 1. For this reason. we onl y need 10 calculate one of tllesc [\\10 qll:lnlitics. Because tlV .. 0 along the p;lth between states 2: and 3.1i12_J .. O. lherefore. tlu,_J .. II:- J' Again ..... e only need to calculate one ofthcse two qll:lnlities. Because the lOcal process is cyclic. the chanSe in an)' state fUllCliQn is zero. lhel\'fore. tlU .. tl ll .. 0 for the cycle. no matter which direction is dKr sen. We now deul wilh each seg.ment individuall y.

Segment 1 ..... 2 Tltc \'il lues of n. Pt. VI' P1 and VI are known. Tltcrefore. TI and T1 can be calculated usi ng the ideal gas law. We usc these tempe",lures 10 calculate tlU

a.~ follows:

"Cy .. tlUI_ 1 .. nCy . .. (Tl - Ttl - ~(PIVI - PIV!l

20.79 J mol- I K- I

O.083 14Loor K- l nlOl l

X ( 16.6bar X 2:5.0 L - 16.6 bar X I .OO L) - 99.6 kJ

lhe process ta kes place al constant pres.~ure. So IrfNm 1

1V " -p.~ ... .-I(Vl - VI ) " - 16 .6bar X ~ X ( 25.0 X 1O- 3mJ - 1.00 X 10-JmJ)

.. - 39.8 kJ

Using the firs! I!lW

II .. tlU - 111 - 99.6 kJ + 39.8 kJ = 139A kJ We neJlt calcubte Tl . and lhen :l. HI_ 2,

T, =!l!l .. 16.6tx.r X25.0L 2.00x lol K nR 2.50 mol X 0.0831 4 L barK I mot I

We ne~1 calculate T) .. TI

TI .. P1VI .. 16.6 bar X 1.00 L liN 2.50 mol X 0 .08J 14 L bar mol I K I = 79.9 K

tl lll _ : .. :l.Vl_2 + tl ( PV) .. tl UI_ 1 + IIN(T! - TI) >= r~"'T_t'l~EtIgoInlPIIIIIp_. ~t'I~l.....-.g~~02012t'1 __ ~1oo:.

34 C H A PT E R 2 HUI. WorK. Intll1'~1 Erot'O'I'. {",I\a IP'!', and the fl~t law of The, For thi s segment. d.U}-1 = 0 and !l.HJ_ 1 - 0 as noted earlier and lIiJ-1 == - qJ-I' Because this is a reversible isotlle:rmal , ompress ion.

1111_1 = -IIRTln ~ = - 2.50 mol x 8.314Jmol IK I X 7'.1.'.I K V,

x 1,~I~.OO,,-,X,-,:I0~-'~m~' 25.0 x 10 1m)

= 5.35 kJ

1lIe results for the indi\'iduaJ segmems and for the cyd e in the indicated direction are given in tile: following Ulble. If the q'CIe is tr.m:rsed in the re\'erse fashion. the rTI:lgnitudes of all quantiti~ in the table remain the Slllm:. but all signs ,hange.

I'alh g (kJ J to (l;:J ) !l.U (kJ ) ~/ (kJ ) 1 _ 2 13'.1.4 3'.1.8 '/'J.6 139.4 2 - 3 -

l to THE REVERSIBU AD IABATIC EXPAN~ON AND CO MPRESSION OF AN IDEAl GAS 35

2.10 THE REVERSIBLE ADIABATIC EXPANSION AND COMPRESSION OF AN IDEAL GAS ll>e adiab;l1ic expansion and compressioo of gases is an impor1:ul1 meleorological process, For e.tamplc , the cooling of a cloud as it IDO\"C.'l UpW'dnJ in Ihe almosphere can be modeled as an ad iabal ic process because the heal transfer belWeen the cloud and lhe rest of the aUllosphere is slow on the limescale of its upW'.m:I motion_

Consi:pansion of an ideal gas. Because q ., O. the first law takes the form

iiU - 111 or Cl'dT " -P,~ ..... dV For a reversible adiabatic process. P - P.~ .. nouI. and

dV

36 C H ... PTE R 2 He.l, W",k, Ime" .... 1 Ene,9'\', Enthalpy. iI

Q2.!! Whal is wrong wilh Ihe following Slalemenl~ Buur.u lite ... f'IN,wlllllf'1f "(JUst' ~r(Jrf'1f u 10/ u/III'U/. II" Il'mpnmurt Ifidn ,/u/lllluch .... hl'n ,I,f' /UNfUCI' /ui/l'd. Rewrile lhe .'lenlence 10 COO\1:)' Ihe same infonnati.,n rorTC'tl)'. Q2. 12 Explain how a nUiss of water in the 5urmundings can be used 10 dclenni!le q for 3 process. Cakulale q if the

lemper~m~ of a I.()().kg water balh in Ihe surmuOOings in ,~ascs by 1.25OC. Assume that the surmuOOings aoe at a con-Slant pre:ssure:.

Problems Problem numbers in nod indicate tMlthe svlution to the prob lem is given in the Smi/,'ms SulmiUll1 M",,,,,,I.

1'2.1 -1.25 molc.~ of an ideal gas with Cr ... - 3R/2 initially al a tcmper~IU rC TI - 325 K am.l P; - 1.00 bar is enclosed in an ad iabatic piston anJ cy linJer asscmbly. The gas is com presscd by placi ng a 575kg mas~ on the piston of diameler 20.0 em. Calculale the work done in this process and the dis tance that Ihe piston IrJvcls. Assume that the mass of the pis ton is negligible.

1'2.2 TIle ICmpcrulUre of 2.50 moles of an ideal gas increas es from 13.50 to SS.IOC as the g3S is cOlnpressed adiabalical Iy. Calculate q. tv. dUo aOO ilH for this process assumi ng th3t Cr ... - 3N/ 2. 1'2.3 1.65 moles of an ide31 gas. for which Cr ... = 38/ 2. is subjecled!O two succCSli i--c changes in Sl:Ite: (I) From 39.O"C aOO 100. X 10J P:t. lhe gas is cxlf:ndcd iSOlhennally againSl: a constant prc:ssurc: of 16.5 X 10 Pa to twice lhe ini li31 \olume. (2) AI Ihe coo of the pre\"il)lls process. the gas is cooled at conSlam ,olume from 39.O"C 10 - 2S.ifC. Calculale q. w. t1U. aOO ilH for cadi of lhe Stages. AI.\.O calculate q. 1t'. ilU. aOO 1111 for the COmplele pnxess.

1'2.4 A hi ker c3ughl in a thunderstorm 1000S heat when her clOIliing becOlnes weI. She is P"Cking emergency r.ltions which if completely IllClabolizcJ will release 30. kJ of heat per gmm of r~tiOIlS con.'.ulllcd . How much r~lions must Ihe hiker consume to avoiJ a reduction in bod)' lempcrmurc: of -1.0 K as a re:sult of hem loss'! Assume the heat c3pacit)' of the body equals that of w~ter anJ Ihat Ihe hi~er weighs 55 kg. State any addilionul a~SU lllptions. 1'2.5 Cl)llnt RUlnford observed th3t u~ing CantlOn bori ng machinery a single horse could hc~t I 1.6 ~g of cold water at T = 273 K IU T - 355 K in 2.5 lIoul1l. Assuming the same ... ~te of work. how high eoulJ a horse "'Jisc u IS{}. kg weighl in I minUle?

1'2.6 2.25 moles of an ide~ 1 g3S 31 35.6"C cxpilnds isother-m311y ffOlll an in it ial volullle of 26.0 JmJ IU u fi n31 volume of 70.0 dml. Calculate 1V for Ihis process (a) for c.lpansioo against a cOIlstanl cXlemal pressure of 1.00 )( lOS Pa 300 (b) for a n:,crsibic eXpilnsion.

PfWBl(MS 37

Q2.13 A chemical n:acl ion occurs in a const:lllt volume cn closu~ separ~tcd from the su rroundings by diathcrmal walls. Can )'011 say whelher the tempeT:;ltun: of the surroundings in cn':Isl's. deCn';L'leS, or re:main~ the same in this process? Explain. Q2. 14 Explain (he n'lal ionship bctwttn lhe Icnns ~.tur' dif /~rf'n'ial aOO SIlJI~ /un("liun. Q2. IS In lhe cxperiment sh"",n in Figurc: 2Ab. the weighl drops in a ,cry shon lime. How will the lempcT:;lture of lhe waler d.ange wilh time'!

P2.7 Calculate q. w. I1U. and 1111 if 1.65 mol of an ideal gas with C1, ... - 3N/ 2 undergoes a reversible adiabuti c ex pansion from an inilial volume v.. '" 7.75 Ill J to a final volume Vj = 20.511\). Tile initiullcmpemture is 300. K. P2.11 C3lcul3te W for Ihe aJ iabatic expansion of I mol of an ideal gas at an initia l pressure of 2.25 bar from an initial tem perature of -175 K to h finallelllpc ... Jture of 322 K. Wri te3n expression for the work done in tllC isotherm~ 1 reversible ex pansion of Ihe gas at 322 K from an ini tial prcs.>ure of 2.25 bar. What value of the fin~1 pressure: wO\lld give the s:lme \":.llue of was the first part of thi s problem? Assume that CP._ - SR/ 2.

1'2.9 AI 298 K aOO I bar pressure. lhe density of water is 0.9910 g cm-1 aOO CI' ... '" 75.3 J K- 'mol- t The change in \"oIume wilh tcmpcr~lUrc is !;i"en by I1V - V ... "",p il T where p. the coefficicnt of therma l e.~pansion. is 2.01 x 10~ K- I. If lhe lemperJlure of!SO.g of water is in creased by 38.0 K, calculate 11'. q . .lH. and j,U. P2.1O A muscle fiber contr.lCtS by 2.0 em ~oo in ooing.\.O lifts a weighl. Calculate the wOtt performed by Ihe fiber. AssunlC the muscle fiber obe)s Hookcs law F = - k .. with a force ,onst3nt. k. of goo. N m- I. 1'2. 11 A cylindrical vessel with rigid adiab~lic walls is scpo a ... ~ted inlo lWO parIS by a fri'lioniess uJi~balic piston. &lch part contains 50.0 L of an ideal monalomic g3S wilh Cr ... - 3R/2. [niti ~lI y, r.. - 32~ K and P, _ 2.50 X 101 Pa in each pan. Heal is slowly inlroouccJ into Ihe left liar! using all electrical he~ler IIntil the I)istoll has moved sufficiemly to the righltl! re'liit in a final pre.~sure Pj = 7.50 b3r in the ri ght pan. Consider the ,ompression of thc gas in thc right pmtlo be ~ reversible process. a. Calculate Ihe work done on the ri ghl par! in this j)fOCCSS

and the final tcmpcr~IUre in the riglll part . b. Calculate the finaltcmpemturc: in the left p;.n and lhe

al11O\lnt of heat that flowed into this pan. P2.!2 In the revcrsible adiabatic cxpansiOll of 2.25 mol of an ideal gas from an iniliallempc"'JlUre of 32.O"C, the work dORe 0

38 C H A PT E R 2 Hut. WorK. Intll1'~1 Erot'O'I', {",I\aIP'!', and the fl~t law of Thec end of the process. the pressure is 2.50 x IrI' p. .. , Calculate the finaltemperJture of the gas. Calculate q. w. :lU. and :ll l (Of Ih is process. 1'2,17 A vessel containing 2,25 mol o( an ideal gas wilh P; '" 1.00 bar and C,. .... "" SR/ 2 is in II"mnal contact with ~ Waler balh. Treat the vesse l. gas, ~nd water roth as being in thermal equilibrium. initially at J 12 K. and as sc:par.lIed by adiabatic walls fll.lll1 lhe rest uf the universe. 11te \'e5scl. g~ and Wolle>' rolh h,l\'e an averuge heal capacity of C,. .. 6250. J K - I, llte gas is compressed reversibly 11.1 Pf : 10.5 bar. What is the tcmperJture of lhe s~lem afrer lhermal equ ilibrium has been established'!

1'2. 18 An ideal gas und.ergoes an e.~pansion (rom the initial state de_"Cri bcd by Pj, Vj, T 10 a final state described by Pf' Vf' T in (a) a prtKcs.~ at the constant e.~ ternal pfC.\sun: Pf and (b) in a reversible pnxess. Deri\'c e.\pressions for the largcst mass that can be lifted through a height" in the sorroundings in these pnxesscs ,

1'2. 19 An ideal gas described by Tj .. 300. K, P; - 1.00 b:Jr. and \oj '" 10.0 L is heated at con.~lJnt volume unti l f' : 10.0 ror. It then undergoes a reversible isothermal c>.:pansion unti l P .. 1.00 bar. It is then restored to its uriginal state by the extr.>elion of heal al constant pressure. Depict this closcdeycle pnxess in a P- V diagr.lm. Calculate 11' fl)l' each step and for the lOcal process. What values for w Would you calculJte if the cycle were trJvcrsed in the opposite direction'!

P2.20 In an adiabatic compression of one mole uf an ideal gas with C" .... = 5R/ 2.1hc: IcmperulUre rises from 293 K to 325 K. Calculate q, 11'. J.H. and ::'U.

Pl.l1 The heat capacity of solid lead o~ide is gi\'en by

C,. ... = 44.35 + 1.47 x IO- l f in units o( J K - t mol - t Calculate the change in cmhalpy of 3.25 mol of PbO(s) if it is cooled from 750. to 300. K at constant pressure. 1'2.22 Carbon dioxide. for which CP . .. .. 37. 1 J K - t mol - I at 298 K. is expanded re\'ClSibly andadiabalically from a vol ume of 2.85 Land temper.lture of 300. K 10 a final \'OIumc uf 16.5 L. Calculate the final temper .. ture. q, w. J.II. and J.U. Assume IItat Cp , .. is constam ()\'er the temper .. ture intenlal. P2.23 One mole of an ideal gas for which P = 1.00 bar and T - 300. K is e.\panded adiabalicall y ag~inst an external pressure of 0.100 bar unlilthe final pres-sure;s 0.100 bar. CJlculatc the finaltemper.llUre, q, W. ::' 11. and:lU for (a) Cy ... .. 3R/ 2, alld (b) Cy ... .. 5R/ 2. Pl.24 2.25 molcs of Nl in a Male defined by 1i .. 300. K and Yo '" 1.00 L undergoes an isothcntlJl reversible expansion unti l Vj '" 20.5 L. Calculate ttl assuming (a) thaI the gas is described by the idea l gas law and (b) that the gas is de -scribed by the Van der Waa ls ellUation of S13te. What is Ihe percent elTor in using the idea l gas law instead of the \'lIn der Waals equation? Tltc van der Waa ls pam meters for Nl are list ed in Table 7.4.

1'2,25 A major I~aguc pilcher throws a baiiCball with a speed of 150. ki lometers per hour. I( tl>c baseball weighs 220. gr.lms and its heat capacity is 2.0 J g - I K- 1 calculate the temper .. ture rise of Inc ball wl>cn it is Stopped by the eateher's min. Assume 00 heat is tr .. nsferred to lite emcltcr's mill and that the catcher's arm does no!: re

1'2.29 A ne~rly flJt bicyde tire becomes noticeably warmer after it h~s been pumped up. Appro~imatc this process as ~ reversible adiurotic compress ion. Assume the initial pressure and temperJture of the air befure it is put in the tire to be P, '"' J.OO \);Ir ~nd T; - 30S K. The final pressure in the tire is PI = 5.75 bar. Calculate the fi n~ l temper.lIure of the ~ir in the tire. Assume tl!;at Cy . .. - SRI 2. P2.311 For 2.25 mol of ~n ideal gas. p~ ... """ = P = 200. x Uy Pa. The lemperJture is cl!;anged from 121"C 10 28.5"

40 C H A PT E R 2 Hut. WorK. Inlll1'~1 Erot'O'I'. {ml\aIP'!', and the fl~t law of The

CHAPTER OUTLINE

... THE MATHEMATICAL PROPfRTIES OF STAH fUNCTIONS ,~ THE OEPENDENCE OF U ON

V AND r

U DOES THE INHRNAL ENERGY DEPfND MORE STRONGLY ONV ORn

... THE VARIATION Of ENTHALPY WfTH HMPERATURE AT CONSTANT PRESSURE

... HOW ARE C~ AND Cv RELAHDl

... THE VARIATION Of ENTHALPY WITH PRESSURE AT CONSTANT TEMPERATURE

.., THE JOULE- THOMSON EXPERIMENT

... LIOUEFYING GASES USING AN ISENTHALPIC EXPANSION

3 The Importance of State

Functions: Internal Energy and Enthalpy

The mathematical p roperties of state functions are utilized to express the infinit esimal quantities dU and dH as exact differentials. By doing so, expressions ca n be derived that relate the change of U with T and V and t he change in H with T and P to experimentally accessible quan tities such as t he heat capacity and the coefficient of the rmal expan

sion. Although both U and H are functions of any two of the variab les

P, V, and T, the dependence of U and H on temperature is generally far greater than the dependence on P (J( V. As a result, for most

processes involving gases, liquids, and wlids, U and H can be regarded as funct ions of T only. An exception to this statement is the cooling on the isenthalp ic expansion of real gases, which is commercially used in the liquefaction of N2, O2, He, and Ar.

3.1 THE MATHEMATICAL PROPERTIES OF STATE FUNCTIONS In Chapler 2 we dcmonSlrau:d that U and H are ~Iale functions and that 111 at>d 'I an: IXIth fUrlCtions. We alw di.'\Cus,;cd how to cakulate changes in these quantities for an ideal gas. In th is chupter, the path independence of state funct ions is exploited to derive rela-tionships with which au and aH ~'an be calculated as fu nctions of p. V. and 1" for real ga,;cs. liquid.s. and solids. In doing w . we develup the fonna l a.spects of thennodynam_ ics. II will be secn that the formal structure of thennodynamics provides a powerful aid in linking thCQl)' and experiment. However. before these topics are di scussed. the math-emati cal propenies of state functions ~d to be outlined.

n.e thcrnlodynamic state funct ions of in terest here are defined by two v..u1ab le.s from the sel P. V. and T. In formulating changes in state functions. we will make exten-sive useof pi.mia l derivatives. which are ""viewed in fhe Math Supplement (Appendi~ BJ. n.e following discussion docs nUl apply tu path functions such :J.S til and q becau.

/

"

, " , (-"') \t '~: .. ("*), .. , (-#), .. : : I ,/)/ ,dJr : , , : :/ ," , I , I t ' " ,. ~'I :I"'m :/ AbitHiIl: 4(Jm , ' / ~:~ ' .... ;:;i 10m See !eYe1

.,

FIGURE 3.1 Starting al (~puinllallelcd : 00 It>. hill . you fi ..... move in the fIOSi!;\'e~ dirlion and then along lhe )' direniOfl , If dx and dy arc: sulflCiolllly small. Ihe .... h:.nge in heigh! d: is ,iwn by

"' : (,,) ,,. (,,) ", ..

3 1 HlE MATHEMATICAL PflOPR lIES Of SlATE fUNC110 NS 43

df is c~l led ~n txllCI d irrert'nli lli. An example of ~ state function and il~ exael ditTeren. tia l is U and dU - 4q - p,.1I"'tIt,IfV.

r-;XAMPLE PROBLEM

I ll. Calcubte ]. ,

for thcfunction f(x.y) - yr + xy + .l In)". b. Determi ne if f(.r, y) is a state function of tIM: variables .r and ,'. c. If f(.r. y) is a state fU llctioo of the variables.f and .,'. what is the total

differentia l df?

Solution

,. (Y.) _ Ji' +)' + In)". (I" , 1

(::{ ), = yc\

1 (,(*), \ -,' + (I, ' 1 + -. ,

b. Bccause we have shown that

('f) _ ~' + x + !. a;: , )"

(,(*),\ (,(*),) a)" 1 (Ix 1

1 + -

,.

ft.x. y) is ~ Statc function of tIM: variables.r and )'. Noce lIlat any wdlbella\w function trot e:m be e.~~d in analytical form is a statc functioo.

c. The tota l differential is g.i\cn by

L df - (Y.) dd (Y.) d, (Ix 1 (I)" ,

- (ye'+ )"+ In)")dx + (i'" + x +~)'I)" Two othcr imponant resu l l~ from ditTerential calculus will be used frequently. Con

sider a function . z ,. f(.r.y). which Call be rcarr.mged to X = g(y. z) or y = h(x. z). For cxamplc, ifP - II RT/ V.thcnV - li NT/ P undT = PV/ nR. In this case

(~), . (~) ax ,

(3.6)

The cycl ic rul e will also be used:

(*l,(W~), - 1 (3.7) It is called me cyclic rule because x. y. and: in the three terms follow theonlcr .r. y.:.. , ', Z and Z . r, y. Eqwtioos 0.6) ~nd 0.7)can be used to reformulate Equation (3.3) given below:

ti P = (.!!) ,IT + (.!!) d V aT y ilV T

__ .. n. )""" .... TlWdo.....~t:>r~"'T_t'l~EtIgoInlPIIIItp_. ~by"--'~~~02012..,. __ ~1oo:.

Suppose this expression needs to be evaluated for a .e lected solids ~Ild liquids arc Shown in Tables 3.1 ~Ild 3.2. re.spec[ively.

Equ~[ion 0.8) is ~n u~mple of how seemingly ah$tr.IC[ p.:Ini~1 !kri\"~[i\'es can be directly linked [0 nperimem.ally dl"lC1Tllined quaJl!jlie..~ u~ing [he nl;llhcma[irul prupo!l1ics of stale functions. Using the definitions of fJ and K. EqU:Jlion (3.3)c;ln be wrinen in the foml

JP = !!..JT - ....!... dV (3.10) .V

which can be imegr.l\oo 10 give Tj v,

f~ f' " tJ.P = -JT- -JV"=-(T.,-T) K KV Ie ' T; v;

, ~ -- In -

TA8LIE 3 .1 ISOBARIC VOLUMETRIC THERMAL EXPANSION COEFfiCIENT fOR SOLIDS AND LIQUIDS AT 298 K

(3.1 1)

Elt mt nl U~/l(K - I ) "mtn l or CUlllptlulld 10' IJ (K- ') 1\8(') 57.6 II g(l) 1.~1 1\1(5) "'., CClil) 11.4 I\u(.,) 42.6 CII )COCII )(I) 14.6 Cu(.) 49.~ CIIPIl(l) 14.9 Fc(f) "'., CllI~OIl(I) 11.2 Mg(J) 7lU C61 1~Cl I )(I)

'" Si(s) 7.5 C6Iitt') IIA W(~) 130M II lO(I)

"" l.n(J) ... , II lO(~) , ... s...,,.... . ~. v.I. Itarri .. J W . SIoe..",.. It. ond L.-.r. ll..llmodbtd.,PIo,SIa. SptuoIlCI. Now Yurt. 2001: Lode. () 1I:.Ed.. If_"'~Iry_PIo."J"'J.3.lnll. CRC ~ lIu;:o II-.I't.. 2001: Bbdwuk. R . Ed. O'AIu Uu T~Jiir CItI-ur...wl PIo,JWr. ;j(/I cd. Sp:wIF. 8goInlPIIIIp_. ~t'I~~~~02012t'1 __ ~1oo:.

3 1 HlE MATHEMATIC AL PflOPRlIES Of SlATE fUNCTIONS

TABLE 3.2 ISOTHERMAL COMPRESSIBILITY AT 298 K

SubSla M

M

ro

.,

~ '"

, , " J '"

"

"

FIGURE 1,2

"'

200 400 600 600 1000 Tempera1llteiK

Molar heal cap",,;'ies C~ . are .h"",n for a number of ,.oes. Al

32 THEOE~NOENC(OfUONV/l.NDT 47

In thermodyn~lI1ics. thc origin for the ~ubst~nce depelldence of CII ... is not ~ 1I1~((er of inqui ry. becJuse thermod}'n~rnics is nO! concerned with the microscopic structure of the systcm. To obtain numerical ~suhs in themlOdynJrnics. s}'Stem-depcndcnt proper-ties such a.~ CII ... a~ obtained frum e~periment or theory. A microscopic modd is re-quilCli to upl~in why Cv ... for a panicular sob>t:u1oCt has its measured \;due. For

e~ample. why is CII ... s m~lIer for g;ncoos Hc than for gaSCOlls methanol? To increase the temper-ature by an amount tiT for a system containing helium. the translation~1 energy of the ~toms is increased. By contrast. to give the same incrcase tiTin a syStem containing methanol. the rotational. vibr.uiooal. and Inlnslational energies of the mole:-cules are all increased simultaneously. because all these degrees offrecdom are in equi-librium with one anolher. 11w:refore. for a giw n temperature incrcmem tiT. mr~"'T_t'l~EtIgoInlPI1IIIp_. ~t'I~l.....-.g~~02012t'1 __ ~101o:.

48 C H A PTE R 1 T~ ImJXlft"l'I(~ of State FlII'I(100ns: Inle", .. 1 Ene'~ and EMhaipy

V" T, V , T,

I

V" T,

v

FIGURE ],l Becau>e U is ~ ilUle fun"il,", ull palhs connecting V .. T, ~nd VI' TJu"", c~ually valid in calculuting au, Thcr(fo""" ~ sP"CifkatiOll of 1"-: !XlIII is n,M 1\C('~.uaI')',

In Ihis equ;uion, lhe symbols dUv and dUrha\'e been u.>ed, where lhe suhscripl indicates which variable is COf1!i;tam, Equ!ltioo (3.20) is an importafll resuh that ~pplit:S to systcms contain ing gases. liquids. or solids in ~ single pIlase (or mi.\ed pIlases al a COOSlant com-position) if no chemical reaClions or ph.;t."" changes occur. 1be ad\"dllIage of writ ing dU in the form givcn by Equation (3.20) owr Ih:u in Equ~liOll (3.1 2) is th~t (iJU/ iW)r can be e\1lluated in terms of the system variables P. V. and T ~nd Iheir dcri\1ll i,es. all of which are cxperimenwlly accessible.

Ooce (iJU / iJV )r and (iJU/ iJT)v are I.':nown. the.w quantities can be used to determine dUo Because U is a .wte function. the path taken between the initial and final Slates is unimportalll. Th"", different paths ~re shown in Figure 3.3. and dU is the same for these and any OIher paths c{)IUl('Cting V~ T; and VI' Tf To sinlplify the cakulatioo. the ]XIth chosen coosists of two segments. in wllicll ooly one of tile variables ch~ngcs in u given p;ltll segment. An e:\umple of sucll ~ ]XIth is Vj T1- V,. Tj - VI' Tf Because T is con-swnt in the first segment.

Because V is cons\.;ln( in tile second segnICIlI. dU = ilUv - Cv ,IT. Finally. Ille tot~1 ':hang~ in U is the sum of the ch~ngcs in the two segmenlS. dU",,"1 - IIU\. + dUro

3 3 DOES THE INTERNAL ENERGY DEPEND MORE STRONGLY ON V OR n Chapler 2 demonslr~ted tllat U is a function of T ulone for ~n idcJI gas, liov.'cver. lhis stJlemcnt is noltl1le for real gases, liquids. and solids for wh icll the change in U with V must be considered. In this section. we ask if the tcmperJture or the "olume dcpendence of U is most impon Jnt in determining dU for a process of interest. To answer th is ques-lion. systems consiSling of an idcJI gas. ~ real gas. a liquid. und a solid JT\! considered !iCparJtcly. &ample Problem 3.3 shows ,hal EquJlion (3.1'}) leadS to a simple result fur a system consisting of an ide~1 gas.

EXAMPLE PROBLEM 3.3 E'OIluJte (iJU/ iJV )r for an ideal gas aoo modif)' Equation (3.20) acwrding ly for the specIfic ease of an ide~1 gas J Solution

T(!!:) _ P = T( iJ[nRT/ v l ) _ P _ liNT _ P _ 0 oT v iJT I' V

1ltcreforc dU = CvdT . showmg that for an Ideal ga.'. U" u function of Tonly

Example Problem 3.3 shvws that U is a funct ion of Tooly fur an ideu l ga.~. Spccific~ l Iy. U is not a function of V. This resu lt is understandable in temlS of the potential function of Figure 1.7. Bcrause ideal gas molecules do not attmct or repe l one ~ nOlhcr. no energy is required to ch:mge their awrJge diswncc of _o;ep"rJt ion (increase or deerea."" V):

,

dU - J Cv(T) tiT T.

(3.2 1)

Bccause U is:l fuoction of T ooly. Eqo~tion (3.21) holds for an ideal gas even if V is Il(){ constant.

Next consider the variJlioo of U wilh T:u\d V for a real gas, 1be C.lperimental detcr-minatioo of (iJIJ/ iJV)T W'olS carried out by J:unes Joule using an appar.l1us consisting of two glass flasks sep.mlted by a stopcock. all of wh icll were immefliCd in a WOlter

__ .. n. ) ,,""""'. llWIIo.-~""~"'T_b\l~EtIgoI"""ptJllpRtid. ~b\I"--'l....-.;J~~02012~ __ Educali

3 3 OOES THE INTERNAl ENERGY DEPEND MORE STRONGLV ON VOR n 49

ooth. An i

EXAMPLE PROBLEM 3.5 For a gas described by the Van der Waals equation of swte . P - nNT/ (V - nb) - unl/ Vl. Usc thiscquation to,onlplcte these Woks: a. Calculale (iJIJ/ iJV)T usi ng (iJIJ/ iJV )T - T(iJP/ iJTh , - P. b. Derive an expression for the ,hange in internal energy.

/lUT = I:"(iJIJ/ iJV )T dV. in compressing a \'lin der Waals gas from an init ial molar volume: Vi to a final molar volume Vf ~t 'OIl.tam tcmpcruturc .

,.

" NT "NT "la lila ~ --- - ---+- = -

V - lib V - "b V1 VI

l

We see tlmt tlUr = 0 if the uttr.lctive Piut of tl~ intcmlolccular potc111ial is negli gible.

Example Problem 3.5 shows that in general (iJU/iJV )T "" O. !lUT can be'akulated if the equa~ioo of state of the real gas is k~wn . This allows the re lativc imponance of tlu,. - Iv/ (iJU/ iJV )T dV and tlUv - IT/ CI. dT to be d' termined in ~ proccss in which both T and V cllangc. as shown in Example Problem 3.6.

EXAMPLE PROBLEM 3.6

One mol of Nl gas undergoes a , hange fTOlTl an ini tial Stale described by T = 200. K and Pi = 5.00 bar to a final state descri bed by T - -100. K and Pf = 20.0 bar. Treat N j as a van der Waals gas with the p:1 ..... lnetcrs (' = 0. 137 Pa m6 mol-I and b = 3.87 X 10- ' ml nw.ll- I. We usc tile p:1th N}(g. T - 200. K. P = 5.00 bar) -> Nz Nz

34 THE VARlAnON Of ENTHALPY WITH TEM PERATURE AT CONSTANT ~ESSURE 51

- (4.50 - 0.7 12 + 0.447 - 0.0610)k.f = .U7k.f :lUr is 3.2% of :lUI' for Ihis case. In Ihis e.~ampk. alld for mOSI processes. I :lUr can be BCglI:Cled relat;\"

Ahhoogh the imegral of 4 'I is in gene,al PJth dependent. it h~ s a unique \'~Iue in this case because lhe palh is speeified. n~mel y, P - p'JOt"..,J - COf1SI~m. Imcgrating both sides of Equ~tion (3.26).

f f f

/"U= /4'/1' - jP dV 01' U,-U/=q,.- P(V,- V;) (J.27) j j i

Because P = P, = Pi. this equation can be rewritten as

(V, + P, V,) - (Ui + P'1Ij) = 'I,. or 1111 = 'I,. (US) The preceding equJtio n slwws that th.e \'a lue of 1111 can be determined (01' an arbi t,ary process at constam P in a closed system in whkh only P- V work occurs by simply measuri ng 'I ,.. the heat transfe rred between Ihe syslem and surroundings in a constanl pressure process. Note the simi larity between Equations (3.28) and (3. 18). FOI' an arbitrary process in a closed system in which there is no work OIher than P_V work. I1U = 'Iv if the process t~kes place at conMant V. and 1111 - 'I,. if Ihe process takes pla 0 in both ea>es and Cp .... r~"'T_t'l~Et

T, T,

All,. = !C,.(T)dT=n!C,. ... (T)dT (3.33) T, T,

If lhe: lempe"'.I!Ure inle r ... al is small enQU~h. il can usually be assumed IIiaI C,. is eon-stant In Ihal case.

(3.34) 111c calculation of AH for chemical rea.:tiQrls and changes in phase will be discussed in Chapters 4 and 6.

!eXAMPLE PROBLEM 3.7 I I : 14J.().g Sffil:s

-dll,. " CvdT + (iiU) dV + PdV av , (3.36)

3 5