Secton 4 Thermodynamcs Hendrck C. Van Ness, D.Eng., Howard. Isermann Department of Chemcal Engneerng, Rensselaer olytechnc Insttute; Fellow, Amercan Insttute of Chemcal Engneers; Member, Amercan Chemcal Socety Mchael M. Abbott, h.d., Howard. Isermann Department of Chemcal Engneerng, Rensselaer olytechnc Insttute; Member, Amercan Insttute of Chemcal Engneers INTRODUCTION ostulate 1................................................. 4-3 ostulate................................................. 4-3 ostulate 3................................................. 4-4 ostulate 4................................................. 4-4 ostulate 5................................................. 4-4 VARIABLES, DEFINITIONS, AND RELATIONSHIS Constant-Composton Systems................................ 4-5 Enthalpy and Entropy as Functons of T and.................. 4-5 Internal Energy and Entropy as Functons of T and V............ 4-6 Heat-Capacty Relatons.................................... 4-6 The Ideal Gas............................................ 4-6 Systems of Varable Composton............................... 4-7 artal Molar ropertes.................................... 4-7 Gbbs/Duhem Equaton.................................... 4-7 artal Molar Gbbs Energy................................. 4-8 The Ideal Gas State and the Compressblty Factor............. 4-8 Resdual ropertes........................................ 4-8 SOLUTION THERMODYNAMICS Ideal Gas Mxtures.......................................... 4-8 Fugacty and Fugacty Coeffcent.............................. 4-9 Fundamental Resdual-roperty Relaton........................ 4-9 The Ideal Soluton.......................................... 4-10 Fundamental Excess-roperty Relaton......................... 4-10 Summary of Fundamental roperty Relatons.................... 4-11 roperty Changes of Mxng................................... 4-11 Behavor of Bnary Lqud Solutons............................ 4-1 EVALUATION OF ROEIES Resdual-roperty Formulatons............................... 4-14 Lqud/Vapor hase Transton................................. 4-15 Lqud-hase ropertes...................................... 4-15 ropertes from VT Correlatons.............................. 4-15 tzer s Correspondng-States Correlaton..................... 4-16 Alternatve roperty Formulatons........................... 4-16 Vral Equatons of State.................................... 4-19 Generalzed Correlaton for the Second Vral Coeffcent......... 4-0 Cubc Equatons of State................................... 4-0 Benedct/Webb/Rubn Equaton of State...................... 4-1 Expressons for the Excess Gbbs Energy........................ 4- EQUILIBRIUM Crtera................................................... 4-4 The hase Rule............................................. 4-4 Example 1: Applcaton of the hase Rule..................... 4-5 Vapor/Lqud Equlbrum.................................... 4-5 Gamma/h Approach..................................... 4-5 Data Reducton........................................... 4-6 Solute/Solvent Systems..................................... 4-7 K-Values................................................ 4-8 Equaton-of-State Approach................................ 4-8 Lqud/Lqud and Vapor/Lqud/Lqud Equlbra................ 4-30 Chemcal-Reacton Stochometry.............................. 4-31 Chemcal-Reacton Equlbra................................. 4-31 Standard roperty Changes of Reacton....................... 4-31 Equlbrum Constants..................................... 4-3 Example : Sngle-Reacton Equlbrum...................... 4-3 Complex Chemcal-Reacton Equlbra....................... 4-33 Example 3: Mnmzaton of Gbbs Energy..................... 4-34 THERMODYNAMIC ANALYSIS OF ROCESSES Calculaton of Ideal Work..................................... 4-34 Lost Work................................................. 4-35 Analyss of Steady-State, Steady-Flow rocesses.................. 4-35 Example 4: Lost-Work Analyss.............................. 4-36 4-1
4- THERMODYNAMICS Nomenclature and Unts Symbols are omtted that are correlaton- or applcaton-specfc. U.S. customary U.S. customary Symbol Defnton SI unts unts Symbol Defnton SI unts unts A Helmholtz energy J Btu â Actvty of speces n Dmensonless Dmensonless soluton B d vral coeffcent, cm 3 /mol cm 3 /mol densty expanson C 3d vral coeffcent, cm 6 /mol cm 6 /mol densty expanson D 4th vral coeffcent, cm 9 /mol 3 cm 9 /mol 3 densty expanson B d vral coeffcent, ka 1 ka 1 pressure expanson C 3d vral coeffcent, ka ka pressure expanson D 4th vral coeffcent, ka 3 ka 3 pressure expanson B Interacton d vral cm 3 /mol cm 3 /mol coeffcent C k Interacton 3d vral cm 6 /mol cm 6 /mol coeffcent C Heat capacty at constant J/(molK) Btu/(lb molr) pressure C V Heat capacty at constant J/(molK) Btu/(lb molr) volume E K Knetc energy J Btu E Gravtatonal potental energy J Btu f Fugacty of pure speces ka ps fˆ Fugacty of speces n ka ps soluton G Molar or unt-mass Gbbs J/mol or J/kg Btu/lb mol energy or Btu/lbm g Acceleraton of gravty m/s ft/s g G E / H Molar or unt-mass enthalpy J/mol or J/kg Btu/lb mol or Btu/lbm K Equlbrum K-value, y /x Dmensonless Dmensonless K Equlbrum constant for Dmensonless Dmensonless chemcal reacton k Henry s constant ka ps M Molar or unt-mass value of any extensve thermodynamc property of a soluton M Molar or unt-mass value of any extensve property of pure speces M artal molar property of speces n soluton M roperty change of mxng M Standard property change of reacton m Mass kg lbm ṁ Mass flow rate kg/s lbm/s n Number of moles n Number of moles of speces Absolute pressure ka ps c Crtcal pressure ka ps Saturaton or vapor pressure ka ps of speces p artal pressure of speces ka ps n gas mxture (y ) Q Heat J Btu Q Rate of heat transfer J/s Btu/s R Unversal gas constant J/(molK) Btu/(lb molr) S Molar or unt-mass entropy J/(molK) Btu/(lb molr) or J/(kgK) or Btu/(lbR) T Absolute temperature K R T c Crtcal temperature, K R U Molar or unt-mass nternal J/mol or J/kg Btu/lb mol energy or Btu/lbm u Velocty m/s ft/s V Molar or unt-mass volume m 3 /mol ft 3 /lb mol or m 3 /kg or ft 3 /lbm W Work J Btu W s Shaft work for flow process J Btu Ẇ s Shaft power for flow process J/s Btu/s x Mole fracton n general Dmensonless Dmensonless or lqud-phase mole fracton of speces n soluton y Vapor-phase mole fracton Dmensonless Dmensonless of speces n soluton Z Compressblty factor Dmensonless Dmensonless z Elevaton above a datum m ft level E d g l lv R t v C c H r rev Superscrpts Denotes excess thermodynamc property Denotes value for an deal soluton Denotes value for an deal gas Denotes lqud phase Denotes phase transton from lqud to vapor Denotes resdual thermodynamc property Denotes a total value of a thermodynamc property Denotes vapor phase Denotes a value at nfnte dluton Subscrpts Denotes a value for a colder heat reservor Denotes a value for the crtcal state Denotes a value for a hotter heat reservor Denotes a reduced value Denotes a reversble process Greek letters α,β As superscrpts, dentfy phases β Volume expansvty, speces K 1 R 1 ε Reacton coordnate for mol lb mol reacton Γ (T) Defned by Eq. (4-7) J/mol Btu/lb mol γ Heat-capacty rato, C /C V Dmensonless Dmensonless γ Actvty coeffcent of speces Dmensonless Dmensonless n soluton µ Chemcal potental of speces J/mol Btu/lb mol ν, Stochometrc number of Dmensonless Dmensonless speces n reacton ρ Molar densty mols/m 3 lb moles/ft 3 σ As a subscrpt, denotes a heat reservor Φ Defned by Eq. (4-83) Dmensonless Dmensonless φ Fugacty coeffcent of pure Dmensonless Dmensonless speces φˆ Fugacty coeffcent of Dmensonless Dmensonless speces n soluton ω Acentrc factor Dmensonless Dmensonless
GENERAL REFERENCES: Abbott, M.M., and H.C. Van Ness, Schaum s Outlne of Theory and roblems of Thermodynamcs, d ed., McGraw-Hll, New York, 1989. Tester, J.W. and M. Modell, Thermodynamcs and ts Applcatons, 3d ed., rentce-hall, Englewood Clffs, N.J., 1996. rausntz, J.M., R.N. Lchtenthaler, and E.G. de Azevedo, Molecular Thermodynamcs of Flud-hase Equlbra, d ed., rentce-hall, Englewood Clffs, N.J., 1986. Red, R.C., J.M. rausntz, and B.E. olng, The ropertes of Gases and Lquds, 4th ed., McGraw-Hll, New York, 1987. Sandler, S.I., Chemcal and Engneerng Thermodynamcs, d ed., Wley, New York, 1989. Smth, J.M., H.C. Van Ness, and M.M. Abbott, Introducton to Chemcal Engneerng Thermodynamcs, 5th ed., McGraw-Hll, New York, 1996. Van Ness, H.C., and M.M. Abbott, Classcal Thermodynamcs of Nonelectrolyte Solutons: Wth Applcatons to hase Equlbra, McGraw-Hll, New York, 198. INTRODUCTION Thermodynamcs s the branch of scence that embodes the prncples of energy transformaton n macroscopc systems. The general restrctons whch experence has shown to apply to all such transformatons are known as the laws of thermodynamcs. These laws are prmtve; they cannot be derved from anythng more basc. The frst law of thermodynamcs states that energy s conserved; that, although t can be altered n form and transferred from one place to another, the total quantty remans constant. Thus, the frst law of thermodynamcs depends on the concept of energy; but, conversely, energy s an essental thermodynamc functon because t allows the frst law to be formulated. Ths couplng s characterstc of the prmtve concepts of thermodynamcs. The words system and surroundngs are smlarly coupled. A system s taken to be any obect, any quantty of matter, any regon, and so on, selected for study and set apart (mentally) from everythng else, whch s called the surroundngs. The magnary envelope whch encloses the system and separates t from ts surroundngs s called the boundary of the system. Attrbuted to ths boundary are specal propertes whch may serve ether (1) to solate the system from ts surroundngs, or () to provde for nteracton n specfc ways between system and surroundngs. An solated system exchanges nether matter nor energy wth ts surroundngs. If a system s not solated, ts boundares may permt exchange of matter or energy or both wth ts surroundngs. If the exchange of matter s allowed, the system s sad to be open; f only energy and not matter may be exchanged, the system s closed (but not solated), and ts mass s constant. When a system s solated, t cannot be affected by ts surroundngs. Nevertheless, changes may occur wthn the system that are detectable wth such measurng nstruments as thermometers, pressure gauges, and so on. However, such changes cannot contnue ndefntely, and the system must eventually reach a fnal statc condton of nternal equlbrum. For a closed system whch nteracts wth ts surroundngs, a fnal statc condton may lkewse be reached such that the system s not only nternally at equlbrum but also n external equlbrum wth ts surroundngs. The concept of equlbrum s central n thermodynamcs, for assocated wth the condton of nternal equlbrum s the concept of state. A system has an dentfable, reproducble state when all ts propertes, such as temperature T, pressure, and molar volume V, are fxed. The concepts of state and property are agan coupled. One can equally well say that the propertes of a system are fxed by ts state. Although the propertes T,, and V may be detected wth measurng nstruments, the exstence of the prmtve thermodynamc propertes (see ostulates 1 and 3 followng) s recognzed much more ndrectly. The number of propertes for whch values must be specfed n order to fx the state of a system depends on the nature of the system and s ultmately determned from experence. When a system s dsplaced from an equlbrum state, t undergoes a process, a change of state, whch contnues untl ts propertes attan new equlbrum values. Durng such a process the system may be caused to nteract wth ts surroundngs so as to nterchange energy n the forms of heat and work and so to produce n the system changes consdered desrable for one reason or another. A process that proceeds so that the system s never dsplaced more than dfferentally from an equlbrum state s sad to be reversble, because such a process can be reversed at any pont by an nfntesmal change n external condtons, causng t to retrace the ntal path n the opposte drecton. Thermodynamcs fnds ts orgn n experence and experment, from whch are formulated a few postulates that form the foundaton of the subect. The frst two deal wth energy: OSTULATE 1 There exsts a form of energy, known as nternal energy, whch for systems at nternal equlbrum s an ntrnsc property of the system, functonally related to ts characterstc coordnates. OSTULATE (FIRST LAW OF THERMODYNAMICS) The total energy of any system and ts surroundngs s conserved. Internal energy s qute dstnct from such external forms as the knetc and potental energes of macroscopc bodes. Although a macroscopc property characterzed by the macroscopc coordnates T and, nternal energy fnds ts orgn n the knetc and potental energes of molecules and submolecular partcles. In applcatons of the frst law of thermodynamcs, all forms of energy must be consdered, ncludng the nternal energy. It s therefore clear that ostulate depends on ostulate 1. For an solated system, the frst law requres that ts energy be constant. For a closed (but not solated) system, the frst law requres that energy changes of the system be exactly compened by energy changes n the surroundngs. Energy s exchanged between such a system and ts surroundngs n two forms: heat and work. Heat s energy crossng the system boundary under the nfluence of a temperature dfference or gradent. A quantty of heat Q represents an amount of energy n transt between a system and ts surroundngs, and s not a property of the system. The conventon wth respect to sgn makes numercal values of Q postve when heat s added to the system and negatve when heat leaves the system. Work s agan energy n transt between a system and ts surroundngs, but resultng from the dsplacement of an external force actng on the system. Lke heat, a quantty of work W represents an amount of energy, and s not a property of the system. The sgn conventon, analogous to that for heat, makes numercal values of W postve when work s done on the system by the surroundngs and negatve when work s done on the surroundngs by the system. When appled to closed (constant-mass) systems for whch the only form of energy that changes s the nternal energy, the frst law of thermodynamcs s expressed mathematcally as du t = dq + dw (4-1) 4-3
4-4 THERMODYNAMICS where U t s the total nternal energy of the system. Note that dq and dw, dfferental quanttes representng energy exchanges between the system and ts surroundngs, serve to account for the energy change of the surroundngs. On the other hand, du t s drectly the dfferental change n nternal energy of the system. Integraton of Eq. (4-1) gves for a fnte process U t = Q + W (4-) where U t s the fnte change gven by the dfference between the fnal and ntal values of U t. The heat Q and work W are fnte quanttes of heat and work; they are not propertes of the system nor functons of the thermodynamc coordnates that characterze the system. OSTULATE 3 There exsts a property called entropy, whch for systems at nternal equlbrum s an ntrnsc property of the system, functonally related to the measurable coordnates whch characterze the system. For reversble processes, changes n ths property may be calculated by the equaton: ds t = dq rev /T (4-3) where S t s the total entropy of the system and T s the absolute temperature of the system. OSTULATE 4 (SECOND LAW OF THERMODYNAMICS) The entropy change of any system and ts surroundngs, consdered together, resultng from any real process s postve, approachng zero when the process approaches reversblty. In the same way that the frst law of thermodynamcs cannot be formulated wthout the pror recognton of nternal energy as a property, so also the second law can have no complete and quanttatve expresson wthout a pror asserton of the exstence of entropy as a property. The second law requres that the entropy of an solated system ether ncrease or, n the lmt, where the system has reached an equlbrum state, reman constant. For a closed (but not solated) system t requres that any entropy decrease n ether the system or ts surroundngs be more than compened by an entropy ncrease n the other part or that n the lmt, where the process s reversble, the total entropy of the system plus ts surroundngs be constant. The fundamental thermodynamc propertes that arse n connecton wth the frst and second laws of thermodynamcs are nternal energy and entropy. These propertes, together wth the two laws for whch they are essental, apply to all types of systems. However, dfferent types of systems are characterzed by dfferent sets of measurable coordnates or varables. The type of system most commonly encountered n chemcal technology s one for whch the prmary characterstc varables are temperature T, pressure, molar volume V, and composton, not all of whch are necessarly ndependent. Such systems are usually made up of fluds (lqud or gas) and are called VT systems. For closed systems of ths knd, the work of a reversble process may always be calculated from dw rev = dv t (4-4) where s the absolute pressure and V t s the total volume of the system. Ths equaton follows drectly from the defnton of mechancal work. OSTULATE 5 The macroscopc propertes of homogeneous VT systems at nternal equlbrum can be expressed as functons of temperature, pressure, and composton only. Ths postulate mposes an dealzaton, and s the bass for all subsequent property relatons for VT systems. The VT system serves as a sfactory model n an enormous number of practcal applcatons. In acceptng ths model one assumes that the effects of felds (e.g., electrc, magnetc, or gravtatonal) are neglgble and that surface and vscous-shear effects are unmportant. Temperature, pressure, and composton are thermodynamc coordnates representng condtons mposed upon or exhbted by the system, and the functonal dependence of the thermodynamc propertes on these condtons s determned by experment. Ths s qute drect for molar or specfc volume V, whch can be measured, and leads mmedately to the concluson that there exsts an equaton of state relatng molar volume to temperature, pressure, and composton for any partcular homogeneous VT system. The equaton of state s a prmary tool n applcatons of thermodynamcs. ostulate 5 affrms that the other molar or specfc thermodynamc propertes of VT systems, such as nternal energy U and entropy S, are also functons of temperature, pressure, and composton. These molar or unt-mass propertes, represented by the plan symbols V, U, and S, are ndependent of system sze and are called ntensve. Temperature, pressure, and the composton varables, such as mole fracton, are also ntensve. Total-system propertes (V ț U ț S t ) do depend on system sze, and are extensve. For a system contanng n moles of flud, M t = nm, where M s a molar property. Applcatons of the thermodynamc postulates necessarly nvolve the abstract quanttes nternal energy and entropy. The soluton of any problem n appled thermodynamcs s therefore found through these quanttes. VARIABLES, DEFINITIONS, AND RELATIONSHIS Consder a sngle-phase closed system n whch there are no chemcal reactons. Under these restrctons the composton s fxed. If such a system undergoes a dfferental, reversble process, then by Eq. (4-1) du t = dq rev + dw rev Substtuton for dq rev and dw rev by Eqs. (4-3) and (4-4) gves du t = T ds t dv t Although derved for a reversble process, ths equaton relates propertes only and s vald for any change between equlbrum states n a closed system. It may equally well be wrtten d(nu) = T d(ns) d(nv) (4-5) where n s the number of moles of flud n the system and s constant for the specal case of a closed, nonreactng system. Note that n n 1 + n + n 3 + = n where s an ndex dentfyng the chemcal speces present. When U, S, and V represent specfc (unt-mass) propertes, n s replaced by m. Equaton (4-5) shows that for the sngle-phase, nonreactng, closed system specfed, nu = u(ns, nv) Then d(nu) = (nu) nv,n d(ns) + (nu) ns,n d(nv) (ns) (nv) where the subscrpt n ndcates that all mole numbers n (and hence n) are held constant. Comparson wth Eq. (4-5) shows that (nu) = T (4-6) nv,n (ns) (nu) = (4-7) (nv) ns,n
VARIABLES, DEFINITIONS, AND RELATIONSHIS 4-5 Consder now an open system consstng of a sngle phase and assume that nu = (ns, nv, n 1, n, n 3,...) Then (nu) (nu) (nu) d(nu) = nv,n d(ns) + ns,n d(nv) + dn (ns) (nv) n ns,nv,n where the summaton s over all speces present n the system and subscrpt n ndcates that all mole numbers are held constant except the th. Let (nu) µ ns,nv,n n Together wth Eqs. (4-6) and (4-7), ths defnton allows elmnaton of all the partal dfferental coeffcents from the precedng equaton: d(nu) = T d(ns) d(nv) + µ dn (4-8) Equaton (4-8) s the fundamental property relaton for snglephase VT systems, from whch all other equatons connectng propertes of such systems are derved. The quantty µ s called the chemcal potental of speces, and t plays a vtal role n the thermodynamcs of phase and chemcal equlbra. Addtonal property relatons follow drectly from Eq. (4-8). Snce n = x n, where x s the mole fracton of speces, ths equaton may be rewrtten: d(nu) T d(ns) + d(nv) µ d(x n) = 0 Upon expanson of the dfferentals and collecton of lke terms, ths becomes du T ds + dv µ dx n + U TS + V x µ dn = 0 Snce n and dn are ndependent and arbtrary, the terms n brackets must separately be zero. Then du = T ds dv + µ dx (4-9) U = TS V + x µ (4-10) Equatons (4-8) and (4-9) are smlar, but there s an mportant dfference. Equaton (4-8) apples to a system of n moles where n may vary; whereas Eq. (4-9) apples to a system n whch n s unty and nvarant. Thus Eq. (4-9) s subect to the constrant that x = 1 or that dx = 0. In ths equaton the x are not ndependent varables, whereas the n n Eq. (4-8) are. Equaton (4-10) dctates the possble combnatons of terms that may be defned as addtonal prmary functons. Those n common use are: Enthalpy H U + V (4-11) Helmholtz energy A U TS (4-1) Gbbs energy G U + V TS = H TS (4-13) Addtonal thermodynamc propertes are related to these and arse by arbtrary defnton. Multplcaton of Eq. (4-11) by n and dfferentaton yelds the general expresson: d(nh) = d(nu) + d(nv) + nv d Substtuton for d(nu) by Eq. (4-8) reduces ths result to: d(nh) = T d(ns) + nv d + µ dn (4-14) The total dfferentals of na and ng are obtaned smlarly: d(na) = ns dt d(nv) + µ dn (4-15) d(ng) = ns dt + nv d + µ dn (4-16) Equatons (4-8) and (4-14) through (4-16) are equvalent forms of the fundamental property relaton. Each expresses a property nu, nh, and so on, as a functon of a partcular set of ndependent varables; these are the canoncal varables for the property. The choce of whch equaton to use n a partcular applcaton s dctated by convenence. However, the Gbbs energy G s specal, because of ts unque functonal relaton to T,, and the n, whch are the varables of prmary nterest n chemcal processng. A smlar set of equatons s developed from Eq. (4-9). Ths set also follows from the precedng set when n = 1 and n = x. The two sets are related exactly as Eq. (4-8) s related to Eq. (4-9). The equatons wrtten for n = 1 are, of course, less general. Furthermore, the nterdependence of the x precludes those mathematcal operatons whch depend on ndependence of these varables. CONSTANT-COMOSITION SYSTEMS For 1 mole of a homogeneous flud of constant composton Eqs. (4-8) and (4-14) through (4-16) smplfy to: du = T ds dv (4-17) dh = T ds + V d (4-18) da = S dt dv (4-19) dg = S dt + V d (4-0) Implct n these are the followng: T = U V = H (4-1) S S = U S = A T (4-) V V V = H S = G T (4-3) S = A V = G (4-4) T T In addton, the common Maxwell equatons result from applcaton of the recprocty relaton for exact dfferentals: T = S V SV (4-5) T = S V S (4-6) = V S T VT (4-7) V = S T (4-8) T In all these equatons the partal dervatves are taken wth composton held constant. Enthalpy and Entropy as Functons of T and At constant composton the molar thermodynamc propertes are functons of temperature and pressure (ostulate 5). Thus H H dh = dt + T d (4-9) T ds = S dt + S T d (4-30) T The obvous next step s to elmnate the partal-dfferental coeffcents n favor of measurable quanttes. The heat capacty at constant pressure s defned for ths purpose: H C (4-31) T It s a property of the materal and a functon of temperature, pressure, and composton. Equaton (4-18) may frst be dvded by dt and restrcted to constant pressure, and then be dvded by d and restrcted to constant temperature, yeldng the two equatons:
4-6 THERMODYNAMICS H S = T H S = T T T + V In vew of Eq. (4-31), the frst of these becomes T T and n vew of Eq. (4-8), the second becomes Heat-Capacty Relatons In Eqs. (4-34) and (4-41) both dh and du are exact dfferentals, and applcaton of the recprocty relaton T T leads to S C = (4-3) H V = V T T (4-33) T Combnaton of Eqs. (4-9), (4-31), and (4-33) gves V dh = C dt + V T d (4-34) T and n combnaton Eqs. (4-30), (4-3), and (4-8) yeld C V ds = dt d (4-35) T T Equatons (4-34) and (4-35) are general expressons for the enthalpy and entropy of homogeneous fluds at constant composton as functons of T and. The coeffcents of dt and d are expressed n terms of measurable quanttes. Internal Energy and Entropy as Functons of T and V Because V s related to T and through an equaton of state, V rather than can serve as an ndependent varable. In ths case the nternal energy and entropy are the propertes of choce; whence U U du = V dt + T dv (4-36) T V S S ds = V dt + T dv (4-37) T V The procedure now s analogous to that of the precedng secton. Defne the heat capacty at constant volume by U C V V (4-38) T It s a property of the materal and a functon of temperature, pressure, and composton. Two relatons follow mmedately from Eq. (4-17): U S = T V T TV U S = T T T V V As a result of Eq. (4-38) the frst of these becomes S C V T T and as a result of Eq. (4-7), the second becomes V = (4-39) U = T T V (4-40) V T Combnaton of Eqs. (4-36), (4-38), and (4-40) gves du = C V dt + T V dv (4-41) T and Eqs. (4-37), (4-39), and (4-7) together yeld C V ds = dt + V dv (4-4) T T Equatons (4-41) and (4-4) are general expressons for the nternal energy and entropy of homogeneous fluds at constant composton as functons of temperature and molar volume. The coeffcents of dt and dv are expressed n terms of measurable quanttes. C = T T V (4-43) T C = T T V V (4-44) T V Thus, the pressure or volume dependence of the heat capactes may be determned from VT data. The temperature dependence of the heat capactes s, however, determned emprcally and s often gven by equatons such as C =α+βt +γt Equatons (4-35) and (4-4) both provde expressons for ds, whch must be equal for the same change of state. Equatng them and solvng for dt gves T V T dt = d + dv C C V T C C V T V However, at constant composton T = T(,V), and T T dt = V d + dv V Equatng coeffcents of ether d or dv n these two expressons for dt gves V C C V = T (4-45) T T V Thus the dfference between the two heat capactes may be determned from VT data. Dvson of Eq. (4-3) by Eq. (4-39) yelds the rato of these heat capactes: C ( S/ T) = ( S/ V) = ( V/ T) CV ( S/ T)V ( S/ )V ( / T) V Replacement of each of the four partal dervatves through the approprate Maxwell relaton gves fnally C γ V = T (4-46) CV V S where γ s the symbol conventonally used to represent the heatcapacty rato. The Ideal Gas The smplest equaton of state s the deal gas equaton: V = where R s a unversal constant, values of whch are gven n Table 1-9. The followng partal dervatves are obtaned from the deal gas equaton: R = = V = 0 T V V R V = = V = 0 T = T V V The general equatons for constant-composton fluds derved n the precedng subsectons reduce to very smple forms when the relatons for an deal gas are substtuted nto them: U = T H T = 0 V T T V S R S = T = T du = C V dt T T V R V
VARIABLES, DEFINITIONS, AND RELATIONSHIS 4-7 dh = C dt ds = dt + dv ds = dt d C V C = T T = 0 V C V T C T C C C V = R γ ln = ln VS CV These equatons clearly show that for an deal gas U, H, C, and C V are functons of temperature only and are ndependent of and V. The entropy of an deal gas, however, s a functon of both T and or of both T and V. SYSTEMS OF VARIABLE COMOSITION The composton of a system may vary because the system s open or because of chemcal reactons even n a closed system. The equatons developed here apply regardless of the cause of composton changes. artal Molar ropertes Consder a homogeneous flud soluton comprsed of any number of chemcal speces. For such a VT system let the symbol M represent the molar (or unt-mass) value of any extensve thermodynamc property of the soluton, where M may stand n turn for U, H, S, and so on. A total-system property s then nm, where n = n and s the ndex dentfyng chemcal speces. One mght expect the soluton property M to be related solely to the propertes M of the pure chemcal speces whch comprse the soluton. However, no such generally vald relaton s known, and the connecton must be establshed expermentally for every specfc system. Although the chemcal speces whch make up a soluton do not n fact have separate propertes of ther own, a soluton property may be arbtrarly apportoned among the ndvdual speces. Once an apportonng recpe s adopted, then the assgned property values are qute logcally treated as though they were ndeed propertes of the speces n soluton, and reasonng on ths bass leads to vald conclusons. For a homogeneous VT system, ostulate 5 requres that nm = (T,, n 1, n, n 3,...) The total dfferental of nm s therefore (nm) d(nm) =,n dt + (nm) (nm) T,n d + dn T n T,,n where subscrpt n ndcates that all mole numbers n are held constant, and subscrpt n sgnfes that all mole numbers are held constant except the th. Ths equaton may also be wrtten M M (nm) dn T,,n d(nm) = n,x dt + n T,x d + T n where subscrpt x ndcates that all mole fractons are held constant. The dervatves n the summaton are called partal molar propertes M; by defnton, M (nm) T,,n (4-47) n The bass for calculaton of partal propertes from soluton propertes s provded by ths equaton. Moreover, the precedng equaton becomes d(nm) = n M,x dt + n M T,x d + M dn (4-48) T Important equatons follow from ths result through the relatons: d(nm) = n dm + M dn dn = d(x n) = x dn + n dx R V R Combnng these expressons wth Eq. (4-48) and collectng lke terms gves dm M M dt,x T,x d T M dx n + M M x dn = 0 Snce n and dn are ndependent and arbtrary, the terms n brackets must separately be zero; whence dm =,x dt + T,x d + M dx (4-49) and M = x M (4-50) Equaton (4-49) s merely a specal case of Eq. (4-48); however, Eq. (4-50) s a vtal new relaton. Known as the summablty equaton, t provdes for the calculaton of soluton propertes from partal propertes. Thus, a soluton property apportoned accordng to the recpe of Eq. (4-47) may be recovered smply by addng the propertes attrbuted to the ndvdual speces, each weghted by ts mole fracton n soluton. The equatons for partal molar propertes are also vald for partal specfc propertes, n whch case m replaces n and the x are mass fractons. Equaton (4-47) appled to the defntons of Eqs. (4-11) through (4-13) yelds the partal-property relatons: H = U + V A = U TS G = H TS ertnent examples on partal molar propertes are presented n Smth, Van Ness, and Abbott (Introducton to Chemcal Engneerng Thermodynamcs, 5th ed., Sec. 10.3, McGraw-Hll, New York, 1996). Gbbs/Duhem Equaton Dfferentaton of Eq. (4-50) yelds dm = x dm + M dx Snce ths equaton and Eq. (4-49) are both vald n general, ther rght-hand sdes can be equated, yeldng M dt +,x M T,x d T M T x dm = 0 (4-51) Ths general result, the Gbbs/Duhem equaton, mposes a constrant on how the partal molar propertes of any phase may vary wth temperature, pressure, and composton. For the specal case where T and are constant: x dm = 0 (constant T, ) (4-5) Symbol M may represent the molar value of any extensve thermodynamc property; for example, V, U, H, S, or G. When M H, the dervatves ( H/ T) and ( H/ ) T are gven by Eqs. (4-31) and (4-33). Equatons (4-49), (4-50), and (4-51) then become dh = C dt + V T,x d + H dx (4-53) H = x H (4-54) V T M V T C dt + V T,x d x dh = 0 (4-55) Smlar equatons are readly derved when M takes on other denttes. Equaton (4-47), whch defnes a partal molar property, provdes a general means by whch partal property values may be determned. However, for a bnary soluton an alternatve method s useful. Equaton (4-50) for a bnary soluton s M = x 1 M 1 + x M (4-56) Moreover, the Gbbs/Duhem equaton for a soluton at gven T and, Eq. (4-5), becomes x 1 dm 1 + x dm = 0 (4-57)
4-8 THERMODYNAMICS These two equatons can be combned to gve dm M 1 = M + x dx1 (4-58a) The followng equaton s a mathematcal dentty: ng 1 ng d d(ng) dt dm M = M x 1 (4-58b) dx1 Thus for a bnary soluton, the partal propertes are gven drectly as functons of composton for gven T and. For multcomponent solutons such calculatons are complex, and drect use of Eq. (4-47) s approprate. artal Molar Gbbs Energy Implct n Eq. (4-16) s the relaton (ng) µ = T,,n n In vew of Eq. (4-47), the chemcal potental and the partal molar Gbbs energy are therefore dentcal: µ = G (4-59) The recprocty relaton for an exact dfferental appled to Eq. (4-16) produces not only the Maxwell relaton, Eq. (4-8), but also two other useful equatons: µ (nv) = T,n T,,n = V (4-60) n µ =,n (ns) T,,n = S (4-61) T n In a soluton of constant composton, µ =µ(t,); whence µ dµ dg = µ,n dt + T,n d T or dg = S dt + V d (4-6) Comparson wth Eq. (4-0) provdes an example of the parallelsm that exsts between the equatons for a constant-composton soluton and those for the correspondng partal propertes. Ths parallelsm exsts whenever the soluton propertes n the parent equaton are related lnearly (n the algebrac sense). Thus, n vew of Eqs. (4-17), (4-18), and (4-19): du = T ds dv (4-63) dh = T ds + V d (4-64) da = S dt dv (4-65) Note that these equatons hold only for speces n a constantcomposton soluton. Substtuton for d(ng) by Eq. (4-16) and for G by H TS (Eq. [4-13]) gves, after algebrac reducton, ng nv nh µ d = d dt + dn (4-66) Equaton (4-66) s a useful alternatve to the fundamental property relaton gven by Eq. (4-16). All terms n ths equaton have the unts of moles; moreover, the enthalpy rather than the entropy appears on the rght-hand sde. The Ideal Gas State and the Compressblty Factor The smplest equaton of state for a VT system s the deal gas equaton: V g = where V g s the deal-gas state molar volume. Smlarly, H g, S g, and G g are deal gas state values; that s, the molar enthalpy, entropy, and Gbbs energy values that a VT system would have were the deal gas equaton the correct equaton of state. These quanttes provde reference values to whch actual values may be compared. For example, the compressblty factor Z compares the true molar volume to the deal gas molar volume as a rato: V V V Z = = = V g / Generalzed correlatons for the compressblty factor are treated n Sec.. Resdual ropertes These quanttes compare true and deal gas propertes through dfferences: M R M M g (4-67) where M s the molar value of an extensve thermodynamc property of a flud n ts actual state and M g s the correspondng value for the deal gas state of the flud at the same T,, and composton. Resdual propertes depend on nteractons between molecules and not on characterstcs of ndvdual molecules. Snce the deal gas state presumes the absence of molecular nteractons, resdual propertes reflect devatons from dealty. Most commonly used of the resdual propertes are: Resdual volume V R V V g Resdual enthalpy H R H H g Resdual entropy S R S S g Resdual Gbbs energy G R G G g SOLUTION THERMODYNAMICS IDEAL GAS MIXTURES An deal gas s a model gas comprsng magnary molecules of zero volume that do not nteract. Each chemcal speces n an deal gas mxture therefore has ts own prvate propertes, unnfluenced by the presence of other speces. The partal pressure of speces n a gas mxture s defned as p = x ( = 1,,..., N) where x s the mole fracton of speces. The sum of the partal pressures clearly equals the total pressure. Gbbs theorem for a mxture of deal gases may be stated as follows: The partal molar property, other than the volume, of a consttuent speces n an deal gas mxture s equal to the correspondng molar property of the speces as a pure deal gas at the mxture temperature but at a pressure equal to ts partal pressure n the mxture. g Ths s expressed mathematcally for generc partal property M by the equaton g g M (T, ) = M (T, p ) (M V) (4-68) For those propertes of an deal gas that are ndependent of, for example, U, H, and C, ths becomes smply g g M = M g where M s evaluated at the mxture T and. Thus, for the enthalpy, g g H = H (4-69) The entropy of an deal gas does depend on pressure: g ds = R d ln (constant T) Integraton from p to gves g g S (T, ) S (T, p ) = R ln = R ln = R ln x p x Whence S g (T, p ) = S g (T, ) R ln x
SOLUTION THERMODYNAMICS 4-9 Substtutng ths result nto Eq. (4-68) wrtten for the entropy gves g g S l = S R ln x (4-70) g where S s evaluated at the mxture T and. For the Gbbs energy of an deal gas mxture, G g = H g TS g ; the parallel relaton for partal propertes s g g g G = H TS In combnaton wth Eqs. (4-69) and (4-70), ths becomes g g g G = H TS + ln x g g g or µ G = G + ln x (4-71) g Elmnaton of G from ths equaton s accomplshed by Eq. (4-0), wrtten for pure speces as: g g dg = V d = d = d ln (constant T) Integraton gves g G =Γ (T) + ln (4-7) where Γ (T), the ntegraton constant for a gven temperature, s a functon of temperature only. Equaton (4-71) now becomes g µ =Γ (T) + ln x (4-73) FUGACITY AND FUGACITY COEFFICIENT The chemcal potental µ plays a vtal role n both phase and chemcalreacton equlbra. However, the chemcal potental exhbts certan unfortunate characterstcs whch dscourage ts use n the soluton of practcal problems. The Gbbs energy, and hence µ, s defned n relaton to the nternal energy and entropy, both prmtve quanttes for whch absolute values are unknown. Moreover, µ approaches negatve nfnty when ether or x approaches zero. Whle these characterstcs do not preclude the use of chemcal potentals, the applcaton of equlbrum crtera s facltated by ntroducton of the fugacty, a quantty that takes the place of µ but whch does not exhbt ts less desrable characterstcs. The orgn of the fugacty concept resdes n Eq. (4-7), an equaton vald only for pure speces n the deal gas state. For a real flud, an analogous equaton s wrtten: G Γ (T) + ln f (4-74) n whch a new property f replaces the pressure. Ths equaton serves as a partal defnton of the fugacty f. Subtracton of Eq. (4-7) from Eq. (4-74), both wrtten for the same temperature and pressure, gves g G G = ln g Accordng to the defnton of Eq. (4-67), G G s the resdual Gbbs energy, G R. The dmensonless rato f / s another new property called the fugacty coeffcent φ. Thus, G R = ln φ (4-75) f where φ (4-76) The defnton of fugacty s completed by settng the deal-gas state fugacty of pure speces equal to ts pressure: g f = Thus, for the specal case of an deal gas, G R = 0, φ = 1, and Eq. (4-7) s recovered from Eq. (4-74). The defnton of the fugacty of a speces n soluton s parallel to the defnton of the pure-speces fugacty. An equaton analogous to the deal gas expresson, Eq. (4-73), s wrtten for speces n a flud mxture: µ Γ (T) + ln f ˆ (4-77) where the partal pressure x s replaced by f ˆ, the fugacty of speces f n soluton. Snce t s not a partal molar property, t s dentfed by a crcumflex rather than an overbar. Subtractng Eq. (4-73) from Eq. (4-77), both wrtten for the same temperature, pressure, and composton, yelds g fˆ µ µ = ln x Analogous to the defnng equaton for the resdual Gbbs energy of a mxture, G R G G g, s the defnton of a partal molar resdual Gbbs energy: g g G R G G =µ µ Therefore G R = ln ˆφ (4-78) fˆ where by defnton ˆφ (4-79) x The dmensonless rato ˆφ s called the fugacty coeffcent of speces n soluton. Eq. (4-78) s the analog of Eq. (4-75), whch relates φ to G R. For an deal gas, G R s necessarly 0; therefore ˆφ g = 1, and g ˆf = x Thus, the fugacty of speces n an deal gas mxture s equal to ts partal pressure. ertnent examples are gven n Smth, Van Ness, and Abbott (Introducton to Chemcal Engneerng Thermodynamcs, 5th ed., Secs. 10.5 10.7, McGraw-Hll, New York, 1996). FUNDAMENTAL RESIDUAL-ROEY RELATION In vew of Eq. (4-59), the fundamental property relaton gven by Eq. (4-66) may be wrtten d ng nv nh G = d dt + dn (4-80) Ths equaton s general, and may be wrtten for the specal case of an deal gas: ng g nv g nh g dn Subtracton of ths equaton from Eq. (4-80) gves R d ng R nv R nh R G = d dt + dn (4-81) where the defntons G R G G g g and G R G G have been mposed. Equaton (4-81) s the fundamental resdual-property relaton. An alternatve form follows by ntroducton of the fugacty coeffcent as gven by Eq. (4-78): d ng R nv R nh R = d dt + ln ˆφ dn (4-8) These equatons are of such generalty that for practcal applcaton they are used only n restrcted forms. Dvson of Eq. (4-8) by d and restrcton to constant T and composton leads to: V R = (G R /) T,x (4-83) Smlarly, dvson by dt and restrcton to constant and composton gves H R = T (G R /),x (4-84) T Also mplct n Eq. (4-8) s the relaton ln ˆφ (ng = R /) T,,n (4-85) n Ths equaton demonstrates that ln ˆφ s a partal property wth respect to G R /. The partal-property analogs of Eqs. (4-83) and (4-84) are therefore: d = d dt + G g
4-10 THERMODYNAMICS R ln ˆφ V = (4-86) T,x R ln ˆφ H = (4-87),x The partal-property relatonshp of ln ˆφ to G R / also means that the summablty relaton apples; thus = x ln ˆφ (4-88) THE IDEAL SOLUTION The deal gas s a useful model of the behavor of gases and serves as a standard to whch real gas behavor can be compared. Ths s formalzed by the ntroducton of resdual propertes. Another useful model s the deal soluton, whch serves as a standard to whch real soluton behavor can be compared. Ths s formalzed by ntroducton of excess propertes. The partal molar Gbbs energy of speces n an deal gas mxture g s gven by Eq. (4-71). Ths equaton takes on new meanng when G, the Gbbs energy of pure speces n the deal gas state, s replaced by G, the Gbbs energy of pure speces as t actually exsts at the mxture T and and n the same physcal state (real gas, lqud, or sold) as the mxture. It then becomes applcable to speces n real solutons; ndeed, to lquds and solds as well as to gases. The deal soluton s therefore defned as one for whch d G G + ln x (4-89) where superscrpt d denotes an deal-soluton property. Ths equaton s the bass for development of expressons for all other thermodynamc propertes of an deal soluton. Equatons (4-60) and (4-61), appled to an deal soluton wth µ replaced by G, can be wrtten G d V d = T,x and S d =,x T Approprate dfferentaton of Eq. (4-89) n combnaton wth these relatons and Eqs. (4-3) and (4-4) yelds d V = V (4-90) d S = S R ln x (4-91) d d d Snce H = G + TS, substtutons by Eqs. (4-89) and (4-91) yeld d H = H (4-9) The summablty relaton, Eq. (4-50), wrtten for the specal case of an deal soluton, may be appled to Eqs. (4-89) through (4-9): G d = x G + x ln x (4-93) V d = x V (4-94) S d = T G R G d x S R x ln x (4-95) H d = x H (4-96) A smple equaton for the fugacty of a speces n an deal soluton follows from Eq. (4-89). Wrtten for the specal case of speces n an deal soluton, Eq. (4-77) becomes d d µ G =Γ (T) + ln f ˆ d When ths equaton and Eq. (4-74) are combned wth Eq. (4-89), Γ (T) s elmnated, and the resultng expresson reduces to fˆd = x f (4-97) Ths equaton, known as the Lews/Randall rule, apples to each speces n an deal soluton at all condtons of T,, and composton. It shows that the fugacty of each speces n an deal soluton s proportonal to ts mole fracton; the proportonalty constant s the fugacty of pure speces n the same physcal state as the soluton and at the same T and. Dvson of both sdes of Eq. (4-97) by x and substtuton of ˆφ d for f ˆ d /x (Eq. [4-79]) and of φ for f / (Eq. [4-76]) gves an alternatve form: d ˆφ =φ (4-98) Thus, the fugacty coeffcent of speces n an deal soluton s equal to the fugacty coeffcent of pure speces n the same physcal state as the soluton and at the same T and. Ideal soluton behavor s often approxmated by solutons comprsed of molecules not too dfferent n sze and of the same chemcal nature. Thus, a mxture of somers conforms very closely to deal soluton behavor. So do mxtures of adacent members of a homologous seres. FUNDAMENTAL EXCESS-ROEY RELATION The resdual Gbbs energy and the fugacty coeffcent are useful where expermental VT data can be adequately correlated by equatons of state. Indeed, f convenent treatment of all fluds by means of equatons of state were possble, the thermodynamc-property relatons already presented would suffce. However, lqud solutons are often more easly dealt wth through propertes that measure ther devatons from deal soluton behavor, not from deal gas behavor. Thus, the mathematcal formalsm of excess propertes s analogous to that of the resdual propertes. If M represents the molar (or unt-mass) value of any extensve thermodynamc property (e.g., V, U, H, S, G, and so on), then an excess property M E s defned as the dfference between the actual property value of a soluton and the value t would have as an deal soluton at the same temperature, pressure, and composton. Thus, M E M M d (4-99) Ths defnton s analogous to the defnton of a resdual property as gven by Eq. (4-67). However, excess propertes have no meanng for pure speces, whereas resdual propertes exst for pure speces as well as for mxtures. In addton, analogous to Eq. (4-99) s the partalproperty relaton, d M E = M M (4-100) where M E s a partal excess property. The fundamental excessproperty relaton s derved n exactly the same way as the fundamental resdual-property relaton and leads to analogous results. Equaton (4-80), wrtten for the specal case of an deal soluton, s subtracted from Eq. (4-80) tself, yeldng: E ng d E nv E nh E G = d dt + dn (4-101) Ths s the fundamental excess-property relaton, analogous to Eq. (4-81), the fundamental resdual-property relaton. The excess Gbbs energy s of partcular nterest. Equaton (4-77) may be wrtten: G =Γ (T) + ln f ˆ In accord wth Eq. (4-97) for an deal soluton, ths becomes d G =Γ (T) + ln x f By dfference d fˆ G G = ln x f The left-hand sde s the partal excess Gbbs energy G E ; the dmensonless rato f ˆ /x f appearng on the rght s called the actvty coeffcent of speces n soluton, and s gven the symbol γ. Thus, by defnton, fˆ γ (4-10) x f and G E = ln γ (4-103) E Comparson wth Eq. (4-78) shows that Eq. (4-103) relates γ to G exactly as Eq. (4-78) relates ˆφ to G R. For an deal soluton,g E =0, and therefore γ = 1.
SOLUTION THERMODYNAMICS 4-11 An alternatve form of Eq. (4-101) follows by ntroducton of the actvty coeffcent through Eq. (4-103): ng E d = d dt + ln γ dn (4-104) SUMMARY OF FUNDAMENTAL ROEY RELATIONS For convenence, the three other fundamental property relatons, Eqs. (4-16), (4-80), and (4-8), expressng the Gbbs energy and related propertes as functons of T,, and the n, are collected here: d(ng) = nv d ns dt + µ dn (4-16) d ng nv nh G = d dt + dn (4-80) d ng R nv R nh R = d dt + ln ˆφ dn (4-8) These equatons and Eq. (4-104) may also be wrtten for the specal case of 1 mole of soluton by settng n = 1 and n = x. The x are then subect to the constrant that x = 1. If wrtten for 1 mole of a constant-composton soluton, they become: dg = V d S dt (4-105) d = d dt (4-106) d = d dt (4-107) d = d dt (4-108) These equatons are, of course, vald as a specal case for a pure speces; n ths event they are wrtten wth subscrpt affxed to the approprate symbols. The partal-property analogs of these equatons are: dg = dµ = V d S dt (4-109) G d = d = d dt (4-110) G R d = d ln ˆφ = d dt (4-111) G E d = d ln γ = d dt (4-11) Fnally, a Gbbs/Duhem equaton s assocated wth each fundamental property relaton: V d S dt = x dµ (4-113) V H d V R d V E nv E G G R G E H R µ H E d V V R V E nh E V dt = G x d (4-114) dt = x d ln ˆφ (4-115) dt = x d ln γ (4-116) Ths depostory of equatons stores an enormous amount of nformaton. The equatons themselves are so general that ther drect applcaton s seldom approprate. However, by nspecton one can wrte a vast array of relatons vald for partcular applcatons. For example, Eqs. (4-83) and (4-84) come drectly from Eq. (4-107); Eqs. (4-86) and (4-87), from (4-111). Smlarly, from Eq. (4-108), V R V E H H R H E H H R H E V E (G = E /) T,x (4-117) H E (G = T E /),x (4-118) T and from Eq. (4-104) (ng ln γ = E /) T,,n (4-119) n The last relaton demonstrates that ln γ s a partal property wth respect to G E /. The partal-property analogs of Eqs. (4-117) and (4-118) follow from Eq. (4-11): E ln γ V = (4-10) T,x E ln γ H = (4-11),x T Fnally, an especally useful form of the Gbbs/Duhem equaton follows from Eq. (4-116): x d ln γ = 0 (constant T,) (4-1) Snce ln γ s a partal property wth respect to G E /, the followng form of the summablty equaton s vald: G E = x ln γ (4-13) The analogy between equatons derved from the fundamental resdual- and excess-property relatons s apparent. Whereas the fundamental resdual-property relaton derves ts usefulness from ts drect relaton to equatons of state, the excess-property formulaton s useful because V E, H E, and γ are all expermentally accessble. Actvty coeffcents are found from vapor/lqud equlbrum data, and V E and H E values come from mxng experments. ROEY CHANGES OF MIXING If M represents a molar thermodynamc property of a homogeneous flud soluton, then by defnton, M M x M (4-14) where M s the property change of mxng, and M s the molar property of pure speces at the T and of the soluton and n the same physcal state (gas or lqud). The summablty relaton, Eq. (4-50), may be combned wth Eq. (4-14) to gve M = x M (4-15) where by defnton M M M (4-16) All three quanttes are for the same T,, and physcal state. Eq. (4-16) defnes a partal molar property change of mxng, and Eq. (4-15) s the summablty relaton for these propertes. Each of Eqs. (4-93) through (4-96) s an expresson for an deal soluton property, and each may be combned wth the defnng equaton for an excess property (Eq. [4-99]), yeldng G E = G x G x ln x (4-17) V E = V x V (4-18) S E = S x S + R x ln x (4-19) H E = H x H (4-130) In vew of Eq. (4-14), these may be wrtten G E = G x ln x (4-131)