Topics covered: Multicomponent systems, chemical potential. Instructor/speaker: Moungi Bawendi, Keith Nelson. First Law of Thermodynamics Adding heat Q to a crystal increases its internal energy U: dU dQ (indicates 'proportional') but if the crystal is allowed to expand, some of the added energy will be consumed by expansion dV, so.
![]() Software Solutions to Problems on Heat Transfer Conduction: Part IDescription. This book contains solutions to problems in the area of Heat Transfer, as per the syllabus of B. E. Comments are included generously in the codes so that the logic behind the solution is clear. Use of computer software helps in solving the problems fast and accurately. More importantly, parameter analysis (what . Once a particular type of problem is solved, solving similar problems later becomes extremely easy. In addition, one can plot the data, curve fit, write functions for various properties or calculations and re- use them. These possibilities create interest, curiosity and wonder in the minds of students and enthuse them to know more and work more. Content. About Mathcad. What is Mathcad? Symbols in Mathcad worksheet. ![]() ![]() Thirumaleshwar graduated in Mechanical Engineering from Karnataka Regional Engineering College, Surathkal, Karnataka, India, in the year 1. He obtained M. Sc (cryogenis) from University of Southampton, U. K. Conceicao Rodrigues Institute of Technology, Vashi, Navi Mumbai, India for eight years. He also worked as Head of Dept. Joseph Engineering College, Vamanjoor, Mangalore, India. A book entitled . Mathcad, EES, FEHT and EXCEL. This book, containing about 2. He has also authored free e- books on Thermodynamics entitled . Mathcad, EES, and TEST. Each of these titles is presented in 5 parts and all the books can be downloaded for free from www. He has also written and published three booklets entitled as follows: Towards Excellence. Thirumaleshwar has attended several National and International conferences and has more than 5. Lecture 1. 4: Multicomponent systems, chemical potential . Butyoucangetvolumeout,becausevolumeis thederivativeof. G, withrespect topressure. Keepingthetemperatureconstant. Andyou'vegot. Snow,yougotvolume. Do youknowwhereyou,howdidyoudefine. Ginthefirstplace? Wedefine. Gas. Hminusd. S. Oneofthemanydefinitionsofg. Reversethat,you'vegotnow. Hasafunctionof. Gandtemperatureandentropy. Well,you'vegotanexpressionfor. G,wejustcalculated,wejustshowedwecouldgetanexpressionfor. S,whichissittingrighthere. Temperature isavariablehere,sonowwehaveanexpressionfor. H. Andyou cangoonlikethatwitheveryvariablethatyou'velearnedinthisclassalready. Forinstance,uis. Hminusp. V. Well,there'sthe. Hhere. Wehavethatequationfor. H. Wehaveanequationfor. V,comingfromhere. Sothere'snothingunknownhere. Ifwehaveanequationfor. Ghereintermsoftemperatureandpressure. Samethingforthe. Helmholtzfreeenergy. You canevengettheheatcapacitiesout. Everysingleoneoftheseinteresting,usefulquantitiesthatonewouldwanttocalculatefallsoutfromthe. Gibbsfreeenergyhere. Anyquestionsonthatimportantstep? And,really,Ican'tbelievehowclueless. Iwaswhen. Istarteddoingresearch. Because. Iwouldgothroughtheprocessofcalculating. G, andgetting. G,andet cetera,et cetera. Iwrotepapers,you know, Gequalsblahblahblah. And. Ididn'trealizethatthat'swhypeoplewantedtoknow. G. Anyway,Iknowbetternow. So,thereare afewthingswecansayabout. Gthatarefairlyeasytocalculate. Forinstance,if. Ilookatliquidsorsolids. And. Iwant toknowhow. Gchangeswithpressure. So,Iknowthatthevolumehered. G/dp,thatd. G/dpisthevolumehere. Soif. Ilookunderconstanttemperature,Ipickmyfundamentalequationunderconstanttemperature,and. Iwant toknowhow. Gischanging,Iintegrate. So I haved. Gis equalto. Vdp. Soif. Ichangemy,and. Ilookatper mole,and if. Ichangemypressurefromp. Iintegratefromp. 1top. Vdpper mole. Andwhatcan. Isay,for aliquidorasolid,thevolumeper mole,over aliquid orasolid,issmallanditdoesn'tchangeverymuch. So. Vissmall. Andusuallythesesolidsandliquids,you canassumetobeincompressible. Meaning,asyouchangethepressure,the volumedoesn'tchange. It'sagoodapproximation. Sowhenyoudoyourintegralhere,you getthat. Gatthenewpressureis. Gattheoldpressure,thenifthisisn'tchangingverymuchwithpressure,ornotatall,thenyoucantakeitout. It'sjustaconstant. Plus Vtimesp. 2minusp. Andso,thisisthe incompressiblepart,youtakeitout. Thefactthatit'ssmallmeansthatyoucanassumethatthis iszero,thiswholethingiszero,thatit'ssmallenough. Andthenyouseethat. G,approximatelydoesn'tchangewithpressure. Tellsyouthat. G,fora liquidorsolid,mostofthetimeyoucanassumethat it'sjustafunctionoftemperature. Justlikewesawforanidealgas,that theenergyandtheenthalpywerejustfunctionsoftemperature. Andthat'sausefulapproximation. It'suseful,butit'snotcompletelytrue. Andifitweretrue,thentherewouldnotbeanypressuredependentstophasetransitions. Andweknowthat'snotthecase. Weknowthatifyoupressonwaterwhenit'sclosetothewaterliquid- solidtransition,thatyoucanlowerthemeltingpointofice. Youpressonice,andyou presshardenough,andicewillmelt,thetemperature iscloser tomeltingpoint. And we'llgothroughthat. So,thatmeansthattherehastobesomesortofpressuredependence,eventually. Andwe'llseethat. Thisisjustanapproximation. Whatelsecanwedo? Wecancalculate,also,foranidealgas. Liquid andsolid,wecandoanidealgas. Soforanidealgas,again,startingfromthefundamentalequation,wehaved. Gequals. Vdp. Wecandoitpermole. Sointegratebothsides,G(T, p. Gattheinitialpressure,plustheintegralfromp. Soinsteadofputtingthevolume,thisisan idealgasnow,wecanputtheidealgaslaw. So. Visreally. RTover p. RToverpdp. Wecanintegratethis. Getalogtermout. G(T, p. RTlogp. 2overp. 1. Andthenit'sveryusefultoreferenceeverythingtothecenterstate. Soifyoutakep. 1equalsonebarasourreferencepoint,andgetridofthelittlesubscripttwohere,wecanwrite. Gof. Tatsomepressurep,thenis. Gandthelittlenaughtontopheremeansstandardstateonebarplus. RTlogpdividedbyonebar. And. Iputinalittledottedlineherebecauseveryoftenyoujustwriteitwithouttheonebar and bar. Andyouknowthattherehastobeaonebar,becauseinsidealogyoucan'thavesomethingwithunits. It hastobeunitless. So youknowif you havesomethingwithbarhere,you'vegottodividewithsomethingwithbar,and therehappenstobeonebar here. Sopressurepis. Gatitsstandard stateplus. RTlogp. And thisbecomesaveryuseful,veryuseful,quantitytoknow. OK,so. Gissoimportant. And. Gper moleissofundamental,thatwe'regoingtogiveitaspecialname. Nottomakeyourlifemorecomplicatedbutjustbecauseit'sjustsoimportant. We'regoingtocall it thechemicalpotential. So. Gpermole,we'regoing tocallmu. And that'sgoingtobeachemicalpotential. We'regoing todoalotmorewiththechemicalpotentialtoday. Andthereasonwhywecallpotentialisbecausewealreadysawthatif you'vegotsomethingunderconstantpressure,temperature,thatyouwant touse. Gasthevariabletotellyouwhethersomethingisgoing tobespontaneousor not. Soyouwant. Gtogodownhill. Andso,we'regoing tobetalkingaboutchemicalspecies. Andinsteadofhavingacarupanddownmountains,tryingtogodowntothevalleys,we'regoing tohavechemicalspeciestryingtofindthevalleys. Thepotentialvalleys. Togettoequilibrium. Andsowe'regoingtobelookingatthe. Gibbsfreeenergy,orthe. Gibbs freeenergyper mole atthatparticularspecies,andit'sgoingtowanttobeassmallaspossible. We'regoing towanttominimizethechemicalpotentials. Andthat'swhyit'scalledpotential. It'slikeanenergy. So,that'stheendoftheonecomponent,thermodynamicbackground,beforewegettomulti- components. Soit'sagoodtimetostopagainandseeifthere'sanyquestions. Anyissues. OK. So,sofarwe'vedoneeverythingwithonespecies. Oneidealgas,oneliquid,onesolid. Wehaven'tdoneanythingwithmixtures,exceptformaybelookingattheentropyofmixing. Wesawtheentropyofmixingwasreallyimportant,becauseitdroveprocesseswhereenergywasconstant. Butmostofwhatwecareaboutinchemistry,atleastinchemicalreactions,specieschange. Theygetdestroyed. Newspeciesgetcreated. There aremixtures. It'sprettycomplicated. Forinstance,if. Itakeareactionofhydrogengaspluschlorinegastoformtwomolesof. HClgas,I'mdestroyinghydrogen,I'mdestroyingchlorine,I'mmaking. HClin thegasphase. Igetabigmixtureattheend. Igetthreedifferentkinds ofspeciesattheend. Sothefundamentalequationsthat. I'vebeentalkingabout,thatwe'vebeentalkingabout,they'retoosimpleforsuchasystem. Becausetheyallcareaboutonespecies. Evenmorecomplicated,forinstance,if Itakehydrogengasandoxygengas and. Imixthemtogethertomakewater,liquid,forinstance,notonlydo. Ihavespeciesthatarechanging,thataregettingdestroyedorcreated,inthiscaseherethetotalnumberofmolesischanging. Andthephase ischanging. Got allsortsofchangesgoingonhere. Andsoif. Iwanttounderstandequilibrium,if. Iwant tounderstandthedirectionoftimeforthesemorecomplicatedprocesses,Ihavetobeabletotakeintoaccount,inaneasyway,thesemixingprocesses,thesephasechanges,thesechangesin thenumberof moles. And that'swhatwe'regoing totalkabouttoday. We'regoing totrytochangeourfundamentalequationstomake themalittlebitmorecomplicatedsothatwecandealwiththesesortsofproblems. Becausethosearetherealproblemsweneedtokeeptrackof. Andtheultimategoal,then,ofchangingourfundamentalquestionsistoderiveequilibriumfromfirstprinciples. To reallyunderstandchemicalequilibrium,whichyou'veallseenbefore. You'veallusedthe chemicalequilibriumconstant. K, you'vedoneproblems. Butyou'vebeengiven,basically,theequilibriumconstant,and notreallyderivedit,understoodwhereitcamefrom. OK,anothersimpleexamplehere,whichisactuallytheonethatwe'regoingtobelookingatinthefirstcase. Wherethere'sachangegoingon,isjusttolookataphasechange. H2. Oliquidgoingto. H2. Osolid. There'saphasechange,youcanthinkofitasonespecies,the. H2. Oliquidsortofchangingintoan. H2. Oasolid. It's thesamechemicalinthiscasehere,there'snochangeinthemolecules. Butit'sstillachangethatwehavetoaccountfor. Anotherexamplethat'salsosimplelikethis,thatyouall,I'msure,haveseenbefore,suppose. Itakeacell. My cellhere,full ofwater. Andthen. Iputmy cell,let'ssay. I takeahumancell. Myskinorsomething. And. Itake itand. Iputitindistilledwater. What'sgoing tohappentothecell? Is itgoing tobehappy? What'sgoing tohappentoit? It'sgoingtoburst,right? Whyisitgoing toburst? Anybodyhaveanideawhy it's going to burst? Yes. STUDENT: . Itrushesintothecod,andwell,thecoddoeswhatthecelldoes,whenyouputitindistilledwater. It sortofbloats. It isn'tveryhappy. OK,sobutallthesethingsarebasicallythesameideahere. Whereyouhaveacomplicatedchange,wherespeciesaremixing,andthingslikethis. Anditturnsoutthechemicalpotentialisgoingtotellusallabouthowtothinkaboutthat. That'swhythechemicalpotentialis soimportant. Sowe'regoing togobacktothesetwoexamplesheremanytimes. Solet'stakethesimplestexamplehere. Let'sgobackandderivesomeequations. Let'stakeoursimplestexamplethat'snotaonespeciessystem,buthastwospecies. Species. 1and. 2. Andn. 1andn. 2arethenumberofmolesofspecies. Andthenwe'regoingtoseeif. Imakeaperturbationinmysystem. Ichangethenumberofmolesof. Howdoesthisaffectthe. Gibbs freeenergy? That'sthequestionwe'regoingtopost. Andourgoalistofindanewfundamentalequationfor. Gthatincludesthenumberofmolesofthedifferentspeciesastheychange. Becauseinchemistrythey'regoingtobechanging. They're notgoingtobefixed. Sowhatwewantisjustpurelymathematicallyformally,takethedifferentialofthe. Gibbsfreeenergy,whichweknowisd. G/d. T,keepingpressure,the number ofmolesof. T. Thatis,d. G/dpconstanttemperature,n. Andthenwehaveourtwomorevariablesnow,d. G/dn. 1,rememberthisisjustaformalstatementkeepingtemperatureandpressureandn. G/dn. 2, dn. 2keepingtemperature andpressureandn. I'm notwritinganythingnewhere,I'mjusttellingyouwhatthedefinitionofthedifferentialishere,for. G. Wealreadyknowwhatsomeofthesequantitiesare. Weknowthatthisistheentropy,minustheentropy. Thishereisthevolume. And. Iknowtheansweralready. But. I'mgoing todefine itanyways. Andwe'regoingtoproveit. I'mgoingtodefinethisasthechemicalpotentialforspecies. I'mgoingtodefinethis,I'mgoing togiveitasymbol,chemicalpotentialmu,forspecies. Sothat. Icanwritemynewfundamentalequationasd. Gasminus. Sd. T. Plus. Vdpplus,andif. Ihavemorethantwospeciespresent,thesumofallspeciesinmymixturetimesthechemicalpotentialofthatspecies,dni. Thechangeinthenumberofmolesofthatspecies. So,thisquantitymu,that. I'vejustdefined,d.
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