32CHAPTER 3. THE FIRST LA W OF THERMODYNAMICS3.3. ENERGY TRANSFERS33notice thatalthoughwe have been illustrating energyandworkbyusingthebali-in-valley idea (Chapter 2)andthe book-and-table idea (this chapter), whichemphasizespotential energy, thisparticularkind ofenergy/workis actually ir-relevant in thermodynamics, except as an analogy. We will define the energycontentofsystemsofimportanceto us to be thesamewhetherthey are on thefloor or the table.3.2.2AbsoluteEnergyIn discussing energy, we alwaysseemto be talkingaboutchanges in energy.Thebookhasmoreenergy on the tablethanon the floor,andpresumablymoreenergy on the roofthanon the table. And we add energy by warmingthebook,too. But howmuchenergyhasthebookgolin anyparticularstate-say,onthe table at 25°C? What is the absolute energycontentof the book? This was adifficultquestionuntil 1905, when Einsteinpostulatedthe essential equivalenceofmassand energy in his famous equationE.=mc2where Eris therestenergy of a system, m is the mass,andc is thespeedoflight. Therefore, the energy contained in any macroscopicsystemis extremelylarge,andadding energy to asystem(forinstanceby heating it) will in factincrease its mass. However, ordinary (i.e., non-nuclear) energy changesresultin extremely small andunmeasurablechanges in mass, sothatrelativitytheoryis not very useful to us, except in thesensethatitgives energy an absolute kindof meaning, which issometimeshelpful in trying to visualizewhatenergy is.Thusin considering ordinary everyday kinds of changesandchemical reac-tions, we will continue to deal with energychangesonly, never with howmuchenergy is in any particular equilibrium state. This is entirely sufficient forourneeds,butit does introducesomecomplicationsthatwould be avoided if wehad a useful absolute energy scale.3.2.3TheInternal EnergyAllthatis required to developourmodel of energyrelationshipsisthateveryequilibriumstateof asystem(such asourbookonthetable ,or the stick ofdynamite on the table) have a fixed energy content, called theinternalenergy,U (orU, themolarinternal energy) of the system. The numerical value of thisenergycontentisnotknown,andnotneeded. It could bethoughtof as identicalto therestenergyEr,ifthathelps, or as some smallsubsetof Er;itdoesn'treallymatter.Allthatmattersisthatwhen thesystemis at equilibrium, its energycontentor energy level isconstant.Formally, the relation between the total orrestenergy and the internal energy used in thermodynamics isE;= U +constantwherethe value of theconstantisunknown(andunimportant).Since we donotuseabsolutevalues of U or U, wecannotuseabsolute values ofanyquantitieshaving U in theirequationsof definition.Somewhat paradoxically, in spite of being possibly themostfundamentalofthermodynamicquantities, Internal Energy or even changes in U are littleusedin geochemical applications. It is never listed in tables ofthermodynamicvalues, for example,andone rarelyneedsto calculatetlU.Thereasonforthiswillbecomeapparentas we proceed.Ithasto do withthefactthatwe, theusersof thermodynamics, have agreatpredilection for usingtemperature,pressure,andvolume asourprincipleconstraintsormeasuredsystemparameters.ThisrequiresthatweusetlUin slightly modified forms,thatis,tlUmodified bywhatareoftenrelatively small correction factors (such as PtlV),andthesemodifiedformsare given differentnamesandsymbols. It isthenquite possible to rarelythinkabouttlU,since itseemsonly to arise in thedevelopmentoftheFirstLaw. For abetterunderstandingofthesubject, however, it isbestto realizethatinmostenergytransfersin the realproblemsthatwe will be considering,tlUis by far thelargestterminvolved.Justbecausewe donotusually calculateitsvalue doesnotmeanit isnotimportant.3.3ENERGY TRANSfERSIn thediscussionsin the previouschapters,weproposedtheideathatchangesorreactionsoccurbecausesystemscanlowertheirenergy bysuchchanges.However, wementionedthatthemostobvious kind of energy,heatenergy,wasnottheright kind of energy.Thereisanotherverycommonkind-energyexpendedas work, as whendynamiteisusedtobreakrock. However, workenergyisnotthe answer toourquestionseither,noris thecombinationofheatandwork. Nevertheless,theyare extremelyimportant,andtogether"form thebasis of the First Law.• Heat (q) is the energythatflowsacrossasystemboundaryinresponseto atemperaturegradient.• Work(w)is the energythatflows across asystemboundaryinresponseto a force movingthroughadistance(such ashappenswhen asystemchangesvolume).Heatandworkare thereforenotseparateentities assuchbutareformsof energythataretransferredindifferentways.Anenlightening analogyhasbeenoffered by Callen (1960). In Figure 3.2 weconsiderthewaterin a verydeeppond(theamountofwateristhusverygreatbutfiniteandin principlecould be exactlymeasured)tocorrespondto theinternalenergy U of asystem.Water may beaddedandsubtractedfromthepondeitherin the form ofstreamwater(heat) orprecipitation/evaporation(work). Both theinletand34CHAPTER 3. THE FIRST LA W OF THERMODYNAMICS3.4. THE FIRST LAW OF THERMODYNAMICS35o01~+q-q+w-w+w-wSystemSystemL\U=q+w+q-q(b)(a)//////Stream water flowing out,qodeepu/\1VeryEvaporation, w.~l~~L\U--.-._._._..--_._._._._._._._.._._._.-._.-.-.IStream water flowing in,~q,Figure 3.2: Thepondanalogy for the First Law.L\U =q-woutletstreamwater can be monitored by flow gauges,andtheprecipitationmeasuredby a rain gauge. Evaporation would be trickier tomeasure,butwemayassumethatwe have a suitablemeasurefor it. Now if the volume ofstreaminlet water over some period of time isqi,thestreamoutletwater qo,the rain Wr,andthe evaporationWe,thenifthereare nootherways of addingor subtracting water, clearly~u=(qi- qo) +(wr-we)where~Uisthechange in theamountof water in thepond,thatcould bemonitored by a level indicator as shown. ThusFigure 3.3: The two commonlyused conventions forthesign of q andw,leadingto two formulations of the First Law.Anotherimplication orassumptioninourpondanalogy isthatwater is con-served,thatis, itcannotsimplydisappearas if by magic. The samepropositionregarding energy is known as the First Law of thermodynamics.3.4 THE FIRST LAW OF THERMODYNAMICSAnother convention (Figure 3.3b) is to saythatheataddedto asystemis pos-itive,butthatwork done by asystemis also positive, orthatworkdoneonThe First Law of thermodynamics isthelaw of conservation of energy. If Uis the energycontentof a system,anditmaygain or lose energy only by theflow ofheat(q)or work(w),thenclearly, as in thepondanalogy,~Umustbe the algebraicsumof q and w. Inorderto express this algebraically, wemusthave some