Part II Thermodynamic properties Properties of Pure Materials Chapter 8

Transcription

1 Part II Thermodynamc propertes Propertes of Pure Materals Chapter 8 Prmary objectve s to evaluate changes n state n terms of prmtve and measurable propertes Secondary objectve s to connect molecular propertes and nteractons to macroscopc propertes and processes

2 Propertes of Pure Materals Chapter 8 Connectons to the Fundamental Equaton va G varous forms for equatons of state reference state condton for S = S o and the Thrd Law of Thermodynamcs Derved property estmaton U and U, etc. Role of departure or resdual functons Consttutve PVTN volumetrc property models Ideal gas law Theorem of Correspondng States Flud behavor from the Boyle pont to the trple pont Zeno condton Pressure and volume explct sem-emprcal EOSs Correlated expermental data Ideal gas state heat capacty models translaton knetc theory - classcal rotaton rgd rotator -classcal vbraton quantzed usng the Ensten model Property estmaton methods Molecular group contrbutons Correspondng States Conformal flud theory Molecular smulatons

3 Connectons to the Fundamental Equaton 3 U = TS PV+ µ N = f ( S,V, N) du = TdS PdV+ µ dn G = U + PV TS = H TS = y () = f(t,p, N ) dg = -S dt +VdP + µ dn C p S H T = P P κ T α P P V V P V V T P T P

4 Consttutve PVTN Volumetrc Property Models 4. Ideal gas law PV = NRT or PV = RT Z = PV/RT =. Theorem of Correspondng States Z = f ( T r, P r, Z c, ω, ) scalng to reduced coordnates fluds from the Boyle pont at low densty to the trple pont at hgh densty the Zeno lne 3. Cubc type EOS P = f(t,v) van der Waals P = RT/(V-b) a/v Redlch-Kwong (RK) Redlch- Kwong-Soave (RKS) Peng-Robnson (PR) 4. Vral type EOS Z = + B/V + C/V + BWR Starlng Martn-Hou

5 5 Combnatorcs When workng wth partcles on a lattce (whether t be,, or 3 dmensonal), use the equatons below to determne Ω, the number of possble confguratons the system can take. For M = # of stes N = # of partcles n system Case #: Partcles are : non-nteractng (more than one can occupy same ste) dstgushable N Ω = M Case #: Partcles are : exclusonary (only one can occupy a gven ste) dstgushable M! Ω= (M N)! Case #3: Partcles are : exclusonary (only one can occupy a gven ste) ndstgushable M! Ω= N!(M N)! Case #4: Partcles are : non-nteractng (more than one can occupy same ste) ndstgushable ( M + N )! Ω= N!(M )! These equatons can be used to determne the entropy for the mcrocanoncal ensemble. For ths ensemble, snce each mcrostate has the same energy, the entropy s determned solely by the number of possble confguratons of the system: S = S = k ln Ω confg

6 DEFINITIONS SYSTEM/SURROUNDINGS (ENVIRONMENT) STATE OF A SYSTEM - IDENTIFIED BY PROPERTY VALUES REQUIRED REPRODUCE THE SYSTEM SIMPLE SYSTEM - DEVOID OF ANY INTERNAL ADIABATIC, RIGID, IMPERMEABLE BOUNDARIES, NO EXTERNAL FORCE FIELDS OR INERTIAL FORCES, CAN BE SINGLE OR MULTI-PHASE COMPOSITE SYSTEM - OR MORE SIMPLE SYSTEMS PHASE - REGION OF UNIFORM PROPERTIES EXTENSIVE/INTENSIVE PROPERTIES - FIRST/ZERO ORDER IN MASS PRIMITIVE PROPERTY - MEASURABLE DERIVED PROPERTY - DEFINED IN TERMS OF CHANGES IN THE STATE OF A SYSTEM OPEN VERSUS CLOSED SYSTEMS - WITH RESPECT TO MASS FLOW STATE VERSUS PATH FUNCTIONS ISOLATED SYSTEM - NO INTERACTIONS WITH SURROUNDINGS QUASI-STATIC, REVERSIBLE, AND IRREVERSIBLE PROCESSES BOUNDARIES -ADIABATIC/DIATHERMAL -PERMEABLE/IMPERMEABLE/SEMI-PERMEABLE -RIGID/MOVABLE

7 PRIMARY NOMENCLATURE/SYMBOLS VARIABLE INTENSIVE EXTENSIVE GENERAL PROPERTY B B ENERGY E E HEAT Q HEAT CAPACITY C v,c p WORK W TEMPERATURE T TIME t PRESSURE P VOLUME V V INTERNAL ENERGY U U ENTHALPY H H ENTROPY S S HELMHOLTZ FREE ENERGY A A GIBBS FREE ENERGY G G CHEMICAL POTENTIAL µ MOLE FRACTION/MOLES x, y n, N FUGACITY f FUGACITY COEFFICIENT φ ACTIVITY a ACTIVITY COEFFICIENT γ - - -

8 8 Energy and the st law Treatment Summary of mathematcal forms System Expresson. closed, general d E = δq + δw. closed, smple d E = du = δq + δw 3. closed, smple, du = δq PdV only PdV work 4. open, smple du = δqσ + δwσ + Hnδnn Houtδnout 5. open, nonsmple PE+KE only (use mass bass) 6. steady state, open, nonsmple d E = δqσ + δwσ n + H n n + gz ( v ) m d E = d U + mg z + d E = 0 dm = 0 or dn = 0 d E dm dn = = = 0 dt dt dt δnn δnout = = n δt δt plus expresson (5) d E dt H δq δt δw + δt σ σ = 0 ss = H out H n PE n out n v + δm n H out (f no reactons) = ( H + PE + KE )n ss ss = g ss ( z z ) out n ss KE ss out v = + gz out out v out v + n δm out 7. non-smple, open dstrbuted nteracton along σ-surface ( ρ E) v q na ρ dv = d + W H + gz + v n d t V a a σ σ σ, σ σ a

9 9 Four Postulates can be used to defne classcal thermodynamcs:. For closed smple systems wth gven nternal restrants, there exst stable equlbrum states that can be characterzed completely by two ndependently varable propertes n addton to the masses of the partcular chemcal speces ntally charged. - "closed, smple systems" - "ndependently varable propertes" - masses of each component known - consstent wth the Gbbs phase rule - restatement of Duhem's theorem P S L T Crt. Pt. V L - V coexs t P S F = n+-π = 3- π for pure n= system S+ L S+V V L L+ V

10 0. In processes for whch there s no net effect on the envronment, all systems (smple and composte) wth gven nternal restrants wll change n such a way that they approach one and only one stable equlbrum state for each smple subsystem. In the lmtng condton, the entre system s sad to be at equlbrum. - apples to all solated systems - descrbes the natural tendency of processes to approach equlbrum at a mnmal energy and maxmal entropy state - lnkng postulates I and II provdes a means for mathematcally descrbng thecrtera of phase and chemcal equlbra

11 3. For any states () and (), n whch a closed system s at equlbrum, the change of state represented by () to () and/or the reverse change () to () can occur by at least one adabatc process and the adabatc work nteracton between ths system and ts surroundngs s determned unquely by specfyng the end states () and (). - adabatc work nteractons of any type n closed systems are state functons. - adabatc form of the st law : de = δw adabatc = δq + δw where δw and δq are the actual heat and work nteractons that occur for non-adabatc processes between the same end states () and().

12 4. If the sets of systems A,B, and A,C each have no heat nteracton when connected across nonadabatc walls, there wll be no heat nteracton f systems B and C are also connected. -"Zeroth law of thermodynamcs" - concept of thermal equlbrum and the absence of a heat nteracton - temperature dfferences are needed for heat transfer to occur

13 3 Specfc type of work nteracton δw = F s dx pressure-volume PdV (expanson or compresson of system volume) surface deformaton σ da ( σ= surface tenson, a = surface area) electrc charge transfer dq ( (or ε) = electromotve force or potental, q = charge) electrcal polarzaton E dd ( E = electrc feld strength, D = electrc dsplacement) magnetc polarzaton MdB ( M = magnetzaton, B = magnetc nducton) frctonal F f dx stress-stran V o ( F x /a)dω (F x /a = stress, dω = lnear stran = dx/x o and V o dω/a=dx) general shaft work δw s

14 4 Supplementary Notes for Chapters -3 Context and Approach st Law: Concepts and Applcatons These notes are ntended to summarze and complement the materal presented n our textbook the 3 rd edton of Thermodynamcs and Its Applcatons and dscussed n our graduate thermodynamcs class (0.40). For the most part, we use the same notaton and make references to llustratons and equatons contaned n the text. Hopefully you wll fnd these notes helpful for self study and revew. We strongly urge you to consder the example problems and other problems that appear at the end of each chapter from Chapter 3 onwards. We begn by summarzng the learnng objectves for Chapter -3. Chapter Learnng Objectves* After fnshng ths chapter you should be able to: Descrbe the three general types of problems that are encountered n thermodynamcs Lst the basc problem solvng steps and dentfy the types of nformaton requred to solve thermodynamcs problems Understand the dfference between the postulatory and hstorcal approach to teachng/learnng thermodynamcs Chapter After fnshng ths chapter you should be able to: Clearly and wthout jargon defne: system, boundary, envronment, prmtve propertes, derved propertes, thermometrc temperature, event nteracton, adabatc, dathermal, phase, restrant, thermodynamc processes, path and state Explan the dfference between an open, closed and solated system Gve an example of an adabatc wall and explan why t s useful Descrbe and gve examples of a smple and a composte system Understand what s meant by a stable equlbrum state and how t may be characterzed Explan the dfference between ntensve and extensve propertes Descrbe stuatons where thermodynamc propertes are not ndependently varable Use the nomenclature and unt systems employed n engneerng thermodynamcs *contrbuted n part by Professor Brad Eldredge, Unversty of Idaho

15 5 Chapter 3 After fnshng ths chapter you should be able to: Clearly defne and gve examples of work, adabatc work nteractons, heat nteractons, deal gas, energy, nternal energy, enthalpy and heat capacty Be able to calculate PdV work nteractons for quas-statc, closed system expansons and compressons Understand the connecton between Joule s experments and adabatc work nteracton Wrte the complete general form of the st Law and show how to smplfy t for smple open and closed systems and for steady state and transent stuatons Usng the st Law, calculate the change n energy for a closed systems gong from State to State and determne the heat and/or work requred to effect ths change Usng the st Law, calculate the change n energy for open system undergong a change from State to State and determne the heat, work and/or mass flows requred to make ths change Postulatory versus Hstorcal Approach Postulatory axoms that cannot be proved from frst prncples are stated they represent the mnmum set of rules, consstent wth the expermental body of knowledge, that determne how systems behave n terms of the nterchange of energy (heat, work,... ) and mass. Hstorcal chronologcal development of concepts (and msconceptons) based on an evolvng set of expermental knowledge. Postulates tend to be abstract but wth contnued exposure they become the logcal set of rules needed to formulate the thermodynamc laws and relatonshps that govern the behavor of systems that transfer energy by heat and work nteractons and by mass transfer. In addton, the postulates provde a means of specfyng the dstrbuton of components n phase and chemcal equlbrum as well as the stablty of specfc phases. Wthn the context of the hstorcal development, our predecessors faced problems and stuatons that they couldn t explan wth exstng prncples n physcs and mathematcs. Consequently, they proceeded to create new theores and laws verfyng ther hypotheses wth experments. The amalgamaton of ther work s captured n the four man postulates of our text. In table I and the two fgures that follow, major contrbutors to classcal chemcal and statstcal thermodynamcs are brefly descrbed n a hstorcal context.

16 6 Major Early Contrbutors to Classcal Chemcal and Statstcal Thermodynamcs Scentst Black Count Rumford (B. Thompson) Carnot Mayer Joule Years Contrbutons Calormetry Mechancal equvalent of heat Heat engnes and converson of heat nto work Interconverson of heat and work Interconverson of heat and work W. Thomson (Kelvn) Clausus W. Thomson (Kelvn) Clausus Maxwell Boltzmann Gbbs Gbbs Gbbs Helmholtz Absolute temperature scale nd law concepts nd law, dsspaton of energy Concept of entropy Heat, statstcal bass for nd law, clarfcaton of many key ssues Thermodynamc propertes from dstrbuton functons Chemcal thermodynamcs Chemcal potental; phase rule; fundamental equaton Defned classcal statstcal thermodynamcs; created ensembles Theory of equlbrum; free and bound energy ( A defned) van=t Hoff Duhem Caratheodory Nernst Smon Chemcal thermodynamcs; theory of equlbrum constant; solutons Mxtures Adabatc work and st law Heat theorem (3 rd law) Improved verson of 3 rd law Adapted from Ladler, K.J. (993). The World of Physcal Chemstry, Oxford. Image by MIT OCW. 3

17 7 Poneer and nventor - of classcal statstcal and chemcal thermodynamcs, analyss of multcomponent, multphase systems, ensembles, Fundamental Equaton, phase rule, chemcal potental, etc. Gbbs Mayer, HCB, Chandler, McQuarre, et al. modern statstcal mechancs Guggenhem Fowler Redlch-Kwong, Soave, Peng-Robnson, Martn, et al. cubc EOS van der Waals Beatte Brdgeman BWR Starlng et al. Hlderbrand Scatchard Redlch-Kster Prausntz Messner Mayer Ptzer Wlson Renon et al. Tat, Hartman, Sanchez-Lacombe, et al. Cluster Integrals Vral EOS Onnes G EX - models electrolyte models polymer EOS models Vral theorem van Laar Margules G.N. Lews Randall Debye Huckel Flory Huggns Clausus Modern Classcal Era Contrbutors Image by MIT OCW. 4

18 8 Preclasscal Era Classcal Era Modern Classcal Era Mostly physcs - focus on experments Galleo Black Count Rumford Mostly physcs, math, and mechancal engneerng - focus on laws and postulates Maxwell Clausus Lord Kelvn Boltzmann Joule & Carnot Hstorcal Progresson of Classcal Thermodynamcs Chemcal and molecular thermodynamcs - focus on non-deal fluds, phase and chemcal equlbrum and stablty and so forth ? Work and Heat Concepts Gbbs Statstcal and Chemcal Thermodynamcs Image by MIT OCW. 5

19 9 Dscusson of Postulates (lsted n Appendx A). For closed smple systems wth gven nternal restrants, there exst stable equlbrum states that can be characterzed completely by two ndependently varable propertes n addton to the masses of the partcular chemcal speces ntally charged. - "closed, smple systems" - "ndependently varable propertes" - masses of each component known - consstent wth the Gbbs phase rule - restatement of Duhem's theorem P S Crt. Pt. L T V L - V coexst F = n+-π = 3- π for pure n= system P S L S+L L+V S+V V. In processes for whch there s no net effect on the envronment, all systems (smple and composte) wth gven nternal restrants wll change n such a way that they approach one and only one stable equlbrum state for each smple subsystem. In the lmtng condton, the entre system s sad to be at equlbrum. - apples to all solated systems - descrbes the natural tendency of processes to approach equlbrum at a mnmal energy and maxmal entropy state - lnkng postulates I and II provdes a means for mathematcally descrbng the crtera of phase and chemcal equlbra 3. For any states () and (), n whch a closed system s at equlbrum, the change of state represented by () to () and/or the reverse change () to () can occur by at least one adabatc process and the adabatc work nteracton between ths system and ts surroundngs s determned unquely by specfyng the end states () and (). - adabatc work nteractons of any type n closed systems are state functons. - adabatc form of the st law : de = δ W adabatc = δ Q + δ W where δ W and δ Q are the actual heat and work nteractons that occur for nonadabatc processes between the same end states () and (). 4. If the sets of systems A,B, and A,C each have no heat nteracton when connected across nonadabatc walls, there wll be no heat nteracton f systems B and C are also connected. -"Zeroth law of thermodynamcs" - concept of thermal equlbrum and the absence of a heat nteracton - temperature dfferences are needed for heat transfer to occur 6

20 0 Work Interactons In general, all work nteractons are path dependent and are defned to occur at the boundary of a system. The symbol ds used here to desgnate a path dependent property. Work n general can be represented by the dot product of a boundary force F s and an extensve dsplacement x whch can be expressed dfferentally usng dx. Thus, Sgn conventon for work: δ W= F s dx = F σ boundary dx W or δ W > 0 f the surroundngs does work on the system. In ths manner t s possble to mantan a consstent conventon that work transferred nto a system wll ncrease ts energy content and work transferred out to the surroundngs wll decrease ts energy content Specfc type of work nteracton δ W = F s dx pressure-volume PdV (expanson or compresson of system volume) surface deformaton σ da ( σ= surface tenson, a = surface area) electrc charge transfer dq ( (or ε) = electromotve force or potental, q = charge) electrcal polarzaton E dd ( E = electrc feld strength, D = electrc dsplacement) magnetc polarzaton MdB ( M = magnetzaton, B = magnetc nducton) frctonal F f dx stress-stran V o ( F x /a)dω (F x /a = stress, dω = lnear stran = dx/x o ) and V o ds/a=dx general shaft work δ W s Pressure-volume work: Consder a pston cylnder geometry wth quas-statc condtons, thus the path can be descrbed dfferentally. Defne the gas nsde the cylnder below the pston as the system, n ths case the work effect at the pston-gas boundary s, δ W system = δ W g =! P gas dv gas = F g dx =! P g a dx We can also descrbe δ W g usng external effects by employng a force balance to connect F g to effects n the surroundngs: 7

21 atmos. Pston of area a F a = P a a gas z mv dv/dz F g mg F f dv Fg = Pa a + mg + mv + F dz f δ Wgas = Fg d x = Pa a dz + mgdz + mvdv + F where d Watmos. = Pa a dz ; δ Wpston = mgdz + mvdv and dwwall = Ff dz,.e. frctonal dsspaton at the wall. Integratng, Note that m Wgas = ( Pa a + mg)( z z) + ( v v ) + F F f dz may be small and can safely be neglected n many cases. Alternatve conceptualzaton--a pston-cylnder arrangement wth the gas contaned selected as the system. Small weghts are removed and placed n storage n a gravtatonal feld n such a manner that the P-V hstory of the gas expanson s reproduced exactly. Ths effectvely s a quas-statc process where the net work s equal to the ΡΕ contaned n the stored weghts. Thus we can always equate the work done by a system to the rse or fall of weghts (dfferentally) n the surroundngs In other words, d W + dw = 0 surroundngs f dz f dz massless pston z gas system as a drect result of the dynamc force balance that mantans F = 0 + system F surroundn gs 8

22 Adabatc Work Interactons and Postulate III Havng developed the defntons of varous types of work nteractons, we can now examne the consequences of Postulate III n detal. Frst, we need to establsh a conventon for the sgn of a work nteracton. In 0.40 we assume that work s postve f the surroundngs do work on the system to ncrease ts energy content. δ W > 0 and wth Postulate III de = + δ W adabatc and E = + W adabatc Now what exactly s an adabatc work nteracton? Here we must rely on some rather specfc defntons. We recommend that you carefully read secton 3. to frm up your understandng of adabatc work nteractons. Then proceed to examne the thrd postulate n the context of a change n state from () to () (or from () to ()) for a closed system gven by a change n the total system energy (E) and represented by an adabatc process occurrng at least n one drecton from () to () or from () to (). Hstorcal Perspectve -- Joule's Experments The st Law of thermodynamcs resulted n part from a seres of experments carred out by Joule between 843 and 848. Hs experments dealt wth adabatc work nteractons and showed an almost exact proportonalty between the amount of work expended on a system and the rse n temperature observed n a fxed mass of lqud water. Joule used a paddle wheel (shaft work) n one experment, the compresson of a gas (PdV work) n another, an electrc current (electrcal work) n a thrd test, and frctonal work (F f dx) n a fourth set of experments. He vewed that the change of state of the water resulted from a converson of work nto "heat" and demonstrated that a constant converson factor exsted regardless of the type of work expended, namely that 4.84 J = calore. Snce heat s only transferred across a system boundary, t s ncorrect to regard the temperature ncrease of the water as nduced by a heat nteracton. Strctly speakng, t corresponds to a change n nternal energy for an adabatc system. Heat Interactons Heat n a classcal sense n thermodynamcs s "devod of any mcroscopc (physcal or molecular) sgnfcance" and s defned n terms of the dfference between the adabatc work nteracton and the actual work nteracton that occurs as the system changes from state to state. Q = W adabatc W = E E W or δ Q = de δ W where the energy dfference E - E = E s a state functon whch can be computed from an adabatc work nteracton. Lke δ W, δ Q s a path dependent functon that occurs only at the boundary of a system. 9

23 3 Path versus State Functons In the above treatment two dfferent dfferental operators d and δ have been used to ndcate exact and path dependent functonal behavor, respectvely. δ refers to the path dependent transfer of a dfferental amount of work or heat (and mass as well). d refers to a dfferental change n a state functon such as E or U. When these functonal dfferentals are ntegrated mportant dfferences appear. For example, Iδ W = W and Iδ Q = Q (total amount of work or heat as a path dependent lne ntegral) I de = E = E E (total change of the system energy E from state to state as ndcated by the dfference operator ) I de = 0 (total change of E around a closed cycle s zero) I δ W 0 or I δ Q 0 (n general, these lne ntegrals are not necessarly zero) 0

24 4 Frst Law for Thermodynamcs for Smple, Open Systems enterng streams n leavng streams out state ntal condton state fnal condton Although the system A bounded by the σ surface s open, the composte system A + δ n n s closed. If E s the total energy of system A, then by applyng the st law to the closed composte system: Q σ σ surface W σ σ surface open to mass at ths pont nsulaton δn n at P n' V n wth E n E (E + E n δ n n ) = δ Q + δ W + P n V n δ n n If the composte system s smple then E = U n system A and E n = U n for the ncomng stream then we can defne a new property the enthalpy H of the ncomng stream as H n / (U + PV ) n Therefore, the open system form of the st law can be wrtten n dfferental form for ths partcular smple system A as: du = δ Q σ + δ W σ + H n δ n n Generalzng for multple enterng and extng streams: du = δ Q +δ W + Hnδn Houtδn (3-60) σ σ out n n Note that we have retaned the path dependent dfferental operator δ for n n and n out because, strctly speakng, only dn of the system s a state varable wth: or for the case of multple streams Generalzed Open Smple System A Adapted from Tester, J. W. and Modell, Mchael. Thermodynamcs and Its Applcatons. Upper Saddle Rver, NJ: Prentce Hall PTR, 997, p. 46. Image by MIT OCW. dn = δ n n δ n out (3-58) dn = δ n n δ n n out out

25 5 Frst Law for Open, Non-smple Systems wth PE + KE Effects Decomposton of E nto three man components where. KE due to nertal effects (velocty/acceleraton of center of mass). PE due to locaton n gravtatonal feld of the earth 3. Internal (U) everythng else assocated wth mcroscopc energy storage on a molecular level. Ths uncouplng requres that we use consttutve equatons for knetc and potental energy whch are easly obtaned from our experence n physcs and mechancs, namely: E = E + E U (3-6) d E KE PE + = d E + d E du (3-6) KE PE + m < v > mv EKE = = and EPE = mgz here m s the mass of the system or subsystem of nterest and *<v>* s the magntude of the velocty vector <v>. By followng the decomposton of E outlned above and the mathematcal approach that led to Eq. (3-60), we obtan a general expresson for the Frst Law for an open system wth ts center of mass located at a vertcal dstance <z> from a reference plane at z = 0 and travelng wth velocty <v> as shown n Fgure 3.. Note that now we wll employ a mass (rather than mole) bass for the ntensve enthalpy terms n the summatons of Eq. (3-60): that s H n and H out have SI unts of J/kg. The ncomng and outgong streams also need ther approprate potental energy contrbutons (from gz n *n n / and gz out *n out ) and knetc energy contrbutons (from v n*n n / and v out*n out /). For ths case, the general form of the Frst Law for open, non-smple systems s wrtten as: v n v out d E = δ Qσ +δ Wσ + Hn + gzn + δ nn Hout + gzout + δ nout (3-63) n out m v wth de = d U + mg z + For any transent process we can use Eq. (3-58) to obtan the rate of change of mass or moles (dn/dt) for the system and dfferentate Eq. (3-63) to obtan the rate of change of the system s total energy (de/dt). Thus, de δ δ δ = dt δt δt δ t Qσ Wσ v n nn Hn gzn n out H out + gz out out v + δ n δ t out (3-64)

26 6 to avod ambgutes, a mass bass should be selected f knetc and potental energy effects are to be ncluded. The sums run over all ncomng and outgong streams. For operaton at steady state wth sngle nlet and outlet streams, Eq. (3-64) s greatly smplfed because both d E dn = 0 and = 0 dt dt As for the system, the total energy E and mass/moles N are constant. Insulaton σ Surface δw σ Center of Mass δn n <v> σ Surface Open to Mass at ths Pont whch mples that δ n n = δ n out and δ nn δ nout δ n = = n& δ t δ t δ t Therefore, Eq. (3-64) s modfed for steady state condtons to: where δq σ < z > z out δn out at P out' V out wth E out Generalzed open, non-smple system movng at velocty <v> n a gravtatonal feld reference to z = 0. Adapted from Tester, J. W. and Modell, Mchael. Thermodynamcs and Its Applcatons. Upper Saddle Rver, NJ: Prentce Hall PTR, 997, p. 49. Image by MIT OCW. (3-66) δ Qσ δ Wσ + = Q& W& σ + σ = ( H ss + PEss + KEss ) n& (3-67) δ t δ t v out v H ss Hout Hn ; PEss g( zout zn ) ; KEss represent the tme-nvarant changes between the outlet and nlet stream quanttes. In some practcal stuatons, other forms of macroscopc energy storage besdes knetc or gravtatonal potental energy may be mportant. To account for these effects, whch could nclude storage n rotatonal modes, electrc or magnetc felds, and elastc deformaton by stressstran effects, t s relatvely straghtforward to modfy Eqs. (3-63) and (3-64) (see Chapter 8). Further, you may encounter problems where ncomng and outgong stream fluxes are dstrbuted on the σ-boundary that encloses the system. In addton, the heat flux may also be dstrbuted on the σ-surface rather than occurrng at dscrete locatons and there may be multple work nteractons. In these cases, an ntegral form of Eq. (3-64) should be used: z n z = 0 n 3

27 7 V ( ρe) v a? dv = q n d σ + W& σ H + gz+ v n daσ (3-68) t aσ aσ where q and v represent vectors for the heat flux and flud velocty, and n s the unt normal vector, perpendcular to the σ-surface pontng outward. Note that there are two types of ntegrals nvolved, one for the total energy over the volume V of the system and another for the heat and mass flux contrbutons over the surface area of the σ-boudary a F. Note that we have nserted an underbar wth the term a F to emphasze that as used here t s the total extensve area. ρ H + gz + v / and v are spatally dependent. The parameters q, ( ) 4

28 8 Energy and the st law Treatment Summary of mathematcal forms System Expresson. closed, general d E = δ Q + δ W. closed, smple d E = du = δ Q + δ W 3. closed, smple, du = δ Q PdV only PdV work 4. open, smple du = δ Qσ +δ Wσ + Hnδnn Houtδnout 5. open, non-smple PE+KE only (use mass bass) 6. steady state, open, non-smple 7. non-smple, open dstrbuted nteracton along σ-surface d E δ Q +δ W = σ σ n + H n n + gz ( v ) m d E = d U + mg z + d E = 0 dm = 0 or dn = 0 d E dm dn = = = 0 dt dt dt δ nn δ nout = = n& δ t δ t plus expresson (5) d E dt H δ Q δ t δ W + δ t σ σ = 0 ss = H out H n PE n out n v + δ m n H out (f no reactons) = ( H + PE + KE )n& ss ss = g ss ( z z ) out n ss KE ss out v = ( ρe) v q n a & ρ + gz out out v out v + dv = d + W H + gz+ v n da t V a a σ n σ σ, σ σ δ m out 0.40/stlaw.doc 9//0 5

29 9 Supplemental Notes for Chapter 4 Second Law: Concepts and Applcatons Real, Irreversble, Quas-statc, and Reversble Partally quas-statc Quas-statc Real (Irreversble) Quas-statc processes Reversble Internally reversble - Along a quas-statc path all ntermedate states are equlbrum states; thus from postulate I quas-statc paths for closed, smple systems can be descrbed by two ndependent propertes. - From postulate II, f a system progressng along a quas-statc path s solated from ts envronment, then the values of all propertes wll reman constant and equal to those just before the solaton. - Quas-statc processes occur at fnte rates but not so rapdly that the system s able to adjust on a molecular level. There would not be, n general, gradents of any ntensve propertes, such as temperature, pressure, densty, etc. - Expandng a gas contaned n a frctonless pston (mass m, area a)-cylnder s not a quas-statc process:. pull stops P a. rapd expanson durng whch a defnte dp/dz gradent exsts n the gas phase P 3. pston moves rapdly as P gas s greater than gas P a + mg a

30 30 Wthout any frcton present, the gas expanson wll clearly not be quas-statc. If frcton s present so that the expanson process occurs very slowly, dp/dz would be neglgble and the propertes of the gas would reman constant f the expanson process were stopped that s, the system would stay n some stable equlbrum state. Thus, wth frcton present n ths manner, the expanson process s quas-statc. A smlar stuaton s encountered n the gas cylnder blowdown. The valve controls the blowdown rate, resultng n a quas-statc process for the gas contaned n the cylnder. In fact, the adabatc tank blowdown process could be modeled as a closed system gas expanson aganst a massless pston that s frctonally damped to keep P gas = P outsde. In ths case, the process s quas-statc and: d U = Pd V = Pd ( NV ) snce N = constant d U = NC dt v = NPdV for an deal gas wth P = RT/V, by elmnatng V, we get dt/t = R/C v (dv/v) = R/C v ( dt/t dp/p) (R/C v +)dt/t = (R/C v )dp/p If we had elmnated T, then dp/p =-(+R/C v ) dv/v and the same equatons result by treatng the system as open. Upon ntegraton, we obtan the famlar relatonshps for a reversble, adabatc expanson (or compresson) of an deal gas, namely, where PV κ = constant or equvalently T/T = (P/P ) R/Cp = (P/P ) ( Κ -)/ Κ κ C p / C and C p = C v + R Reversble Processes v - Va Postulate II, f any (real or deal) system n a non-equlbrum state s solated, t wll tend toward a state of equlbrum. - All real or natural processes are not reversble. Hence reversble processes are only dealzatons that are very useful n showng lmtng behavor. The performance of real processes s frequently compared wth deal performance under reversble condtons. - "In a reversble process, all systems must be n states of equlbrum at all tmes, that s all subsystems must traverse quas-statc paths." - A system undergong a reversble process s no more than dfferentally removed from an equlbrum state the system passes through a set of equlbrum states.

31 3 - "A process wll be called reversble f a second process could be performed n at least one way so that the system and all elements of ts envronment can be restored to ther respectve ntal states, except for dfferental changes of second order." For example, n a reversble expanson or compresson δ(δw) dpdv - If a cyclc process A B A s reversble, then when the process s carred out, no changes wll occur n any other bodes. For example, f A B nvolves the absorpton of a quantty of heat Q, then B A wll reject the same quantty Q to the envronment. true. - Any reversble process s also quas-statc, but the reverse s not necessarly - Smple systems undergong reversble processes have no nternal gradents of temperature or pressure. - Frcton and other dsspatve forces are not present n reversble processes. A truly reversble process wll always requre an nfntesmal drvng force to ensure that energy transfer occurs wthout degradaton, hence ts rate would be nfntely slow. Therefore, a reversble process always can be shown to requre a mnmum a mount of work or wll yeld a maxmum amount of work. - Heat engnes n reversble processes operate at maxmum effcency Summary of the nd Law - The st law nvolves prmarly the prncple of energy conservaton and s not suffcent to descrbe how a natural process wll proceed. - The nd law s concerned wth descrbng the drecton s whch a process can take place. For example, the flow of heat from a hot to a cold body. - The nd law descrbes, n mathematcal terms, the physcal mpossblty of reversng Joule's experments. It s not possble to convert heat nto an equvalent amount of work some amount of heat must be transferred to a second body or the envronment n the process of convertng heat nto work. - Carnot heat engnes operate cyclcally and reversbly T H between two sothermal reservors at T H and T δq C H δw and a work reservor. The effcency of the c δq C Carnot process for convertng heat nto work s η c = δw c /δq H = (T H T C )/T H T H >T C so 0 < η c.0 T C 3

32 3 Gven a reversble process where temperature changes, t s always possble to fnd a reversble zg-zag path consstng of adabatc-sothermal-adabatc steps such that the heat nteracton n the sothermal step s equal to the heat nteracton of the orgnal process. Defnton of entropy S as an derved state functon ds S = d S and d S = 0 δ Q rev / T (an exact dfferental) Clausus nequalty for descrbng heat nteractons between two sothermal reservors at T A and T B. δ Q A / T A + δ Q B / T B 0 For any fully reversble process, the equalty apples, for all others the nequalty apples A reversble process adabatc process occurs at constant entropy. For any real process; () ds > 0 (system + surroundngs) () S unverse = S system + S surroundn gs > 0 Entropy s a measure of the degradaton of work producng potental. All natural processes n solated, closed systems always occur n a drecton that ncreases entropy. 4

33 33 Combned Frst and Second Laws. Closed, sngle phase, smple systems From the st Law: de = du = δ Q + δ W For an nternally reversble, quas-statc process wth only PdV work: δq = δq = Td S and δw = δw rev = PdV rev Therefore, du= TdS PdV whch also provdes a way to determne entropy changes: Bass: mole of deal gas of constant C p ds=du/t + P/TdV C v P ds = dt + dv T T wth PV=RT P R dp ds = C v dt + T dt RT T P P combnng terms: or C v + R dp ds = dt R T P d S T P = S = C p ln R ln For deal gas only T P 5

34 34. Open, sngle phase, smple systems For an nternally reversble, quas-statc process wth one component enterng and leavng the system, all ntensve propertes must reman the same. Hence, σ Surface δw rev Insulaton δn n ( E, V, S, T, P) T n = T out = T P n = P out = P U n = U out = U S n = S out = S V n = V out = V δq rev δn out Adapted from Tester, J. W. and Mode ll, Mchael. Thermodynamcs and Its Applcat ons. Upper Saddle Rver, NJ: Prentce Hall PTR, 997, p. 87. Image by MIT OCW. From a mole balance, dn = δn n δn out. Now the st Law, whch descrbes an energy balance, can be wrtten wth only PdV work as: d E = d U = δq Pd V + ( U + PV dn and lkewse an entropy balance can be formulated as: rev d S = δq rev / T + SdN + δ S gen = δ Q / T + SdN snce δs gen = 0 for ths reversble case. Now by substtutng δ Q rev nto the st Law expresson: d U = Td S ) ) rev Pd V + ( U + PV TS dn = Td S where μ s the chemcal potental defned as: μ G U + PV TS = H TS Pd V + µ dn Ths result can be generalzed for a multcomponent, sngle phase system that s traversng a quas-statc path as, 6

35 35 n d U = Td S PdV + µ dn where U s a contnuous functon of n + varables: U = f ( S, V, N ) =, n and d U = ( U / δ S ) V, N d S ( d ( + U / V ) S, N V + U / N ) S V n,, N dn j () T -P Whch s often referred to as the Fundamental Equaton of Thermodynamcs. 3. Avalablty (maxmum and mnmum work concepts) As descrbed n Secton 4., consder a process shown at the rght that nteracts wth a work reservor and rejects heat to the surroundngs. Other constrants are: - reversble/quas-statc operaton - steady state δn δ n n = δ n out thus δ N = 0 for prmary system - Carnot heat engne operates cyclcly so δ Q s s at the system temperature T δnout Small carnot engne Secondary system Prmary system δq s δq R Heat reservor at T o δw s δw c δn n Work reservor Adapted from Tester, J. W. and Mode ll, Mchael. Thermodynamcs and Its Applcatons. Upper Saddle Rver, NJ: Prentce Hall PTR, 997, p all heat δq R s rejected sothermally at T # Image by MIT OCW. A dfferental steady state st Law balance around the process (prmary system) yelds: d E = 0 = δq + δw s + ( H H out ) δn () s n Where δ W s represents the shaft work contrbuton. A steady state st law balance around the Carnot heat engne gves: d E = 0 = δq s + δq R + δw c Where the mnus sgn on δ Q s reflects ts drectonal change relatve to the prmary system,.e., the engne s recevng heat from the system. δq = δq R + δw c () s Where δw c represents the Carnot heat engne work contrbuton. A steady state entropy balance for the composte secondary system the process and the heat engne yelds: 7

36 36 d S = 0 + = δq R / T + ( S S out ) δn (3) o n Combnng equatons (), (), and (3) to elmnate δ Q and δ Q R and solvng for the total s work nteracton (Carnot + shaft work) gves wth rearrangement: δ W = δ W total = ( δw c + δw s ) = [( H out H n ) T o ( S out S n )] δn (4) Snce we are dealng wth a reversble process, Eq. (4) gves the maxmum work per mole that could be produced (or the mnmum work requred). Ths s called the avalablty or exergy: B H S T o B H out H n T o ( S out S n ) = H T o S (5) Now Eq. (4) can be rewrtten δw max = ( δw + δw s ) or by takng the tme dervatve to gve maxmum power: c = B δn (6) B B n (7) W max = ( ) δn / δt = ( ) Clearly, the avalablty s a state functon n the strctest mathematcal sense so the maxmum (or mnmum) work assocated wth any steady state process s also ndependent of the path. H:\0.40\ndlaw.doc (9//0) 8

37 37 Supplementary Notes for Chapter 5 The Calculus of Thermodynamcs Objectves of Chapter 5. to understand the framework of the Fundamental Equaton ncludng the geometrc and mathematcal relatonshps among derved propertes (U, S, H, A, and G). to descrbe methods of dervatve manpulaton that are useful for computng changes n derved property values usng measurable, expermentally accessble propertes lke T, P, V, N, x, and ρ. 3. to ntroduce the use of Legendre Transformatons as a way of alternatng the Fundamental Equaton wthout losng nformaton content Startng wth the combned st and nd Laws and Euler s theorem we can generate the Fundamental Equaton: Recall for the combned st and nd Laws: Reversble, quas-statc Only PdV work Smple, open system (no KE, PE effects) For an n component system and Euler s Theorem: du = Td S PdV + du = Td S PdV + n = n = ( H TS ) µ dn dn Apples to all smoothly-varyng homogeneous functons f, f(a,b,, x,y, ) where a,b, ntensve varables are homogenous to zero order n mass and x,y, extensve varables are homogeneous to the st degree n mass or moles (N). df s an exact dfferental (not path dependent) and can be ntegrated drectly f Y = ky and X = kx then Modfed: /9/03

38 38 f(a,b,, X,Y, ) = k f(a,b,, x,y, ) and f x x a, b,..., y,.. + f y y a, b,.., x, = () f ( a, b,... x, y,... ) Fundamental Equaton: Can be obtaned va Euler ntegraton of combned st and nd Laws Expressed n Energy (U) or Entropy (S) representaton U = f or S = f u s [ S, V, N, N,..., Nn ] = T S PV + U T P T n = µ N [ U, V, N, N,..., Nn ] = + V N The followng secton summarzes a number of useful technques for manpulatng thermodynamc dervatve relatonshps Consder a general functon of n + varables ( ) F x, y,z,...,z + 3 n where x z, y z. Then expandng va the rules of multvarable calculus: n = µ T n+ F df = dz = z Now consder a process occurrng at constant F wth z 3,.., z n+ all held constant. Then F F df = 0 = dx + dy x y y,z 3,... x,z 3,...

39 39 Rearrangng, we get: Trple product x-y-z-(-) rule for F(x,y): ( F / x) ( x / y) ( y / F ) = y example: ( H / T ) ( T / P) ( P / H ) = P F H x T Add another varable to F(x,y): ( F / y) x = ( F / φ) x ( y / φ) x ( ) S S / T C p / T P example: F( x, y) = S( P, H )and φ = T then = = = / T H ( H / T ) C Dervatve nverson for F(x,y): ( F / y) x = / ( y / F ) x example: ( T / S) = / ( S / T ) = T / C p Maxwell s recprocty theorem: Apples to all homogeneous functons, e.g. F(x,y,..) ( F / x) dy P y,... ( F / y) x,... x,.. = x P y,.. P or F xy = F yx P p example: du = Td S PdV + n µ dn = / V ) S, N = U SV = U SV = ( P / S) V, N ( T = U = U V S VS 3

40 40 Legendre Transforms: General relatonshp ( x, ξ ) ( S, T ) ( V, P) ( N, µ ) ( x a, F ) (, σ) (extensve, ntensve) Conjugate coordnates Examples (0) y = f [ x,..., x ] ( bass functon) U = f [ S, V, N,... y ( k) k (0) = y ξ x = m N n ( k th transform) y () = A = U T S or by changng varable order to U = f (V, S, N,,N n ), () y = H = U + PV ] 4

41 4 General relatonshp Examples dy ( k) = k = x dξ + m ξ dx = k+ dy or dy () () d A = SdT PdV + d H = Td S + VdP + n = n = µ dn µ dn y ( m) dy ( m) = y = (0) m = m = ξ x dξ x ( + ) y n = 0 = 0 ( total transform wth m = n + ) ( n+ ) = 0 dy = SdT + VdP Ndµ = 0 (Gbbs-Duhem Equaton) Relatonshps among Partal Dervatves of Legendre Transforms y y ( k) j (0) = y ( k) j y = x x ( k) y = x x (0) j y (0) (Maxwell relaton) y = x (0) y () [NB : ξ y x = ξ ξ (0) = y = x = > y (0) (0) n = x [ ] as well for j > ] 5

42 4 Reorderng and Use of Tables Table 5.3 nd & 3 rd order dervatves of [ y () () j and y jk ] n terms of (0) y, etc Table 5.4 Relatons between nd order dervatves of j th Legendre transform ( j) y k and the bass functon y Table 5.5 Relatonshps among nd order dervatves of j th Legendre transform (j-q) transform y ( j q) k (0) k ( j) y k to \0.40\Ch5-Calc. of Thermo. Suppl. Notes Modfed: /9/03 6

43 43 Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below:. Notaton and operatonal equatons for mxtures. PVTN EOSs for mxtures 3. General effects of mxng on heat and work nteractons and state property changes 4. Gbbs-Duhem relatonshp and thermodynamc consstency 5. Mxng functons 6. Ideal gas mxtures and deal solutons 7. Fugacty and actvty concepts 8. Fugacty coeffcents from PVTN EOS property relatonshps 9. Actvty coeffcents from G EX property relatonshps 0. Ideal reversble work effects n mxng or separatng components. Notaton and operatonal equatons for mxtures Partal molar propertes are extensve propertes that determne how derved propertes change as a functon of mole number or composton. For a general property, B, whch could be U, H, S, V, or A, the partal molar B s defned as: B B = (9-8) N TPN,, j [] There are a few mportant relatonshps n Chapter 9 that allow you to calculate partal molar propertes gven the extensve form B of the property or the ntensve form B. An mportant one of these s: B B = B x j (9-53) j x j TPxj,,, Note that for a bnary, two component system that Eq.(9-53) quckly yelds the tangentntercept method of evaluatng partal molar propertes from a graph of B versus x j at constant T and P. For example, see Fgure 9. and the dscusson n the text.. PVTN EOSs for mxtures Volume and pressure explct equatons of state are commonly used to represent the volumetrc propertes of flud mxtures of both gases and lquds. The key feature that dstngushes a mxture EOS from ts pure component counterpart s the presence of compostonal dependence. Ths dependence expresses tself n the form of so-called mxng rules that ncorporate pure component EOS parameters and weght them [ ] Modfed: 0/6/003

44 44 proportonally to the concentraton of each component followng a specfc mathematcal recpe. Wth the excepton of the vral equaton of state no rgorous theoretcal approach exsts to specfy a mxng rule recpe. The most common approach s to use some varaton of the Lorentz-Bertlelot rules used by van der Waals and others over a century ago (see Eqs (9-7 and 9-8). In mxtures that exhbt consderable non-dealty, a bnary or hgher order nteracton parameter δ j s ntroduced to capture specfc nteractons between molecules of type and j. Each mxture PVTN EOS wll have some prescrbed recpe for ts mxng rules. For example see the conventons followed for the RK and PR EOS on pages In stuatons where hgh level non-deal effects are present often more complex forms for mxng rules are ntroduced. The Wong-Sandler and Chueh-Prausntz rules dscussed on pages are examples of ths type of mxng rule. 3. General effects of mxng on heat and work nteractons and state property changes There are a few typcal classes of problems that you should be able to solve. These nclude: Calculatng the change n enthalpy as a result of mxng to determne a heat nteracton for mantanng a constant temperature or to follow some prescrbed recpe for temperature, for example, see Problem 9.. Calculatng a change n volume as a result of mxng to determne a change n densty or the magntude of a work nteracton at constant pressure. 4. Gbbs-Duhem relatonshp and thermodynamc consstency For mxture data and correlatons for any property B, the partal molar quanttes are nterrelated through the use of the general Gbbs-Duhem relatonshp: n B B xdb = dt dp = T + Px, P Tx, Note that B s a functon of T, P, and x for =,, n. The Gbbs-Duhem relatonshp can be appled to any partal molar property, such as: H,V, µ, ln ˆφ, ln ˆf,G, ln a, ln γ,etc. For a bnary system, the Gbbs-Duhem relatonshp s frequently used to check thermodynamc consstency of thermodynamc data; for example, actvty coeffcents. Addtonally, for a bnary mxture, f you have a measurement of the partal molar property of one component as a functon of composton you can determne the property for the other component. Modfed: 0/6/003

45 45 5. Mxng functons n ( ) ( ) B B T,P,x, =,...,n x B T,P,x, =,...,n (9-68) mx = whch s n ntensve form. A smlar expresson results for B mx n extensve form wth x replaced by N. You should be famlar wth the concept of a reference state (+). Strctly speakng, reference states are arbtrary, but n practce several common forms appear for x +, T +, and P +, for example: () pure at T and P of the mxture () x + 0 at T and P of the mxture that s n an nfnte dluton state where Henry s law behavor s followed (3) a fxed composton of molal or molar concentraton at T and P of the mxture that behaves n some deal manner commonly used for electrolytes (see Chapter ) 6. Ideal gas mxtures and deal solutons In ths secton of the text, a set of defntons were used to characterze non-deal solutons n terms of a devaton from deal behavor. g deal gas mxture: G =µ = RT ln yp +λ( T ) ID deal soluton: G =µ = RT ln x +Λ ( T,P ) where λ and Λ are constants specfc to component. The key ponts to remember are that partal pressure y P s the deal gas mxture compostonally dependent varable whle for deal condensed phase soluton t s the mole fracton x. 7. Fugacty and actvty concepts For a real mxture or soluton we ntroduce the followng models: real flud mxture: G = µ = RT ln ˆf +λ real soluton: G =µ = RT ln a +µ ( T,P,x ) whch lead to the followng defntons of the fugacty coeffcent and actvty coeffcent ˆφ and γ, respectvely: and ˆ ˆf φ ---- represents a devaton from deal gas/flud mxture behavor yp ˆf f ˆ γ = ---- represents a devaton from deal soluton behavor a ˆ f + x Modfed: 0/6/003 3

46 46 8. Fugacty coeffcent relatonshps from PVTN EOS property models There are two basc approaches one nvolvng pressure explct EOSs lke the PR or RKS, and the other for volume explct EOSs, such as Vral or Correspondng States formulatons, ncludng compressblty expansons n densty or molar volume or smlarly structured equatons. Equatons (9-4) and (9-43) provde convenent forms for pressure explct EOS models for mxture and pure components, respectvely, whle Eqs(9-9) and (9-7) work for volume explct EOS models. For example, for component n a mxture: or V ln ˆ P RT RT φ = dv RT ln Z N V T,V,Nj [] RT ln ˆφ =+ P 0 RT V P dp At ths pont you should know how to calculate the fugacty or fugacty coeffcent for a pure component usng a pressure- or volume-explct EOS or for a component n a bnary mxture usng a sutable PVTN EOS that has been properly formulated wth mxng rules for ts constants (for example, the a mx and b mx constants n the RKS EOS) n terms of pure component propertes and a bnary nteracton parameter, eg δ j. Beng able to do ths provdes a powerful tool for calculatons requred later n the course, for example to estmate the vapor pressure of a pure component you would equate ˆφ for the lqud and vapor phases at a gven temperature by estmatng the P and usng the EOS to calculate lqud and vapor volumes (denstes) untl the fugacty coeffcents for each phase were equal. 9. Actvty coeffcents from G EX property relatonshps The actvty coeffcent s defned n terms of the partal molar excess Gbbs free energy of mxng and can be drectly related to the fugacty usng the defnton of actvty: EX EX G RT ln γ = G = N [] T,P,Nj + where γ a ˆ ˆ x = f f x To evaluate the partal dervatve we need an expresson for G EX : EX ID G Gmx G = f T, P, N ( =,..., n) whch represents the dfference between the actual enthalpy of mxng and the enthalpy of mxng for an deal soluton at the same T, P and composton as the actual mxture. Modfed: 0/6/003 4

47 47 Typcally, one has access to a model that gves G EX = G EX / N as a functon of T, P, and x, and wth Eq. (9-53) one can easly obtan γ for each component, e.g. EX EX G RT ln γ = G x j x j T,P,x j, In addton, the Gbbs-Duhem equaton can be appled to calculate the other actvty coeffcent gven a set of data for one component (e.g. f γ solvent s known as a functon of composton then γ solute can be estmated by ntegratng the Gbbs-Duhem relatonshp). Conversely, f both actvty coeffcents of a bnary mxture are known then the Gbbs- Duhem equaton can be used to check the thermodynamc consstency of a gven set of data. For example, the slope and area tests have been developed specfcally for ths purpose (see pages and Fg. 9.3). A key ssue here s how to deal wth the standard or reference state condton (+), as that wll have a drect effect on the magntude and behavor of γ. If a symmetrc reference state s used then ˆ f + = f( pure ) and the Lews and Randall rule s followed as x goes to.0 wth γ approachng.0. Alternatvely, a unsymmetrcal reference state can be used where the nfnte dluton behavor as defned by Henry s Law determnes that γ ** approaches.0 as x goes to 0. Ths s frequently called the McMllan-Mayer reference condton. Another popular alternatve commonly employed for electrolytes s to use a molal soluton at T and P of the mxture where the mxture follows Henry s Law n dlute solutons as x or m 0 (see Fg. 9.9 for example). The dscusson n the text from pages 360 to 365 should be carefully revewed to see how the actvty coeffcent s related to fugacty behavor for dfferent reference state condtons. Partcular attenton should be pad to Fgures Ideal reversble work assocated wth mxng or separatng components Under sothermal condtons the reversble work s equal to the net change n Gbbs free energy assocated wth the process, more generally the reversble work can be related drectly to the change n avalablty usng the methods ntroduced n Chapter 4. Examples 9.8 and 9.9 should be revewed to see how the concepts are employed. The approach s qute straghtforward f you are dealng wth a steady state process. Thngs are a bt more complcated f the system or process condtons are changng wth tme, but the general concept remans unchanged. [ ] Modfed: 0/6/003 5