Thermodynamics and Thermochemistry for Engineers

Transcription

1 hermodynamcs and hermochemstry for Engneers Stanley M. Howard, PhD South Dakota School of Mnes and echnology Department of Materals and Metallurgcal Engneerng Rapd Cty, SD USA

2 Preface here are four reasons to wrte another textbook on thermodynamcs for students nvolved wth materals. Frst s to add a rcher hstorcal context for the subject than s normally found n texts. Often students thnk thermodynamcs s a collecton of formulas to be memorzed devod of practcal meanng when, n fact, the subject s rch n practcal hstorcal sgnfcance. hermodynamcs s a beautful subject wth a rch hstory of clever people who often had only practcal reasons for ther work. However, as t turned out, they along wth theoretcans, provded tremendous nsght nto the fundamentals of what we now call thermodynamcs. Even though the author has taught the subject nearly 50 tmes n the last 40 years, he s stll fndng new and fascnatng nsghts nto the hstorcal underpnnngs of the subject. hs text attempts to keep these connectons wth the subject so that the student gans a rchness that carres more meanng than just a rgorous treatment so commonly found n texts. he addng of hstorcal context s not meant to dengrate the pure formal treatment of the subject. Indeed, that has ts own appeal and place but more so for those frst ntated n thermodynamcs. Second s to reduce the cost of the textbook for the student. hs s accomplshed by makng ths a dgtal text, whch s also avalable n hard copy through MS for a modest charge. Reduced prcng for Materal Advantage members promotes student membershp n the student chapters. hrd s convenence. Many unverstes requre laptop computers and nearly every student has one. So, t makes sense that a dgtal text s pared wth the laptop. Specal prcng s possble when unverstes preload software resources on laptops. Dgtal form also permts students to prnt selected chapters to carry to the classroom thereby reducng the load they need to carry from class to class. Also, dgtal form obvates the undesrable rtual of sellng textbooks at semester s end. Fourth s purely one of author preference. When one wrtes a text, t s exactly as the author prefers. Chances are small that the author s preferences match another professor s, but t s certanly perfect for the author. he good news s, however, that snce the text s n dgtal form, arrangements can be made to rearrange and add content ncludng changng notaton wth the replace command. hs alludes to an eventual open source textbook, whch s nherently uncontrollable, but ths s as t should be wth a subject so beautfully and powerfully fundamental. he author acknowledges consderable nfluence from three thermodynamc texts that have played a sgnfcant role n hs learnng of the subject: Prncples of Chemcal Equlbrum by Kenneth Denbgh and Chemcal hermodynamcs by Irvng M. Klotz. As a young professor the author used for a few years Physcal Chemstry for Metallurgsts by J. Mackowak and employed some deas from Mackowak n descrbng the Frst Law. For several decades before the wrtng of ths text, the author reled on Davd Gaskell s Introducton to hermodynamcs for Materals Engneers, 5 th edton and the precedng edtons. 2

3 Chapter 0 Basc Concepts he Zeroth Law of hermodynamcs Defntons Mathematcal Requrements he Zeroth Law of hermodynamcs was an afterthought occurrng after the Frst, Second, and hrd Laws of hermodynamcs were stated. It acknowledges that thermal equlbrum occurs between bodes of the same degree of hotness, whch n today s vernacular s equvalent to temperature. Before heat was known to be a form of energy and thought to be a calorc flud that was somewhat mystcally related to objects, the concept of thermal equlbrum nvolved more than degree of hotness or temperature. For example, when small metal chps were made durng the borng of canons, Calorc Flud heory held that the metal chps were hot because the flud flowed nto the smaller peces. In vew of the complex and long-abandoned calorc theory, the Zeroth Law observaton that only degree of hotness fxes thermal equlbrum s elegant n ts smplcty and usefulness. he Zeroth Law of hermodynamcs he Zeroth Law of hermodynamcs states that f two bodes, a and b, are each n thermal equlbrum wth a thrd body, c, then they are n thermal equlbrum wth each other and that the only germane property the bodes have n common s the degree of hotness. hs s a smple Eucldean relatonshp s expressed as follows where the functon h s the measure of hotness. If then h(a) = h(c) and h(b) = h(c) (0.1) h(a) = h(b). (0.2) Hotness and temperature are so ntertwned n today s language that the terms are used nterchangeably; one mmedately thnks bodes wth the same hotness are also at the same temperature. hs s true but temperature s a defned measure - not an observed natural property. Indeed, there was a tme that temperature was unused, undefned and unknown, but thermal equlbrum could stll be defned through observable hotness. emperature scales have been defned n many dfferent ways, ncludng the frst Fahrenhet scale that assgned zero to body temperature and 100 to a salt-ce-water eutectc mxture. hs was later reversed so that 100 F was body temperature (actually 98.6 F) and salt-ce-water eutectc was 0 F (actually -6.0 F). hs flexblty n the defnton of temperature scales llustrates that as temperature scales were beng establshed there was no nherent relatonshp between hotness and temperature. he laws of thermodynamcs arse from observatons of nature; consequently, the Zeroth Law does not rely on a defned temperature scale but only observable hotness. he Absolute hermodynamc emperature Scale s defned n Chapter 2. 3

4 Defntons Other defned concepts that wll be needed throughout ths text are for systems and varables. A system s defned as a part of the unverse selected for consderaton. Everythng that has any nteracton wth the system s termed the surroundngs. Systems may be open, closed, or solated. Open systems can exchange both mass and energy wth the surroundngs; closed systems exchange energy but no mass; solated systems exchange nether mass nor energy. A varable descrbng a partcular pece of matter s sad to be extensve f ts value depends on the quantty of the matter beng descrbed. For example, total heat capacty and mass are both extensve varables as opposed to ntensve varables such as densty, specfc heat capacty, and temperature, whch are ntrnsc to the matter and ndependent of quantty. When Gbb s Phase rule s presented later n the text, t wll be shown that fxng any two ntensve varables for a pure materal wll fx all other ntensve varables. hroughout ths text the terms varables and functons are often used nterchangeably. A very mportant and mutually exclusve dstncton s made between path and state functons. A functon s sad to be a state functon f the change n the functon whle gong from state a to state b s the same regardless of the path taken to move from state a to state b. he change n a path functon does depend on the path taken. Illustratons of path functons wll be gven n Chapter, 1 but suffce t to say here that most varables - but not all - are state functons, ncludng varables such as temperature, pressure, and volume. State functons are exact n the mathematcal sense, whch s to say that they may be dfferentated n any order and the same result s obtaned. A process s sad to be reversble whle gong from state a to state b f t could be returned to state a whle leavng no more than a vanshngly small change n the surroundngs. here s no requrement that the system actually returns to the orgnal state, only that t could whle leavng no more than a vanshngly small change n the surroundngs. An rreversble process s one that requres some sgnfcant change n the surroundngs were the system to be returned to ts orgnal state. hs s llustrated by consderng a compressed gas n a steel gas cylnder ftted wth a frctonless pston. If the valve on the tank s opened so that the compressed gas (state a) nflates a balloon (state b), there s no way to return the gas to ts orgnal state wthout employng some substantal work to force t back nto the hgh-pressure state nsde the tank. Fgure 1 shows n theory a way to reversbly take a compressed gas to a low-pressure. As the gas expands t lfts sand whch s contnuously moved to an nfnte array of shelves. As the gas expands, the sand s lfted and deposted on the shelves. he accompanyng gas expanson s reversble, because the gas can be returned to ts orgnal state by movng the sand stored on the shelves back onto the plate orgnally holdng the sand. Perhaps one addtonal gran of sand mght need to be lfted at the top of the expanson to commence the compresson. hs one extra gran fallng from the top to the bottom of the stroke could be sad to be a vanshngly small change n the surroundngs; therefore, the expanson process s sad to be reversble. It s noteworthy that a gas compresson process s nherently reversble because the work of compressng the gas s always avalable to return the surroundngs to ther orgnal state. Also, frcton necessarly causes a process to be rreversble, because there s no way to reclam the energy lost to frcton for use for returnng a system to ts orgnal state. 4

5 Fgure 0.1 Reversble gas expanson process Requred Mathematcal ools Mathematcal tools from calculus requred for the successful completon of ths text are Dfferentaton and Integraton of rudmentary functons Dfferentaton by the quotent rule and the product rule Integraton by parts otal dfferentaton Propertes of partal dervatves, Numercal ntegraton usng the trapezod rule L Hoptal s Rule 5

6 Chapter 1 he Frst Law of hermodynamcs Conservaton of Energy he word thermodynamcs s from the Greek therme (heat) and dynams (power). It was coned by Joule 1 n 1849 to descrbe the evolvng use and understandng of the converson of heat to produce mechancal power n steam engnes. hese engnes drove the rapd ndustral expanson of the ndustral revoluton. Characterzed by a renewed sprt of nqury and dscovery n a poltcal envronment supportng such nqury, the ndustral revoluton requred power beyond that avalable from manual labor, farm anmals, water wheels, and wnd mlls. Near the end of the 17 th century, nventons by Papn, and Savery ponted a way to use steam s expanson and condensaton to perform useful work. In 1712 the margnally useful steam dgester (Papn) and the water pump (Savery) were superseded by homas Newcomen s atmospherc engne that removed water from mnes. Newcomen s engne was the frst useful engne that produced work from heat. Practcal advances n the steam engne ncreasngly drove the ndustral revoluton. Industry was beng freed from the need for water wheels for ndustral motve power. Industry could locate where ever there was a source of fuel. James Watt and Matthew Boulton n the late 18 th century successfully marketed an mproved effcency steam engne by sharng the energy savngs of ther engnes wth ther customers. Parallelng early advances n the practcal advances n the steam engne was an ll-defned theoretcal understandng of heat, work, and the maxmum work attanable from heat that begged for defnton. Clarty had to wat for several underlyng prncples: the true nature of heat as a form of energy, the Ideal Gas Law, and the absolute temperature scale. Great effort was drected on producng the most work for the least heat snce fuel was an expensve and lmted resource. Many dubous theores were advanced: some clamng there was no theoretcal lmt on how much work could be extracted from heat only practcal lmts. he quest for defnng effcency was fnally realzed, n the theoretcal sense, by the work of Sad Carnot n 1824 when he publshed and dstrbuted n hs deas n an obscure booklet enttled Reflectons on the Motve Power of Fre. hs booklet was dscovered by Clausus by chance at a news stand that stocked one of the few copes of Carnot s work. In the decades after hs work became wdely known, Carnot s deas formed the bass of work by others ncludng Clausus and Clapeyron. Carnot s accomplshment s remarkable because he dd not have the advantage of the Ideal Gas Law advanced by Émle Clapeyron n 1834 or an absolute temperature scale defned by Lord Kelvn n Carnot was educated n Pars at the École Polytechnque. He was a mltary engneer but after Napoleon s defeat n 1815, Carnot left mltary servce and devoted hmself to hs publcaton (n England?). Carnot was a contemporary of Fourer of heat transfer fame, who held a char at the École Polytechnque. 1 Perrot, Perre (1998). A to Z of hermodynamcs. Oxford Unversty Press. ISBN OCLC

7 Carnot s short lfe ( ) concded wth the perod durng whch the Calorc Flud heory of heat was beng debunked. In 1798 Count Rumford proposed that heat was not a flud but rather a form of energy. Although Carnot lved n the world of controversy on these competng vews of heat, the controversy, whch ended wth Joule s work n 1840, t dd not prevent Carnot s progress. oday, we accept heat to be a form of energy, not a calorc flud, and t seems lkely Carnot knew ths as well. However, hs thnkng about steam engne effcency was llustrated n flud terms. He thought of heat as water drvng a water wheel to produce work. he heat s temperature drvng a steam engne was analogous to the water s heght. he farther the water fell over the water wheel, the greater the work produced; the farther the temperature used to power a steam engne dropped, the greater the work produced. oday s student arrves n the classroom wth a better understandng of heat and work than the poneers who bult steam engnes had. ralng the bulders by about a century, theoretcans eventually defned the theoretcal lmtatons of the converson of heat nto work. Steam engnes were followed by the nternal combuston engne and the turbne engne. All convert the heat from fuel combuston nto work and are now generalzed as heat engnes. hey all have the same theoretcal lmtatons descrbed by the early thermodynamcsts. he work of these early thermodynamcsts s mportant for every engneer to know. Even more mportant for the materals and metallurgcal engneer s that the analyss of heat engnes leads to an understandng of entropy, the bass of predctng f a process s possble. Entropy s the arrow of tme, but before t can be presented one must understand heat engnes. Frst Law of hermodynamcs he Frst Law of hermodynamcs may be stated as follows: In an solated system of constant mass, energy may be dstrbuted n dfferent forms but the total energy s constant. he term system means a regon of space under consderaton. he term solated means that no mass or energy s allowed to enter or leave the system. he term constant mass precludes the occurrence of nuclear processes. Fgure 1.1 shows some of the many forms of energy. he Frst Law s a statement of conservaton of energy. he analyss of heat engnes nvolves only three of the many forms of energy shown n Fgure 1.1: nternal energy, heat, and work. he ncluson of heat (q) and work (w) are understandable snce they are the forms of energy under consderaton n heat engnes. he nternal energy, U, s needed to account for the change of the workng flud s (steam) energy. he nternal energy s the total energy of all the molecular moton, bondng energy, etcetera, wthn the workng flud. he workng flud s nternal energy changes as the workng flud s temperature s changed by the addton or removal of heat. herefore, the change n the workng flud s nternal energy s related to the amount of heat exchanged wth the workng flud and the work performed by the workng flud. he Frst Law may be wrtten. Born Benjamn hompson n Woburn, Massachusetts n Left wth the Brtsh n 1776 as a Loyalst. he Bavaran government bestowed the ttle Count of the Holy Roman Empre 7

8 U q w (1.1) he sgns on the q and w terms are completely arbtrary, but once assgned, a sgn conventon s establshed. If heat s added to the workng flud, t wll rase the workng flud s nternal energy. herefore, postve q s heat nto the system. Lkewse, snce a system that performs work, wll decrease the nternal energy, postve w s work done by the system. One could change any sgn n Equaton (1.1) and t would smply change the sgn conventon. Internal = U work = w heat = q KE = ½ mv 2 PE = mgh wave = h Elect pot = ev surface = A rad =A 4 Fgure 1.1 Conservaton of Energy Forms hroughout ths text the sgn conventon set by Eq. (1.1x) wll be used. Forms of energy other than heat, work, and nternal energy are excluded from consderaton n the Frst Law; however, when a system s encountered n whch another form of energy s mportant, such as surface energy (surface tenson) n nanoscence, the Frst Law can easly be amended. When knetc energy, potental energy, and frcton are ncluded, the resultng equaton s called the Overall Energy Balance used n the analyss of fluds n ppng systems. Such amendments are beyond the scope of ths text. he focus of ths chapter s the use of the Frst Law to analyze the behavor of gases undergong selected processes so as to lay the foundaton for understandng entropy the arrow of tme. Entropy change s the bass for predctng by computaton f a processes s at equlbrum or, f not, the drecton of spontaneous change. he typcal processes selected for consderaton are sothermal (constant temperature), sobarc (constant pressure), sochorc (constant volume), adabatc (no heat exchange). Before such processes are consdered, a revew of deal gases propertes and defnton of heat capacty s needed. 8

9 Ideal Gas An deal gas has the followng propertes: Small atoms that have volumes that are neglgble compared to the volume of the contaner enclosng the gas Atoms that store energy by translaton moton only (½ mv 2 ), whch excludes rotatonal, vbratonal, and bond energes assocated wth molecules. Atoms that have perfectly elastc collsons wth each other and wth the contaner walls Atoms have no bondng nteractons Atoms are randomly dstrbuted wthn the contaner holdng them Ideal gases observe the Ideal Gas Law PV = nr (1.2) where P = pressure V = volume n = moles of gas R = the gas constant = absolute temperature. Energy added to an deal gas can only be stored as knetc energy. hs energy s the nternal energy of an deal gas. As the energy of the gas ncreases, the velocty of the atoms ncrease. he nternal energy s ndependent of the volume and, therefore, the pressure of the gas. It s a functon of temperature only. A drect relatonshp between the nternal energy of an deal gas and temperature may be obtaned from the propertes of an deal gas by consderng an deal gas wth n atoms of gas havng speed of c. he mass of one atom s m. If the atom wth mass m moves nsde a cube wth edge of length L wth velocty components v x, v y, and v z, the force exerted by the atom n each drecton s found from the rate of momentum change mv 2mv x y 2mvz F x,f y, F z (1.3) L L L he gas pressure s the sum of all these forces for all atoms dvded by the cube s area of 6L 2 P 2 2nmc 2KE = 3 6L 3V (1.4) where the sum of the squares of the velocty components has been replaced by the atom speed squared, L 3 wth volume, and nmc 2 wth twce the total knetc energy of the gas. hs knetc energy s also the nternal energy of the deal gas, whch for one mole of gas gves PV 2 = U nr (1.5) 3 9

10 Work It takes work to compress a gas. hs work may be determned by ntegratng the defnton of work 1 2 w F ds (1.6) he force, F, s the product of the pressure P of the gas beng compressed and the area A of the pston compressng the gas as shown n Fgure 1-2. he change n pston dstance s related to the gas s volume change as shown. Substtutng these expressons for F and ds n Equaton (1.6) gves wc 1 2 PdV (1.7) Area A P F = PA ds = dv/a Fgure 1.2 Relatonshp between pressure and force and pston dstance and volume change durng gas compresson If a gas undergoes expanson, the computaton of work requres nformaton about the restranng force n Equaton (1.6). For example, t s possble to expand a gas wthout any restranng force. hs s called free expanson and such expanson performs no work. Free expanson s rarely acheved snce t requres that there s no atmospherc pressure, whch requres work to dsplace. he opposte extreme to free expanson s reversble expanson. Under ths condton, the restranng force s only nfntesmally less than the maxmum force exerted by the gas; namely, PA. he reversble work s the maxmum work that a gas can perform on expanson. herefore the work exerted on expanson can range from maxmum or reversble work all the way to the zero work of free expanson. 10

11 w E 2 Irrev PdV (1.8) 1 Rev No such range n work s needed for compresson snce the maxmum work must always be performed on compresson. For convenence and smplcty of wrtng equatons, work wll not be subscrpted as ether compresson or expanson type work. Work wll always be wrtten as though t s maxmum work and the student should must always exercse care when usng the maxmum work expresson and make the requred downward adjustments for expanson occurrng under rreversble condtons. Of course, the same care must be used n computng heat durng expanson snce t s also a path functon changng n concert wth work to equal U. Equaton (1.6) shows that the area under a plot of P versus V equals the work of compressng a gas as shown n Fgure 1-3. If the compresson proceeds along the path labeled a, the work requred to compress the gas equals the cross-hatched area n accordance wth Equaton (1.6).he work done on the gas durng compresson s negatve n agreement wth the sgn conventon of the Frst Law. he sgn s also fxed by the decrease n volume, whch makes the change n volume negatve: a good reason for Frst Law sgn conventon. hs negatve value for work s denoted n the fgure wth negatvely sloped cross hatchng. hs conventon wll be observed throughout ths text. P 2 c a 1 b Fgure 1-3 Work durng gas compresson. V If the compresson proceeds from state 1 to state 2 along path b, the work as represented by the area under the P-V curve would be less than along path a: proceedng along path c would requre more work than along path a. Snce the value of work depends on the path taken from state 1 to state 2, work s a path functon as descrbed n Chapter 0. hs also has mplcatons for heat because heat and work both appear n the Frst Law. Internal energy, whch equals the dfference n heat and work accordng to the Frst Law, depends only on the state of the system. Indeed, n the case of an deal gas, the nternal energy s a functon of 11

12 temperature only. he only way for the qualty of the Frst Law to be mantaned s for heat to change n concert wth the work functon of work so that ther dfference always equals the change n nternal energy, a state functon. hs means that both work and heat are path functons. Enthalpy Enthalpy s term defned for convenence as H U PV (1.9) Snce all terms n the defnton are state functons, enthalpy s also a state functon. he reason enthalpy s convenent s that t s equals to the heat for a reversble sobarc process provdng the only work performed s expanson type work (PdV). Frcton s an example of possble other types of work. Isobarc processes are common snce work occurrng under atmospherc condtons s essentally sobarc. he dervaton of ths useful relatonshp begns by wrtng the defnton n dfferental form dh = du + PdV. (1.10) Substtuton of the ncremental form of the Frst Law gves dh dq dw PdV (1.11) whch for reversble and only expanson type work becomes or dh dq p (1.12) H (1.13) q p Work for selected processes Equaton (1.6) may be ntegrated for deal gases under sothermal and sobarc condtons. o gve p w P V V nr (1.14) dv w nr nr V 2 2 ln (1.15) 1 V V1 here can be no compresson work under sochorc condtons snce there s no volume change. w 0 (1.16) v 12

13 Under adabatc condtons the heat s by defnton zero. herefore, accordng to the Frst Law, work s the negatve of the change n nternal energy. wq 0 U (1.17) Heat Capacty he heat capacty s the amount of heat requred to rase the temperature of a materal. If the heat capacty s for a specfed amount of materal, t s an ntensve varable. Some texts dstngush between the extensve, or total, heat capacty and the ntensve heat capacty. In ths text the heat capacty wll always refer to the ntensve varable. he heat capacty s measured and reported for both sobarc and sochorc condtons as follows: c c v p 1 q n v 1 q n p (1.18) (1.19) where q s the heat per gmole. he value of c v can be determned from Equaton (1.5) snce all the heat added to an deal gas at constant volume becomes nternal energy. herefore, c v 1 q 1U 1KE 1 32 nr 3 R n n n n 2 v v v v (1.20) he value for c p for an deal gas may be found by substtutng the Frst Law nto the defnton of c p and recognzng that knetc energy depends on only to gve 1 q 1 U w 1KE PdV 5 cp cv R R n p n n p p 2 p (1.21) Heat for selected processes he computaton of heat for sochorc and sobarc processes flow drectly from the defntons of c v and c p q nc (1.22) v v q nc (1.23) p p 13

14 For sothermal processes, the Frst Law requres that the heat assocated wth a process equals the sum of the work and nternal energy. However, the nternal energy of an deal gas s a functon of temperature only makng ts change zero. herefore, for sothermal processes nvolvng deal gases q w nrln V V 2 (1.24) 1 Of course, q = 0 for adabatc processes. U and H for selected processes Both nternal energy and enthalpy are state functons. Any equaton that relates a change n a state functon for a partcular change n state s vald for any path. hs means that a dervaton for a change n a state functon that reles on a certan path as part of the dervaton does not encumber the result wth the path constrants. For example, n the case of nternal energy one may wrte based on the defnton of c v dq nc d (1.25) v but dq du snce dw 0 for a constant volume process. herefore, du nc d (1.26) v Now the sgnfcance of state functons and the above comments come nto focus. Even though the dervaton reled on the assumpton of constant volume, there s no such restrcton on the resultng equaton because nternal energy s a state functon. herefore, the change n nternal energy for any process can be computed from U ncv (1.27) In the case of enthalpy dq nc d (1.28) p p but dq dh. herefore, p H ncp (1.29) for all processes. 14

15 emperature-pressure-volume relatonshps for selected processes For a specfed number of moles, the Ideal Gas Law s a constrant between the three state varables V, P, and. Specfyng any two of these three varables fxes the remanng state varable. If the system then undergoes a specfed process (sothermal, sobarc, sochorc, adabatc), only one fnal state varable needs to be specfed to fx all state varables n the fnal state. Useful relatonshps for each of the specfed relatonshps are now gven. Isothermal: PV constant P1 V2 P 2 V1 (1.30) Isochorc: P P1 1 constant P2 2 (1.31) Isobarc: V V2 2 constant V (1.32) 1 1 Reversble adabatc: Snce ths condton nvolves heat rather than any of the three terms n the Ideal Gas Law, an ndrect method for arrvng at -P-V relatonshps s requred. he dervaton begns wth the Frst Law wth dq = 0 du dw (1.33) Substtuton for each term gves ncvd PdV (1.34) Substtuton of the Ideal Gas Law for P and rearrangng gves d R dv (1.35) c V v Upon ntegraton the relatonshp between state 1 and state 2 temperatures and volumes s determned. 2 V 1 V 1 2 R cv (1.36) he Ideal Gas Law can be used to replace ether the temperatures or volumes wth pressures to gve 2 P 2 P 1 1 R cp P 2 V 1 P V 1 2 cp cv (1.37) 15

16 P vs V Plots he P versus V plot wll be used throughout the early chapters of ths text so addtonal dscusson of them s warranted. Fgure 1.4 shows paths for sobarc, sochorc, sothermal, and adabatc compresson of an deal gas. he reversble-adabatc process s steeper than an sothermal process snce the work of compresson ncreases the temperature of the gas whereas the temperature remans the same durng an sothermal process. P dq = 0 (adabatc) dp = 0 (sobarc) dv = 0 sochorc d = 0 (sothermal) V Fgure 1-4 Paths for selected processes. Summary able 1 summarzes the equatons used to compute work, heat, and changes n nternal energy and enthalpy for selected processes as well as the assocated -P-V relatonshps. By the tme students reach ther upper-level classes, they should not navgate thermodynamcs by attemptng to memorze the contents of able 1 but rather learn to use ther prevous knowledge to derve the contents of able 1. he frst method s memorzaton whereas the latter establshes a logcal bass of learnng that s more endurng than the results of memorzaton. o ths end, the followng table s the bass from whch everythng n able 1 s obtaned. able 2 reduces to the Ideal Gas Law, work beng the ntegral of PdV, the defntons of heat capactes, and the Frst Law. he student who can begn wth these four fundamental deas to obtan all of the equatons enumerated n able 1 wll fnd masterng Chapters 1 and 2 greatly smplfed compared to the student who tres to memorze or who must contnually refer to able 1. 16

17 able 1 Summary equatons for selected deal gas compresson processes d=0 dp=0 dv=0 Rev adabatc 2 w nr ln V PV 2 V1nR U V q w ncp U 0 ncv 1 ncv ncv ncv 0 H 0 ncp ncp ncp 2 1 Rcp V2 P2 1 P 2 V 1 1 V 1 P P1 V2 P 2 V1 2 P1 P V 1 P 1 V 1 1 P 2 1 P1 1 2 V2 V 2 P1 2 V1 V 1 P 1 1 V P 2 1 V1 V1 1 2 P2 Rcv cp R cp cv cv R cv cp able 2 Summary of fundamental bases for the equatons for selected deal gas compresson processes Orgnatng concept Addtonal nformaton or result w 2 w PdV ; PV nr 1 1 q cv n ; dv=0; qv ncv U v q 1 q cp n ; dp=0; qp ncp H p U qw 0 ; d=0; q w 1 q 1U U cv n v n ; U ncv 1 q 1H H cp n p n ; H ncp PV 2 2 PV 1 1 For 1 = 2, P 1 =P 2, or V 1 =V 2 2, P 2, V du dw ; ncvd PdV For reversble adabatc processes Example Problems 17

18 Problems here are almost as many formulatons of the second law as there have been dscussons of t. Phlosopher / Physcst P.W. Brdgman, (1941) 18

19 Chapter 2 he Second Law of hermodynamcs he Arrow of me Basc Concepts he Zeroth Law of hermodynamcs Defntons Mathematcal Requrements If we see a flm clp of a match beng struck, burstng nto flame, and slowng burnng out, we thnk nothng beyond the process as vewed; however, f the clp s run n reverse we are mmedately aware that we are observng somethng mpossble. he mpossblty of the process s precsely the reason the process grabs our attenton. Rocks do not run up hll. Explosons do not collect tny fractured bts of debrs together through a huge freball that contnues to shrnk nto some undestroyed structure rgged wth an explosve devce. Each of us has learned from an early age that natural processes have a predctable sequence that s called n ths text the arrow of tme. Learnng ths arrow of tme the drecton of natural processes - provdes the predctablty necessary to our functonng. In the case of chemcal reactons, we understand the drecton of many processes but not all. Processes such as combuston of natural gas wth ar or the reacton of vnegar wth bakng soda are known, but few people would know by experence what, f any, reacton mght occur f alumnum oxde s mxed wth carbon and heated to 1000 C. It turns out no reacton occurs at atmospherc pressure, but the more nterestng queston s can one calculate f a reacton s favorable and what s the bass for makng such a computaton. Often students suggest that predctng the down-hll drecton of rocks nvolves nothng more than determnng the drecton for the reducton of potental energy. hat works for rocks but potental energy has no value n tryng to predct the drecton of chemcal reactons. In that case students often - and ncorrectly thnk that a reacton s favorable f t lberates heat, but f t were so smple then endothermc reactons would never occur: yet they do. herefore, there must be more to the predcton of reactons than smply thermcty. he Second Law of hermodynamcs defnes a quantty called entropy that allows predctng the drecton of chemcal reactons, the drecton rocks roll, and the drecton of all physcal processes. Fgure 2.1 llustrates the drecton of change n three systems. In Fgure 2.1 a) a pendulum moves from poston A towards B whle at poston B t moves toward A. At equlbrum t rests between A and B. In Fgure 2.1 b) spontaneous change s shown n terms of ce and water, whch s at equlbrum at ce s meltng temperature. Fgure c) shows a general spontaneous change and equlbrum n a system comprsed of materal A and B. In all processes, spontaneous change can be shown to be accompaned by an ncrease n total entropy. For an unfamlar reacton, the drecton of the reacton s unknown so one assumes t proceeds as wrtten. For example, a reacton mght be assumed to proceed to the rght when wrtten as 19

20 Ice Water > 0 C A B A Ice Water B < 0 C A B Ice = Water A = B = 0 C a) b) c) Fgure 2.1 hree possble scenaros for a system nvolvng State A and State B. aa + bb cc (2.1) and the total entropy change computed. If the total entropy change > 0, then the assumed drecton s correct. On the other hand f total entropy change < 0, then the actual spontaneous process drecton s the opposte of the assumed drecton and the drecton of the reacton s to the left. A zero total entropy change ndcates the reacton s at equlbrum. he arrow of tme corresponds to ncreases n total entropy. Such processes are spontaneous or natural processes. Now that the usefulness of the entropy s establshed, the value of the Second Law of hermodynamcs from whch entropy arses should be greatly apprecated. he Second Law of hermodynamcs s based on observaton of the natural world and permts the predcton of the processes drecton through the computaton of total entropy change. hs drecton corresponds to the arrow of tme. he Second Law of hermodynamcs n verbal form s stated as follows: It s mpossble to take a quantty of heat from a body at unform temperature and convert t to an equvalent amount of work wthout changng the thermodynamc state of a second body. Most students do not fnd much meanng n ths statement of expermental observaton. Fgure 2.2 s the author s pedagogcal devce for makng the Second Law more understandable. he fgure conssts of a tank of water at unform temperature wth a thermometer along the left sde to measure the water s temperature. A weght on the rght sde s connected to a mxer nsde the tank so that as the weght falls the mxer spns. In the normal course of events, the potental energy of the weght s converted to heat nsde the tank of water and the water s temperature ncreases. However, the Second Law s stated for the mpossble case n whch heat from the water as ndcated on the dagram by the fallng water temperature, s converted to work that lfts the weght. hs s mpossble. It has never been observed and try as one mght there has never been a reproducble experment n whch heat from a body at unform temperature (the water n ths case) has been converted to an equvalent amount of work (rasng the weght n ths case) wthout changng the 20

21 thermodynamc state of a second body. A second body s not needed n the example n Fgure 2.2 snce there s no change n any such body n the mpossble case. Fgure 2.2 he Second Law of hermodynamcs n graphcal format. he mathematcal statement of the Second Law s for a closed system 0 he left term s the defnton of entropy change. (2.2) (2.3) he defnton requres that the change n entropy s computed under the constrant of reversble condtons. here are two prmary processes that wll be of nterest n ths chapter: heat transfer and gas expanson. Reversble means dfferent condtons for each. For heat transfer the entropy change for a sngle body s computed usng the body s temperature for the n the denomnator of the defnton of entropy. In the case of gas expanson, reversble requres that the expanson occurs reversbly, whch means that the gas performs maxmum work durng the expanson. It makes no dfference f the actual expanson occurred reversbly. Gas compresson s always reversble so maxmum work s always performed durng compresson. 21

22 When a gas undergoes expanson, the entropy change for the gas s the same regardless of whether maxmum work was derved from the expanson or not. However, ths does not mean that the total entropy change for the expanson s the same because the total entropy change ncludes the change n the heat snk, whch undergoes a consderably dfferent entropy change dependng on how much work the gas actually does durng the expanson. It s mportant to remember that reversble means dfferent condtons for dfferent knds of processes. For the gas t means maxmum work: for the heat snk t means the n Equaton (2.3) s the temperature of the heat snk and has nothng to do wth the amount of heat transferred to the gas as t undergoes ts expanson reversbly or not. Example 2.1: Fnd the total entropy change for 1000 joules of heat conductng from a massve body of copper at 500K to another massve body of ron at 400 K. Cu 500K q = 1 kj Fe 400K Possble Example 2.2: Fnd the total entropy change for 1000 joules of heat conductng from a massve body of copper at 500K to another massve body of ron at 500 K Equlbrum Example 2.3: Fnd the total entropy change for 1000 joules of heat conductng from a massve body of copper at 400K to another massve body of ron at 500 K, whch s mpossble

23 Impossble, he heat actually goes the other drecton. Example 2.4: Fnd the total entropy change when a rock at 300 K rolls down a hll. Assume the rock s potental energy change, whch ends up as heat n the hll, equals 3,000 Joules and that the rock (after some tme) returns to ts orgnal temperature of 300 K Possble Example 2.5: Fnd the total entropy change for the sothermal expanson of two moles of deal gas at 10 atm and 500 K to 1 atm n contact wth a heat snk at 500 K whle performng a) maxmum work, b) 70 percent of the maxmum work, c) no work. he entropy change for the gas s the same regardless of how the much work the gas actually does durng the expanson. 210 he entropy change for the snk s the amount of heat from the snk dvded by the snk s temperature. he amount of heat from the snk to the gas s the same as the work actually performed by the gas. he gas s heat gan s the snk s loss; therefore and a) Maxmum work

24 Reversble process conducted under equlbrum condtons b) 70 Percent maxmum work Possble and conducted under rreversble condtons c) No work Possble but conducted under rreversble condtons Example 2.6: Fnd the total entropy change for the same condtons as Example 2.5a except for a snk temperature of a) 700 K and b) 300 K. he entropy change for the gas s the same regardless of how the much work the gas actually does durng the expanson. 210 he entropy change for the snk s as before a) Snk =700 K 210 Possble but conducted under rreversble condtons. a) Snk =300 K 210 Impossble because heat cannot flow from a 300 K heat snk nto a gas at 500 K; however, the process could occur n the opposte drecton (.e. - compresson rather than expanson)

25 he Four Propostons he Second Law of hermodynamcs s sad to be stated as many ways as those wrtng about t. he statement used n ths chapter s based on Denbgh 1. Denbgh s treatment of thermodynamcs n Prncples of Chemcal Equlbrum s beautfully concse and formal. It s recommended for students who have completed the correspondng subjects n ths text. Denbgh presents four propostons arsng from the Second Law statement. he proofs for these propostons are gven n Appendx A. he propostons are stated here. Proposton 1 For a Carnot cycle operatng between a heat snk at hotness t 1 and hotter snk at hotness t 2 and exchangng the correspondng heats q 1 and q 2 to the workng flud that, (2.4) Proposton 2 he hermodynamc emperature Scale s defned by the rato of heats exchanged n a Carnot cycle. (2.5) Proposton 3 Entropy defned as (2.6) s a state functon. Proposton 4 he change n entropy s zero for a reversble process and greater than zero for an rreversble process. 0 (2.7) 25

26 Chapter 3 he hrd Law of hermodynamcs and Entropy of Mxng Basc Concepts Heat capacty s varaton wth temperature Entropy of pure crystallne materals at absolute zero Entropy of deal mxng Expermental measurements for the heat capacty of a pure crystallne materal have the form shown n Fgure 3.1. As the temperature approaches absolute zero, the heat capacty not only approaches zero but so does ts slope. he reason for ths s that as the temperature s lowered towards absolute zero the crystallne structure has fewer and fewer ways of storng energy. At absolute zero there s no addtonal way to store or remove energy from the crystal. Accordng to Equaton(2.3), the zero slope of Cp as the temperature approaches absolute zero requres a constant value of entropy. lm lm lm 0 (3.1) Cp 0 0 Absolute Fgure 3.1 Heat capacty s varaton wth absolute temperature In 1923 Glbert N. Lews and Merle Randall stated the hrd Law of hermodynamcs as follows: If the entropy of each element n some (perfect) crystallne state be taken as zero at the absolute zero of temperature, every substance has fnte postve entropy; but at the absolute zero of temperature the entropy may become zero, and does so become n the case of perfect crystallne substances. he value of the hrd Law s that the entropy s a materal property that may be computed from expermental data. Snce the entropy of a pure crystallne substance s zero at absolute zero, the absolute value of entropy may be computed at hgher temperatures by ntegratng Equaton (2.3) where to gve 26

27 0 S (3.2) Entropy of Mxng: confguratonal entropy he total entropy change for spontaneous processes s greater than zero. hs has been shown true for all processes and llustrated for gas expanson processes and for heat exchange. A thrd spontaneous change observed n nature s that of mxng. When two gases are placed n the same contaner, they spontaneously mx. he prevous treatment does not offer a way by whch the entropy of mxng may be computed. Boltzmann consdered mxng from the atomstc perspectve. He consdered the probabltes of varous mxng arrangements or confguratons. For a system of atoms of A and B where each A s ndstngushable from other A atoms and the same beng true for atoms of B, the entropy of the system of atoms s then gven by (3.3) where k = Boltzmann s constant (R/Avogadro s Number) = Number of confguratons In the case of sx atoms of A and sx atoms of B, the number of mxed confguratons s computed as N N! N!N! 12! 924 6! 6! Snce the entropy of the unmxed system has only one confguraton, ts entropy s 0. herefore, the entropy of mxng for ths deal mxng process s S M,ID = kln(924). For systems composed of large numbers of atoms, large factorals become an mposng computatonal obstacle. hs can be overcome by employng Sterlng s Approxmaton for large factorals lnn! NlnN N (3.4) In the case that N A + N B = A Avogadro s Number, the entropy of mxng for one mole of atoms s 27

28 S M, ID NA NB! kln NA! NB! A Bln A B A B ln ln k N N N N N N kn N N kn N N A A A B B B (3.5) whch becomes M, ID S R xa ln xa xbln xb (3.6) where R A k, x A N A N N A B N B, xb NA N B Example What s the confguratonal entropy change for 2 moles of Au mxng wth 8 moles of Ag? Soluton: 2 xau xag S M, ID = R0.2ln ln0.8, per mole Answer: S M, ID =, 10S M ID, for 10 moles 28

29 Chapter 4 he Auxlary Equatons - he Equaton ool Kt Basc Concepts Auxlary Functons Fundamental Equatons for a Closed System hermodynamc Relatonshps dervng from the otal Dfferental Maxwell Relatonshps he Crtera of Equlbrum o Constant and V o Constant and P he Gbbs-Helmholtz Equaton Chemcal Potental Fundamental Equatons for an Open System hs chapter conssts of defnng the auxlary functons, whch are useful combnatons of prevously-known state functons such as U, P, V,, and S. hese auxlary functons are defned for convenence only. hat s to say that the groups of state functons are partcularly useful and so commonly used that assgnng these combnatons of varables ther own names makes usng them easer. Auxlary Functons he auxlary functons are defned as follows: Enthalpy: H U +PV (4.1) Helmholtz Energy: A U -S (4.2) Gbbs Energy: G H -S (4.3) It has been shown n Chapter 2 that a change n enthalpy corresponds to the heat for a process conducted reversbly at constant pressure. Helmholtz energy wll be shown below to be the crteron of equlbrum for a system at constant and V whle Gbbs energy s the crteron for equlbrum n systems at constant and P. Fundamental Equatons for a Closed System he frst of the four Fundamental Equatons for a closed system s obtaned by substtutng nto the Frst Law the defnton of entropy and PdV for maxmum work. du = ds PdV (4.4) Snce the defnton of entropy requres reversble condtons, the correspondng maxmum work s substtuted for the work term. he resultng equaton mght be thought to apply only to reversble processes snce t was derved makng such an assumpton but ths s not a constrant that needs to be mantaned snce all of the terms n the resultng fundamental equaton are state varables. hat s, the changes n the state varables n Equaton (4.4) are the same for a process whether conducted reversbly, or not. 29

30 30 he remanng three fundamental equatons are obtaned by dfferentatng each auxlary equaton and substtutng Equaton (4.4) and the resultng fundamental equatons to obtan dh = ds + VdP (4.5) da = -PdV Sd (4.6) dg = VdP Sd (4.7) hermodynamc relatonshps derved from the total dfferental he total dfferental for a functon z(x, y) s y x z z dz dx dy x y (4.8) If each of the Fundamental Equatons are wrtten n terms of the dfferands, an equaton may be wrtten for each state varable n terms of the slope of the ndependent varable. For example, for the frst Fundamental Equaton, U(S, V), for whch the total dfferental s V S U U du ds dv S V (4.9) Comparson of the coeffcents for ds and dv gves V U S and S U P V (4.10) Contnung ths process for all of the fundamental equatons gves the followng relatonshps: V P U H S S (4.11) S U A P V V (4.12) S H G V P P (4.13) V P A G S (4.14)

31 31 Maxwell Relatons he Maxwell Relatons are a consequence of the qualty f exactness, whch for functon z(x, y) s y x x y z z x y y x (4.15) For the frst Fundamental Equaton, the functon U = f (S, V) the qualty of exactness s wrtten as V S S V U U S V V S (4.16) Substtutng Equaton (4.11) nto Equaton (4.16)gves the frst Maxwell Relaton. S V P V S (4.17) Repeatng for the remanng three fundamental equatons gves the remanng three Maxwell Relatons. S P V P S (4.18) V P S V (4.19) P V S P (4.20) Crtera of Equlbrum he Helmholtz energy was defned to descrbe the crteron on equlbrum at constant and V. hs s shown by consderng a change n A

32 da = du - d(s) (4.21) From the Frst Law and mposng d = 0 gves da = dq - dw - ds (4.22) f the work s comprsed of expanson-type work (PdV), whch s zero, and other work dw, then da = dq ds - dw (4.23) For a reversble process, dq = ds. For an rreversble process, such as an expanson of a gas, dq s always less than ds. Snce and ds > 0 and dq < 0, dq ds s negatve and da - dw (4.24) herefore, a system at constant and V that performs no w, wll reach equlbrum at mnmum Helmholtz energy as shown n Fgure 4.1. Furthermore, the maxmum w that a system at constant and V can produce s - da. he Gbbs energy was defned to descrbe the crteron on equlbrum at constant and P. hs s shown by consderng a change n G dg = dh - d(s) (4.25) Substtutng for dh = du +d(pv) and the Frst Law gves dg = dq PdV - dw+ PdV - ds (4.26) whch reduces to the same rght hand sde as Equaton (4.23). Usng the same logc for arrvng at Equaton (4.24) gves dg - dw (4.27) herefore, a system at constant and P that performs no w, wll reach equlbrum at mnmum Gbbs energy as shown n Fgure 4.2. Furthermore, the maxmum w that a system at constant and P can produce s - dg. 32

33 100 A 0 Equlbrum 0 10 Process Extent Fgure 4.1 Processes at Constant and V wth expanson only work move towards mnmum Helmholtz energy. 100 G 0 Equlbrum 0 10 Process Extent Fgure 4.2 Processes at Constant and P wth expanson only work move towards mnmum Gbbs energy. Gbbs-Helmholtz Equaton Gbbs energy s use frequently because most systems of practcal nterest are at constant temperature and pressure. Snce temperature s a major system varable under the control of the expermentalst or the processng engneer, the queston naturally arses about the effect of changng a system s temperature on G. hs s easly determned by the Gbbs-Helmholtz Equaton, whch s derved by substtutng Equaton (4.14) nto the defnton of Gbbs energy. G G H S H P (4.28) Multplyng by d and rearrangng whle constranng the equaton for constant P gves 33

34 whch may be wrtten dg d or dg Gd Hd (4.29) H (4.30) 2 d G H d 1 (4.31) he Gbbs-Helmholtz Equaton could be wrtten for two states and the dfference n the equatons for the two states would be d G d 1 H (4.32) If H and G are for a chemcal reacton, the Gbbs-Helmholtz Equaton shows that an endothermc ( H > 0) reacton becomes more favorable ( G decreases) wth ncreasng. Chemcal Potental Chemcal potental s as mportant to chemcal processng as voltage s to electrcal engneerng. Chemcal potental s defned as G (4.33) n, P, nother he chemcal potental of a component s the Gbbs energy per mole of that component. It can apply to a component n the pure sold or lqud state, the gaseous state, or n a sold or lqud soluton. o understand the concept, one may thnk of the total Gbbs energy of a system, G, composed of a large number of moles of not only component but also other components, n other. he addton of one mole of component, dn = 1, to the system wll change the systems Gbbs energy by dg. hs change whle keepng and P constant s the Gbbs energy of that one mole of component n the system. Of course one could add less than one mole of component, dn, and the change per mole would reman the same. he chemcal potental s smply the slope of a plot of G vs. n whle keepng, P, and n other reman constant as shown n Fgure

35 100 G Slope =, P, n Other Constant n Fgure 4.3 he chemcal potental s the slope of a plot of the total system G vs. n whle keepng, P, and n other constant. Fundamental Equatons for open systems he frst of the four Fundamental Equatons s derved by wrtng the same functon relatonshp for U as used n the dervaton of Equaton (4.10) but ncludng the moles of the components that may be changng (enterng or leavng) n system U(S, V, n 1 n 2, --- n j ). he total dfferental s then du ds dv dn J U U U (4.34) S V, n V, 1 Other S n n Other V, S, nother he frst two dfferentals are known from Equatons (4.11)-(4.14) to be and P. herefore, U du ds PdV dn J 1 n (4.35) V, S, nother Repeatng ths same procedure for the remanng three Fundamental Equatons gves H dh ds VdP dn J 1 n (4.36) S, P, nother A da PdV Sd dn J 1 n (4.37) V,, nother G dg VdP Sd dn J 1 n (4.38), P, nother 35

36 Each of the summaton terms can be shown to be equal by substtutng Equaton (4.35) nto the defntons of H, A, and G. In the case of H ths gves U dh du PdV VdP ds PdV dn PdV VdP U dh ds VdP dn J 1 n (4.39) V, S, nother J 1 n (4.40) V, S, nother Comparson of Equaton (4.40) and Equaton (4.36) gves the desred result. H U J J dn dn 1 n S, P, n 1 n (4.41) Other V, S, nother By the same procedure, the followng may also be shown A U J J dn dn 1 n V,, n 1 n (4.42) Other V, S, nother and H G J J dn dn 1 n S, P, n 1 n (4.43) Other, P, nother he rght term s the defnton of the chemcal potental defned n Equaton (4.33). herefore, each of the summaton terms may be replaced wth the chemcal potental to gve the Fundamental Equatons for an open system. j du ds PdV dn (4.44) 1 j dh ds VdP dn (4.45) 1 j da PdV Sd dn (4.46) 1 j dg VdP Sd dn (4.47) 1 36

37 he Fundamental Equatons for a closed system are the same except the summaton terms are all zero snce the number of moles of each speces remans constant n a closed system. 37

38 Chapter 5 Enthalpy and Entropy Changes Basc Concepts hree knds of enthalpy and entropy changes sensble, phase transformaton, and reacton Arbtrary-defned system for Enthalpy: H, Enthalpes of formaton:, H Form Entropy: S Enthalpy of reacton at 298 K from heats of formaton: H R, Calculaton schematc for enthalpes and entropes at other than 298 K Gbbs energy of reacton: G R, Uses of the calculaton Schematc o Adabatc flame temperature: AF o ransformaton reacton values at other than the equlbrum Databases: JANAF, hermocalc By ths end of ths chapter, the student wll be able to calculate whether a chemcal reacton wll occur at a specfed temperature. hs wll be accomplshed by calculatng the change n the Gbb s energy of the reacton, whch from Equaton (4.3) s G H S R, R, R, (5.1) at constant. One sees from ths the mportance of beng able to compute H R, and S R,. he means of performng such computatons presented now. hree knds of enthalpy and entropy changes Heat can be stored or released from a materal n three ways. Sensble heat s the heat that changes the temperature of the materal. It s related to the heat capacty of the materal. It s called sensble heat because one can sense the flow of heat from a warm or cold object. In ether case, heat s flowng from or to the materal touched and ether cools down or warms up because of the loss or gan of heat. Equaton (1.19) may be rearranged and ntegrated to fnd sensble enthalpy (heat) changes. Sens 2 H Cpd (5.2) 1 he defnton of entropy gven n Equaton (2.3), may be ntegrated to fnd sensble entropy changes. 2 Cp SSens d (5.3) 1 38

39 Phase transformaton heat s the heat requred or released because of a phase change such the meltng of ce or the vaporzaton (bolng) of water. Meltng ce or bolng water remans at the same temperature durng the phase transformaton process. Phase transformaton heat s also called latent heat frst observed scentfcally by Scottsh Joseph Black n Some clam ths to be the begnnng of hermodynamcs. James Watt used t to mprove the steam engne and the sprt dstllaton ndustry (scotch dstllers) better understood ther craft as well. Fgure 5.1 shows a typcal plot of vs. tme of a pure, molten Pb as t cools to ts meltng pont and then further cools. Note how the coolng rate of the hot molten lead slows as ts temperature decreases followed by a long perod at constant temperature whle the heat of fuson s gven up as the Pb goes from a lqud to a sold. After complete soldfcaton, the temperature drop resumes at the same rate as the coolng lqud just above the meltng pont but slows more slowly as t becomes cooler. Values for the enthalpy (heat) of a phase transformaton must be determned expermentally. Values for computatons are obtaned from thermochemcal databases., C Phase ransformaton Heat Loss Sensble Heat Loss t, mn. Fgure 5.1 Coolng curve for soldfcaton of Molten Pb he heat of reacton s the heat released or requred when a chemcal reacton occurs. he most common example of a heat of reacton s combuston. For example, the burnng of methane produces heat. CH 4 + 2O 2 = CO 2 + 2H 2 O H, kcal/ gmole R K (5.4) Values for the enthalpy (heat) of reacton must be determned from expermental data. Values for computatons are obtaned from thermochemcal databases. he next secton 39

40 descrbes how the data requred to compute the nearly countless possble reactons can be obtaned from a relatve few heats of reacton; namely, from heats of formaton. Heats (enthalpes) of formaton able 5.1 Heats of transformaton for selected compounds and elements Speces State ransformaton H ran,298 K emperature (cal/gmole) K C graphte damond 298 * 453 Cu s l ,170 Cu l g ,743 H 2 O s l 273 1,436 H 2 O l g 373 9,718 Zr * Not at equlbrum Data from JANAF hermochemcal ables and hermocalc Enthalpy of formaton he heat of formaton for any compound or element at a specfed temperature s defned as the enthalpy change assocated wth the reacton to form the element or compound from the most stable form of elements at the specfed temperature. For example the heat of formaton for carbon doxde gas at 500 K s the enthalpy change for the reacton C graphte,500 K + O 2 gas,500 K = CO 2 gas, 500 K H Form,500 K (5.5) Elements n ther most stable form have a zero heat of formaton by defnton, but elements not n ther most stable form are not zero. For example, the heat of formaton reacton for damond at 298 K s 453 cal/gmole. C graphte,300 K = C damond, 300 K H, cal / gmole Form K (5.6) able 5.2 shows selected heats of formaton at 298 K. he entropes values n able 5.2 wll be dscussed a lttle later n ths chapter. All of the enthalpes, be they sensble, transformaton, or reacton, are changes n H: not absolute values of enthalpy. Indeed, one cannot determne the absolute entropy value because t s defned n terms of the nternal energy (H U+PV), whch n turn s not knowable. Such energy would nclude all the energy of bondng, electron moton, as well as the energy of all the subatomc partcles that comprse protons, neutrons, and electrons. Perhaps, at some level the very mass mght be thought of as energy snce the two are related (e = mc 2 ). Consequently, we accept that absolute enthalpy s not measurable, but rather 40

41 changes n enthalpy are. Sensble heat s calculable from the temperature change usng Equaton (5.2); transformaton heats are measured as the change n heat requred to transform a materal from one state to another; reacton heat s the heat assocated wth reactants becomng products. able 5.2 Heats of formaton and entropes at 298 K for selected compounds and elements Speces State H Form,298 K S 298K (cal/gmole) (cal/k gmole) C damond C 60 buckeyballs 578, CH 4 g -17, CO g CO 2 g -94, H 3 PO 4 s -301, H 2 O s H 2 O l -67, H 2 O g -57, Hg s Hg g 14, O 3 g 34, Arbtrary-defned system for Enthalpy hs need to always refer to enthalpy as a change has seemed unnecessarly bothersome by some users of thermodynamc data, and so they have devsed an alternate, but acceptable arbtrary way of thnkng about enthalpy. In ths scheme the enthalpy of a pure element n ts most stable form at 298 K and 1 atm s arbtrarly assgned an enthalpy of zero. hs s not unlke assgnng a pont on the Vrgna coast as sea level: elevaton 0. he Pacfc Ocean sea level s several feet hgher (at Panama) and that tdes move sea level regularly and substantally, but snce elevatons are a relatve measure, any arbtrary zero wll stll provde the sought changes n elevaton. So t s wth enthalpy n the present case. Absolute enthalpes mght be compared to actual elevatons whch would be measured from the center of the Earth: very dffcult to do and a waste of effort snce only changes n elevaton are needed. Entropy Unlke enthalpy, absolute entropy values are known. As descrbed n the chapter on the hrd Law, the entropy of a pure crystallne materal s zero at absolute zero. Gven measured values of heat capacty from near absolute zero to any temperature of nterest, makes possble the computaton of absolute entropy by ntegratng Equaton (2.3). able 5.2 shows selected values of entropy (absolute values) determned by such ntegraton. 41

42 Entropy changes may also be computed for changes n temperature (sensble entropy), transformatons (entropy of transformaton), and for reactons (entropy of reacton). Sensble entropy changes are computed from the defnton of entropy n Equaton (2.3). (5.7) In the case of sensble heats ths s the same as Equaton (3.2) 0 S Sen (5.8) he entropy of transformaton at constant pressure s the enthalpy of transformaton dvded by the equlbrum temperature of the transformaton. he condton of constant pressure assures that the enthalpy change s the heat of the transformaton and the equlbrum temperature conforms to the requred reversble condtons. For example the entropy of transformaton for the fuson of Cu s s Cu s = Cu l (5.9) H cal cal K K fuson, Cu 3,170 Sran, Cu, fuson fuson, Cu 1356 (5.10) hese entropes of transformaton are so easly computed from the enthalpy and temperature of transformaton, they are often not tabulated. It s presumed a person traned n thermodynamcs would know ths fundamental concept. he same dea may be used to compute the reacton entropy change f the reacton f the heat of reacton s known when the reacton s at equlbrum. hs s an unusual condton and the heat of reacton at equlbrum s rarely easly found. Consequently, entropes of reacton are typcally computed by totalng the entropes of the reacton products and subtractng the sum of the reacton reactants. For example, the entropy change for the reacton s CO 2 + C graphte = 2 CO (5.11) S 2S S S R,298 K CO,298 K CO2,298 K Cgrapte,298K (5.12) 42

43 Heats of reacton at 298 K from heats of formaton he last mportant pece of nformaton needed to compute the enthalpy and entropy for any reacton at 298 K s how to use heat of formaton data to fnd the heat of any reacton. Currently the only data presented s for heats of formaton, whch by defnton are reactons from the elements n ther most stable form. hese may be used to determne the heat of any reacton. Consder the reacton gven n Equaton (5.11). One may unform the CO 2 nto ts elements n ther most stable form and then reform the elements nto the products. he sum of all of the deformng and reformng reactons equals the enthalpy change of the reacton. Each of the deformng steps s the negatve of the deformed compound and the reformng enthalpes are the heats of formaton. Fgure 5.2 shows the deformng of CO 2 and the formng of 2CO from these deformng products and the orgnal mole of C. he sum of these enthalpy changes s the heat of reacton H 2H H R,298 K form, CO,298 K form, CO2,298K (5.13) hs s more smply remembered as the sum of the heats of formaton of the products (the fnal state) mnus the sum of the enthalpes of the reactants (ntal state). It s of the same form as Equaton (5.12), except the heat of formaton of graphte s not wrtten snce t s zero beng an element n ts most stable form. CO 2 + C graphte O 2 + C graphte = 2 CO Fgure 5.2 Schematc showng the computaton of the heat of reacton from the heats of formaton Calculaton schematc for enthalpes and entropes at other than 298 K he prevous computatons for enthalpy and entropy or reacton were at the same temperature as the data: namely 298 K. In ths secton, the method of computng reacton enthalpes and reacton entropes at other temperatures usng heat capactes wll be developed. It wll be assumed that the only data avalable conssts of heats of formaton at 298 K, entropes at 298 K, transformaton data at the temperatures of transformaton, and heat capacty data, whch s ftted to the followng functon of temperature Cp = a + b + c -2 (5.14) 43

44 he method wll be developed for enthalpes. It s then easly extended to entropy computatons. he method used s based on the concept of state functon. he enthalpy changes from reactants to products wll necessarly equal the change along any path. he methods presented earler cannot be used for the drect path because there s no heat of formaton data avalable at temperature. herefore, the reactants wll be taken along an ndrect path along whch all the enthalpy changes can be computed that ends wth the products n ther fnal state. hen sum of all the enthalpy changes along the path wll be the enthalpy change for the desred drect path. hs s llustrated f Fgure 5.3 below. he Enthalpy of reacton at 500 K cannot be computed drectly but f the CO 2 and the C are cooled to 298 K and reacted there where the heats of reacton are avalable, the CO product can be heated to 500 K endng where t s ntended to be n the orgnal reacton. Snce the sum of the enthalpy changes are the same by any path, the Heat of reacton at 500 K may be computed usng H R,500 K 500 K CO 2 + C graphte = 2 CO H 1 H 2 H 3 H R,298 K 298 K CO 2 + C graphte = 2 CO Fgure 5.3 he computaton schematc for the enthalpy of reacton at 500 K from heat of formaton data at 298 K. H R,500 K = H R,298 K + 3 H (5.15) 1 he three enthalpes n the summaton are each sensble heats that may be computed as follows: K 298K 500K 2 500K CO2 500K C 298K CO H Cp d Cp d Cp d (5.16) Equaton (5.15) s then smplfed to 44

45 R,500 K R,298K 500K H H Cpd 298K (5.17) where Cp Cp Cp Cp 2 CO CO C 2 he term Cp s smply the heat capacty of products (fnal state) mnus the heat capacty of the reactants (ntal state). One can reduce the computaton effort by combnng the coeffcents n Equaton (5.14), When ntegrated Equaton (5.17) becomes 3 1 b H a( ) c (5.18) where a 2 a ; 2 ; 2 CO a a CO C b bco b bc c cco c cc 2 CO2 CO2 he schematc presented n Fgure 5.3 was easy to construct because as the reactants and products n the partcular example do not go any phase transformatons between 298 and 500 K. In case where phase transformatons occur, they must be ncorporated nto the path constructed. Unfortunately, use of the greatly smplfed Equaton (5.15) s no longer mathematcally vald. An example of such a reacton s the oxdaton of Pb at 1000 K Pb melts at 600 K. he calculaton schematc for the oxdaton s shown n Fgure 5.4. he frst sensble heat for Pb now conssts of two sensble heat 1l and 1s and one heat of transformaton 1t. Start H R,1000 K 1000 K Pb l + O 2 = PbO End H 1l H 2 H K Pb l Pb s H 1t H 1s H R,298 K 298 K Pb s + O 2 = PbO Fgure 5.4 he computaton schematc for the enthalpy of Pb oxdaton at 1000 K from heat of formaton data at 298 K. Some students fnd useful thnkng of the computaton schematc as a rver-crossng task. he reactants are at the start on one bank and the products are at the end locaton on the other bank. he reactants are all moved to the brdge where they react to the products, whch are then moved to the fnal endng locaton. he known heat of formaton s the brdge. As one 45

46 moves reactants or products downstream (lower ) or upstream (hgher ) the phase transformaton steps must be ncluded n the accountng of changes n enthalpy. Entropy of reacton he same calculaton schematc followed to fnd the enthalpy of a reacton may be used to fnd the entropy of the reacton. Entropes are typcally reported at 298 K. he reactants may be moved usng sensble and transformaton entropy changes, as needed, to reach 298 K where they may be reacted to form products that are then rased to the fnal temperature usng the requred sensble and transformaton entropes needed to move the products to the endng temperature. Fgures 5.3 wrtten for entropy computatons s shown n Fgure 5.5 and Equaton (5.17) becomes Cp S S d 500K R,500 K R,298K 298K (5.19) f there are no phase transformatons between 298 K and 500 K. If there are phase transformatons, then the phase transformatons must be taken nto account. he calculaton schematc for entropy of reacton at 1000 K correspondng to Fgure 5.4 s shown n Fgure 5.6. here are four sensble entropy changes (1l,1s,2, and 3); one transformaton (1t) and one entropy of reacton at 298 K. he sum of these entropy changes equals the entropy of reacton at 1000 K. S R,500 K 500 K CO 2 + C graphte = 2 CO S 1 S 2 S 3 S R,298 K 298 K CO 2 + C graphte = 2 CO Fgure 5.5 he computaton schematc for the entropy of reacton at 500 K from entropy data at 298 K. Gbbs energes of reacton he Gbb s energy of reacton can be computed at a gven temperature from the correspondng enthalpy and entropy changes for the reacton as shown n Equaton (5.1) at the begnnng of ths chapter. G H S R, R, R, (5.1) 46

47 Start S R,1000 K 1000 K Pb l + O 2 = PbO End S 1l S 2 S K Pb l Pb s S 1t S 1s S R,298 K 298 K Pb s + O 2 = PbO Fgure 5.6 he computaton schematc for the enthalpy of Pb oxdaton at 1000 K from heat of formaton data at 298 K. he superscrpt o denotes that the reactants and products are all n ther standard state, whch for solds and lquds are the pure materals and gases are at 1 atm (actually at fugacty 1 but that wll be dscussed later). Accordng to the crteron of equlbrum for a system at constant and P, a reacton n whch the reactants and products are n equlbrum wll have a standard Gbbs energy change of zero, whle spontaneous reactons wll be negatve; postve values ndcate a propensty for the reacton to go from rght to left. As a practcal matter, the computaton of Gbbs energy changes for reactons s usually greatly smplfed by fttng the results of the above descrbed lengthy procedures for computng H R, and S R, to smpler functons of temperature: usually lnear functon suffce. able 5.3 shows the lnear ft equatons of the standard Gbbs energy for selected reactons. able 5.3 Standard Gbbs energy lnear data ft for selected reactons Reacton GR, AB Range A B (K) (cal/gmole) (cal/k) C graphte O 2(g) = CO (g) -26, C graphte + O 2(g) = CO 2-94, Fe + 2 O 2(g) = Fe 3 O 4-260, H 2(g) O 2(g) = H 2 O (g) -58, Al (l) O 2(g) = Al 2 O 3(s) From the Makng Shapng and reatng of Steel 10 th ed. Students are cautoned not to conclude that by the Method of Coeffcents that A s H R, and that B s - S R, o make such a concluson s to assume that both H R, and S R, are 47

48 functons of temperature. Yet, so long as Cp s not zero they are functons of temperature and A and B are approxmatons of H R, and S R,. Uses of the calculaton Schematc In addton to the uses descrbed above, there are many other uses for the calculaton schematc. One of great practcal sgnfcance s to use t to determne the heat balance for an ndustral process wth several reactant nput and several product output streams each one wth ts unque temperature. he schematc can be used to descrbe the overall progress from nput to ext by enterng all of the sensble, transformaton, and reacton enthalpes needed to progress from nputs to outputs. he sum of all the enthalpes s the overall heat balance for the reacton process. Such a problem s beyond the scope of an ntroductory course on chemcal thermodynamcs but not beyond the student s capablty to perform after masterng the current subject matter. Adabatc flame temperature: AF A common example for the use of the calculaton schematc s the calculaton of the adabatc flame temperature AF. he AF s the temperature reached by the products of combuston f all the heat of combuston s used to heat the products. Snce no heat escapes from the system, the term adabatc s used to descrbe the computaton. he frst step n constructng the calculaton schematc for the burnng of natural gas at room temperature wth pure oxygen s shown n Fgure 5.7a and conssts of placng the known nformaton on the schematc wth temperature as the vertcal drecton and reacton progress n the horzontal drecton. A path s then constructed along whch all enthalpy changes can be computed. If heats of formaton data are avalable at 298 K, then the reactants can be taken to products at 298 K snce the heat of that reacton can be calculated from the heats of formaton. he CO 2 and H 2 O products are then rased to the fnal state, whch s the unknown but to be calculated AF. he completed calculaton schematc s shown n Fgure 5.7b and conssts of one heat of reacton and two sensble heats. he sum of these three enthalpes must equal zero to mantan the condton of adabatc. he only unknown s the fnal temperature of the reacton products, the AF, as shown n Equaton (5.20). 3 1 AF R,298K 298 CO2 H2O H H Cp 2Cp d 0 (5.20) AF CO H 2 O 298 K CH 4 + 2O 2 (a) 48

49 (b) AF CO H 2 O H1 H2 298 K CH 4 + 2O 2 = CO H 2 O H R,298 K Fgure 5.7 Calculaton schematc for determnng the AF for the combuston of natural gas wth pure oxygen both at 298 K: a) known nformaton, b) computatonal path ransformaton reacton values at other than the equlbrum It can be useful to know the heat of transformaton at temperatures other than the equlbrum temperature. For example, pure sold NaCl melts to form pure lqud NaCl at 1074 K, but t also becomes a lqud when t s dssolved n water at room temperature. he heat of fuson at room temperature s requred to change the NaCl from sold to lqud n a water soluton. here s also the matter of soluton heat, whch s covered n a later chapter. he calculaton schematc can be used to fnd the heat of fuson for NaCl at temperatures other than 1074 K as shown n Fgure 5.8. he sold NaCl at 298 K s heated to 1074 K and then fused to lqud, whch s then cooled to 298 K. he sum of the three enthalpes assocated wth these three changes total the enthalpy of fuson at 298 K. fuson = 1074 K NaCl (s) NaCl (l) H fuson,1074 K H1 H2 298 K NaCl (s) = NaCl (l) H fuson,298 K Databases here are many thermochemcal databases but two of the most commonly used are the JANAF and hermocalc. 49

50 JANAF he acronym name s formed from the frst letters of the Jont Army Navy Ar Force who commssoned ths database n the 1960 s. Wth the wdespread avalablty of hgh speed computers, t became possble to mathematcally process data that could be arrved at by more than one computatonal route or source and a make t nternally consstent. hs nternal consstency feature of JANAF hermochemcal ables combned wth ts careful attenton to detal that made t a major achevement. It has remaned a key source of thermochemcal data and s avalable by hard copy or sometmes dgtally for free. he database uses the enthalpy and Gbbs energy of formaton schema combned wth, of course, absolute entropy. hese values are tabulated from 0 K n 100 K ncrements, plus 298 K. able 5.3 shows some selected datasheets from the JANAF hermochemcal ables. hermocalc hermocalc s a dgtal thermochemcal computatonal applcaton wth user-selectable databases some of whch are hghly specalzed for specfc systems such as those for nckelbased alloy steels. A common database for use n chemcal reactons nvolvng pure substances s the SSUB3 database. he database for each substance contans the followng data: the coeffcents for temperature dependent heat capacty functon s used to compute entropes from absolute zero usng Equaton (5.8). Enthalpes are calculated usng Equaton (5.2) the Arbtrary-defned system for Enthalpy n whch the entropy of a pure element n ts most stable form s zero at 298 K. Enthalpes of formaton and phase changes are ncluded n the database so as to allow the computaton of so-called absolute enthalpes based on the Arbtrary-defned system for Enthalpy. Once the entropy and enthalpy for all reactants and products nvolved n a reacton, one may easly fnd the change n entropy, enthalpy, or Gbbs energy of the reacton by computng the dfference n the values for the products (fnal state) mnus the reactants (ntal state) for the desred quantty. able 5.4 shows selected data sheet output from hermocalc. able 5.3 Selected JANAF hermochemcal ables able 5.4 Selected hermocalc output. 50

51 Chapter 6 Smple System - One component systems Basc Concepts Phase dagrams Clausus-Clapeyron Equaton he effect of P on H and S Before tryng to understand the behavor of a system composed of multple components, one should have a clear understandng of how one-component systems behave. emperature and pressure are the state varables of greatest nterest that determne the state of a component. Phase dagrams Fgure 6.1 shows a one component phase dagram for H 2 O. he Internet has many such dagrams for H 2 O that may be found by search. One such dagram that s of partcular nterest because t extends to extreme pressures that mght be of great nterest to geologsts s shown n Fgure 6.2. Most students are surprsed to learn that H 2 O has many dfferent sold phases. Some exst well above 400 C at tremendously hgh pressure (>100 kbar, ~1.5 mllon ps). lqud sold 1 atm P rple Pont: ( K, atm) gas Fuson (273 K) bolng (373 K) Fgure 6.1 Phase dagram for H 2 O 51

52 Fgure 6.2 he H 2 O phase dagram at extreme pressures showng several sold forms of ce. ( 52