The material in this ebook also appears in the print version of this title: X.

Transcription

1

2 Copyrght 8, 997, 984, 973, 963, 95, 94, 934 by The McGraw-Hll Companes, Inc. All rghts reserved. Manufactured n the Unted States of Amerca. Except as permtted under the Unted States Copyrght Act of 976, no part of ths publcaton may be reproduced or dstrbuted n any form or by any means, or stored n a database or retreval system, wthout the pror wrtten permsson of the publsher The materal n ths ebook also appears n the prnt verson of ths ttle: X. All trademarks are trademarks of ther respectve owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names n an edtoral fashon only, and to the beneft of the trademark owner, wth no ntenton of nfrngement of the trademark. Where such desgnatons appear n ths book, they have been prnted wth ntal caps. McGraw-Hll ebooks are avalable at specal quantty dscounts to use as premums and sales promotons, or for use n corporate tranng programs. For more nformaton, please contact George Hoare, Specal Sales, at george_hoare@mcgraw-hll.com or () TERMS OF USE Ths s a copyrghted work and The McGraw-Hll Companes, Inc. ( McGraw-Hll ) and ts lcensors reserve all rghts n and to the work. Use of ths work s subject to these terms. Except as permtted under the Copyrght Act of 976 and the rght to store and retreve one copy of the work, you may not decomple, dsassemble, reverse engneer, reproduce, modfy, create dervatve works based upon, transmt, dstrbute, dssemnate, sell, publsh or sublcense the work or any part of t wthout McGraw-Hll s pror consent. You may use the work for your own noncommercal and personal use; any other use of the work s strctly prohbted. Your rght to use the work may be termnated f you fal to comply wth these terms. THE WORK IS ROVIDED AS IS. McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYERLINK OR OTHERWISE, AND EXRESSLY DISCLAIM ANY WARRANTY, EXRESS OR IMLIED, INCLUDING BUT NOT LIMITED TO IMLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A AICULAR UROSE. McGraw-Hll and ts lcensors do not warrant or guarantee that the functons contaned n the work wll meet your requrements or that ts operaton wll be unnterrupted or error free. Nether McGraw-Hll nor ts lcensors shall be lable to you or anyone else for any naccuracy, error or omsson, regardless of cause, n the work or for any damages resultng therefrom. McGraw-Hll has no responsblty for the content of any nformaton accessed through the work. Under no crcumstances shall McGraw-Hll and/or ts lcensors be lable for any ndrect, ncdental, specal, puntve, consequental or smlar damages that result from the use of or nablty to use the work, even f any of them has been advsed of the possblty of such damages. Ths lmtaton of lablty shall apply to any clam or cause whatsoever whether such clam or cause arses n contract, tort or otherwse. DOI:.36/757X

3 Ths page ntentonally left blank

4 Secton 4 Thermodynamcs Hendrck C. Van Ness, D.Eng. Howard. Isermann Department of Chemcal and Bologcal Engneerng, Rensselaer olytechnc Insttute; Fellow, Amercan Insttute of Chemcal Engneers; Member, Amercan Chemcal Socety (Secton Coedtor) Mchael M. Abbott, h.d. Deceased; rofessor Emertus, Howard. Isermann Department of Chemcal and Bologcal Engneerng, Rensselaer olytechnc Insttute (Secton Coedtor)* INTRODUCTION ostulate ostulate (Frst Law of Thermodynamcs) ostulate ostulate 4 (Second Law of Thermodynamcs) ostulate VARIABLES, DEFINITIONS, AND RELATIONSHIS Constant-Composton Systems U, H, and S as Functons of T and or T and V The Ideal Gas Model Resdual ropertes ROEY CALCULATIONS FOR GASES AND VAORS Evaluaton of Enthalpy and Entropy n the Ideal Gas State Resdual Enthalpy and Entropy from VT Correlatons Vral Equatons of State Cubc Equatons of State tzer s Generalzed Correlatons OTHER ROEY FORMULATIONS Lqud hase Lqud/Vapor hase Transton THERMODYNAMICS OF FLOW ROCESSES Mass, Energy, and Entropy Balances for Open Systems Mass Balance for Open Systems General Energy Balance Energy Balances for Steady-State Flow rocesses Entropy Balance for Open Systems Summary of Equatons of Balance for Open Systems Applcatons to Flow rocesses Duct Flow of Compressble Fluds pe Flow Nozzles Throttlng rocess Turbnes (Expanders) Compresson rocesses Example : LNG Vaporzaton and Compresson SYSTEMS OF VARIABLE COMOSITION artal Molar ropertes Gbbs-Duhem Equaton artal Molar Equaton-of-State arameters artal Molar Gbbs Energy Soluton Thermodynamcs Ideal Gas Mxture Model Fugacty and Fugacty Coeffcent Evaluaton of Fugacty Coeffcents Ideal Soluton Model Excess ropertes roperty Changes of Mxng Fundamental roperty Relatons Based on the Gbbs Energy Fundamental Resdual-roperty Relaton Fundamental Excess-roperty Relaton Models for the Excess Gbbs Energy Behavor of Bnary Lqud Solutons EQUILIBRIUM Crtera hase Rule Example : Applcaton of the hase Rule Duhem s Theorem Vapor/Lqud Equlbrum Gamma/h Approach Modfed Raoult s Law Example 3: Dew and Bubble ont Calculatons *Dr. Abbott ded on May 3, 6. Ths, hs fnal contrbuton to the lterature of chemcal engneerng, s deeply apprecated, as are hs earler contrbutons to the handbook. Copyrght 8, 997, 984, 973, 963, 95, 94, 934 by The McGraw-Hll Companes, Inc. Clck here for terms of use. 4-

5 4- THERMODYNAMICS Data Reducton Solute/Solvent Systems K Values, VLE, and Flash Calculatons Example 4: Flash Calculaton Equaton-of-State Approach Extrapolaton of Data wth Temperature Example 5: VLE at Several Temperatures Lqud/Lqud and Vapor/Lqud/Lqud Equlbra Chemcal Reacton Stochometry Chemcal Reacton Equlbra Standard roperty Changes of Reacton Equlbrum Constants Example 6: Sngle-Reacton Equlbrum Complex Chemcal Reacton Equlbra THERMODYNAMIC ANALYSIS OF ROCESSES Calculaton of Ideal Work Lost Work Analyss of Steady-State Steady-Flow roceses Example 7: Lost-Work Analyss

6 THERMODYNAMICS 4-3 Nomenclature and Unts Correlaton- and applcaton-specfc symbols are not shown. U.S. Customary U.S. Customary Symbol Defnton SI unts System unts Symbol Defnton SI unts System unts A Molar (or unt-mass) J/mol [J/kg] Btu/lb mol Saturaton or vapor pressure ka ps Helmholtz energy [Btu/lbm] of speces A Cross-sectonal area n flow m ft Q Heat J Btu â Actvty of speces Dmensonless Dmensonless q Volumetrc flow rate m 3 /s ft 3 /s n soluton Q Rate of heat transfer J/s Btu/s a artal parameter, cubc R Unversal gas constant J/(mol K) Btu/(lb mol R) equaton of state S Molar (or unt-mass) entropy J/(mol K) Btu/(lb mol R) B d vral coeffcent, cm 3 /mol cm 3 /mol [J/(kg K)] [Btu/(lbm R)] densty expanson S B artal molar second cm 3 /mol cm 3 G Rate of entropy generaton, J/(K s) Btu/(R s) /mol Eq. (4-5) vral coeffcent T Absolute temperature K R Bˆ Reduced second vral T c Crtcal temperature K R coeffcent U Molar (or unt-mass) J/mol [J/kg] Btu/(lb mol) C 3d vral coeffcent, densty cm 6 /mol cm 6 /mol nternal energy [Btu/lbm] expanson u Flud velocty m/s ft/s Ĉ Reduced thrd vral coeffcent V Molar (or unt-mass) volume m 3 /mol [m 3 /kg] ft 3 /(lb mol) D 4th vral coeffcent, densty cm 9 /mol 3 cm 9 /mol 3 [ft 3 /lbm] expanson W Work J Btu B d vral coeffcent, pressure ka ka W s Shaft work for flow process J Btu expanson W C 3d vral coeffcent, pressure ka ka s Shaft power for flow process J/s Btu/s x Mole fracton n general expanson x D 4th vral coeffcent, ka 3 ka 3 Mole fracton of speces n lqud phase pressure expanson y B j Interacton d vral cm 3 /mol cm 3 Mole fracton of speces n /mol vapor phase coeffcent Z Compressblty factor Dmensonless Dmensonless C jk Interacton 3d vral cm 6 /mol cm 6 /mol z Elevaton above a datum level m ft coeffcent C Heat capacty at constant J/(mol K) Btu/(lb mol R) Superscrpts pressure C V Heat capacty at constant J/(mol K) Btu/(lb mol R) E Denotes excess thermodynamc property volume d Denotes value for an deal soluton f Fugacty of pure speces ka ps g Denotes value for an deal gas fˆ Fugacty of speces n soluton ka ps l Denotes lqud phase G Molar (or unt-mass) J/mol [J/kg] Btu/(lb mol) lv Denotes phase transton, lqud to vapor Gbbs energy [Btu/lbm] R Denotes resdual thermodynamc property g Acceleraton of gravty m/s ft/s t Denotes total value of property g G E / Dmensonless Dmensonless v Denotes vapor phase H Molar (or unt-mass) enthalpy J/mol [J/kg] Btu/(lb mol) Denotes value at nfnte dluton K Equlbrum K value, y /x Dmensonless [Btu/lbm] Dmensonless Subscrpts K j Equlbrum constant for Dmensonless Dmensonless c Denotes value for the crtcal state chemcal reacton j cv Denotes the control volume k Henry s constant for ka ps fs Denotes flowng streams solute speces n Denotes the normal bolng pont M Molar or unt-mass soluton r Denotes a reduced value property (A, G, H, S, U, V) rev Denotes a reversble process M Mach number Dmensonless Dmensonless M Molar or unt-mass Greek Letters pure-speces property α, β As superscrpts, dentfy phases (A, G, H, S, U, V ) M β Volume expansvty K R artal property of speces ε j Reacton coordnate for mol lb mol n soluton (A, G, H, S, U, V reacton j ) Γ (T) Defned by Eq. (4-96) J/mol Btu/(lb mol) M R Resdual thermodynamc property γ Heat capacty rato C /C V Dmensonless Dmensonless (A R, G R, H R, S R, U R, V R ) γ Actvty coeffcent of speces Dmensonless Dmensonless M E Excess thermodynamc property n soluton (A E, G E, H E, S E, U E, V E ) κ Isothermal compressblty ka ps M artal molar excess thermodynamc µ Chemcal potental of speces J/mol Btu/(lb mol) property ν, j Stochometrc number Dmensonless Dmensonless M roperty change of mxng of speces n reacton j ( A, G, H, S, U, V) ρ Molar densty mol/m 3 lb mol/ft 3 M j Standard property change of reacton j ( G j, H j, C j ) σ As subscrpt, denotes a heat reservor m Mass kg lbm m Φ Defned by Eq. (4-34) Dmensonless Dmensonless Mass flow rate kg/s lbm/s φ Fugacty coeffcent of Dmensonless Dmensonless n Number of moles n pure speces Molar flow rate φˆ Fugacty coeffcent of Dmensonless Dmensonless n Number of moles of speces speces n soluton Absolute pressure ka ps ω Acentrc factor Dmensonless Dmensonless

7 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, 989. olng, B. E., J. M. rausntz, and J.. O Connell, The ropertes of Gases and Lquds, 5th ed., McGraw-Hll, New York,. rausntz, J. M., R. N. Lchtenthaler, and E. G. de Azevedo, Molecular Thermodynamcs of Flud-hase Equlbra, 3d ed., rentce-hall TR, Upper Saddle Rver, N.J., 999. Sandler, S. I., Chemcal and Engneerng Thermodynamcs, 3d ed., Wley, New York, 999. Smth, J. M., H. C. Van Ness, and M. M. Abbott, Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., McGraw- Hll, New York, 5. Tester, J. W., and M. Modell, Thermodynamcs and Its Applcatons, 3d ed., rentce-hall TR, Upper Saddle Rver, N.J., 997. Van Ness, H. C., and M. M. Abbott, Classcal Thermodynamcs of Nonelectrolyte Solutons: Wth Applcatons to hase Equlbra, McGraw-Hll, New York, 98. INTRODUCTION Thermodynamcs s the branch of scence that lends substance to the prncples of energy transformaton n macroscopc systems. The general restrctons shown by experence 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 can be an object, a quantty of matter, or a regon of space, selected for study and set apart (mentally) from everythng else, whch s called the surroundngs. An envelope, magned to enclose the system and to separate t from ts surroundngs, s called the boundary of the system. Attrbuted to ths boundary are specal propertes whch may serve ether to solate the system from ts surroundngs or to provde for nteracton n specfc ways between the 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 measurng nstruments such as thermometers and pressure gauges. 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 postulates 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 subject. The frst two deal wth energy. OSTULATE 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 the measurable coordnates that characterze the system. 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 t s 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 postulate depends on postulate. 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. For such systems energy s exchanged between 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 n whch only nternal-energy changes occur, the frst law of thermodynamcs s expressed mathematcally as du t = dq + dw (4-) 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-) 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 or functons of the thermodynamc coordnates that characterze the system. 4-4

8 VARIABLES, DEFINITIONS, AND RELATIONSHIS 4-5 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 that characterze the system. For reversble processes, changes n ths property may be calculated by the equaton ds t = dq rev (4-3) T 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 t, U t, S t ) do depend on system sze and are extensve. For a system contanng n mol of flud, M t = nm, where M s a molar property. Applcatons of the thermodynamc postulates necessarly nvolve the abstract quanttes of 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-) du t = dq rev + dw rev Substtuton for dq rev and dw rev by Eqs. (4-3) and (4-4) gves du t = TdS 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 s equally well wrtten as 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 + 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 a sngle-phase, nonreactng, closed system, nu = u(ns, nv). (nu) (ns) (nu) (nv) Then d(nu) = nv,n d(ns) + ns,n d(nv) where subscrpt n ndcates that all mole numbers n (and hence n) are held constant. Comparson wth Eq. (4-5) shows that ( nu ( n ) S) = T and ( nu) nv,n ( nv) ns,n = For an open sngle-phase system, we assume that nu = U (ns, nv, n, n, n 3,...). In consequence, d(nu) = ( nu ( n ) S) d(ns) + ( nu) nv,n ( nv) ns,n d(nv) + ( nu) n ns,nv,nj dn where the summaton s over all speces present n the system and subscrpt n j ndcates that all mole numbers are held constant except the th. Defne nu) µ ( n ns,nv,nj The expressons for T and of the precedng paragraph and the defnton of µ allow replacement of the partal dfferental coeffcents n the precedng equaton by T,, and µ. The result s Eq. (4-6) of Table 4-, where mportant equatons of ths secton are collected. Equaton (4-6) s the fundamental property relaton for sngle-phase VT systems, from whch all other equatons connectng propertes of

9 4-6 THERMODYNAMICS TABLE 4- Mathematcal Structure of Thermodynamc roperty Relatons For homogeneous systems of rmary thermodynamc functons Fundamental property relatons constant composton Maxwell equatons U = TS V + x µ (4-7) H U + V (4-8) A U TS (4-9) G H TS (4-) d(nu) = Td(nS) d(nv) + µ dn (4-6) d(nh) = Td(nS) + nv d + µ dn (4-) d(na) = ns dt d(nv) + µ dn (4-) d(ng) = ns dt + nv d + µ dn (4-3) du = TdS dv (4-4) dh = TdS+ Vd (4-5) da = SdT dv (4-6) dg = SdT+ Vd (4-7) T S = V S V (4-8) T V = S S (4-9) S V = T T V (4-) V T S = T (4-) U, H, and S as functons of T and or T and V artal dervatves Total dervatves dh = H T H dt + d (4-) T ds = S S dt + T T d (4-3) du = U T U dt + V V dv (4-4) T ds = S S V dt + T T dv (4-5) V H T S = T = C (4-8) T H S = T T + V = V T T V T (4-9) U T S = T V = C V V T (4-3) U V S = T T = T T V V (4-3) T U Internal energy; H enthalpy; A Helmoholtz energy; G Gbbs energy. dh = C dt + V T V T d (4-3) ds = C V dt T T d (4-33) du = C V dt + T V T dv (4-34) ds = C V dt + T V dv (4-35) T 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. (4-6). Because n = x n, where x s the mole fracton of speces, ths equaton may be rewrtten as d(nu) Td(nS) + d(nv) µ d(x n) = Expanson of the dfferentals and collecton of lke terms yeld du TdS+ dv µ dx n + U TS + V x µ dn = Because n and dn are ndependent and arbtrary, the terms n brackets must separately be zero. Ths provdes two useful equatons: du = TdS dv+ µ dx U = TS V + x µ The frst s smlar to Eq. (4-6). However, Eq. (4-6) apples to a system of n mol where n may vary. Here, however, n s unty and nvarant. It s therefore subject to the constrants x = and dx =. Mole fractons are not ndependent of one another, whereas the mole numbers n Eq. (4-6) are. The second of the precedng equatons dctates the possble combnatons of terms that may be defned as addtonal prmary functons. Those n common use are shown n Table 4- as Eqs. (4-7) through (4-). Addtonal thermodynamc propertes are related to these and arse by arbtrary defnton. Multplcaton of Eq. (4-8) of Table 4- by n and dfferentaton yeld the general expresson d(nh) = d(nu) + d(nv) + nv d Substtuton for d(nu) by Eq. (4-6) reduces ths result to Eq. (4-). The total dfferentals of na and ng are obtaned smlarly and are expressed by Eqs. (4-) and (4-3). These equatons and Eq. (4-6) are equvalent forms of the fundamental property relaton, and appear under that headng n Table 4-. Each expresses a total property nu, nh, na, and ng as a functon of a partcular set of ndependent varables, called 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 relaton to the canoncal varables T,, and {n }, the varables of prmary nterest n chemcal processng. Another set of equatons results from the substtutons n = and n = x. The resultng equatons are of course less general than ther parents. Moreover, because the mole fractons are not ndependent, mathematcal operatons requrng ther ndependence are nvald. CONSTANT-COMOSITION SYSTEMS For mol of a homogeneous flud of constant composton, Eqs. (4-6) and (4-) through (4-3) smplfy to Eqs. (4-4) through (4-7) of Table 4-. Because these equatons are exact dfferental expressons, applcaton of the recprocty relaton for such expressons produces the common Maxwell relatons as descrbed n the subsecton Multvarable Calculus Appled to Thermodynamcs n Sec. 3. These are Eqs. (4-8) through (4-) of Table 4-, n whch the partal dervatves are taken wth composton held constant. U, H, and S as Functons of T and or T and V At constant composton, molar thermodynamc propertes can be consdered functons of T and (postulate 5). Alternatvely, because V s related to T and through an equaton of state, V can serve rather than as the second ndependent varable. The useful equatons for the total dfferentals of U, H, and S that result are gven n Table 4- by Eqs. (4-) through (4-5). The obvous next step s substtuton for the partal dfferental coeffcents n favor of measurable quanttes. Ths purpose s served by defnton of two heat capactes, one at constant pressure and the other at constant volume: H C T (4-6) U C V T (4-7) V Both are propertes of the materal and functons of temperature, pressure, and composton.

10 VARIABLES, DEFINITIONS, AND RELATIONSHIS 4-7 Equaton (4-5) of Table 4- may be dvded by dt and restrcted to constant, yeldng ( H/ T) as gven by the frst equalty of Eq. (4-8). Dvson of Eq. (4-5) by d and restrcton to constant T yeld ( H/ ) T as gven by the frst equalty of Eq. (4-9). Equaton (4-8) s completed by Eq. (4-6), and Eq. (4-9) s completed by Eq. (4-). Smlarly, equatons for ( U/ T) V and ( U/ V) T derve from Eq. (4-4), and these wth Eqs. (4-7) and (4-) yeld Eqs. (4-3) and (4-3) of Table 4-. Equatons (4-), (4-6), and (4-9) combne to yeld Eq. (4-3); Eqs. (4-3), (4-8), and (4-) to yeld Eq. (4-33); Eqs. (4-4), (4-7), and (4-3) to yeld Eq. (4-34); and Eqs. (4-5), (4-3), and (4-) to yeld Eq. (4-35). Equatons (4-3) and (4-33) are general expressons for the enthalpy and entropy of homogeneous fluds at constant composton as functons of T and. Equatons (4-34) and (4-35) 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, d, and dv are all composed of measurable quanttes. The Ideal Gas Model An deal gas s a model gas comprsng magnary molecules of zero volume that do not nteract. Its VT behavor s represented by the smplest of equatons of state V g =, where R s a unversal constant, values of whch are gven n Table -9. The followng partal dervatves, all taken at constant composton, are obtaned from ths equaton: r.. Z =.98 Z =. V = = V g V g R R = = = V g T T T T T V V g The frst two of these relatons when substtuted approprately nto Eqs. (4-9) and (4-3) of Table 4- lead to very smple expressons for deal gases: U g = H g S = T g R S = T T g = V V T Moreover, Eqs. (4-3) through (4-35) become g C R dh g = C g dt ds g = dt d T g C R du g = C g V dt ds g = V dt + dv T V g R V g In these equatons V g, U g, C g, V Hg, C g, and S g are deal gas state values the values that a VT system would have were the deal gas equaton the true equaton of state. They apply equally to pure speces and to constant-composton mxtures, and they show that U g, C g, V Hg, and C g, are functons of temperature only, ndependent of and V. The entropy, however, s a functon of both T and or of both T and V. Regardless of composton, the deal gas volume s gven by V g = /, and t provdes the bass for comparson wth true molar volumes through the compressblty factor Z. By defnton, V V V Z = = (4-36) V g The deal gas state propertes of mxtures are drectly related to the deal gas state propertes of the consttuent pure speces. For those propertes that are ndependent of U g, H g, C g V, and Cg the mxture property s the sum of the propertes of the pure consttuent speces, each weghted by ts mole fracton: M g g = y M (4-37) where M g can represent any of the propertes lsted. For the entropy, whch s a functon of both T and, an addtonal term s requred to account for the dfference n partal pressure of a speces between ts pure state and ts state n a mxture: S g g = y S Ry ln y (4-38). 3 4 T r FIG. 4- Regon where Z les between.98 and., and the deal-gas equaton s a reasonable approxmaton. [Smth, Van Ness, and Abbott, Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., p. 4, McGraw-Hll, New York (5).] For the Gbbs energy, G g = H g TS g ; whence by Eqs. (4-37) and (4-38): G g g = y G + y ln y (4-39) The deal gas model may serve as a reasonable approxmaton to realty under condtons ndcated by Fg. 4-. Resdual ropertes The dfferences between true and deal gas state propertes are defned as resdual propertes M R : M R M M g (4-4) where M s the molar value of an extensve thermodynamc property of a flud n ts actual state and M g s ts correspondng deal gas state value at the same T,, and composton. Resdual propertes depend on nteractons between molecules and not on characterstcs of ndvdual molecules. Because the deal gas state presumes the absence of molecular nteractons, resdual propertes reflect devatons from dealty. The most commonly used resdual propertes are as follows: 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 Useful relatons connectng these resdual propertes derve from Eq. (4-7), an alternatve form of whch follows from the mathematcal dentty: G G d dg dt

11 4-8 THERMODYNAMICS Substtuton for dg by Eq. (4-7) and for G by Eq. (4-) gves, after algebrac reducton, d G V H = d dt (4-4) Ths equaton may be wrtten for the specal case of an deal gas and subtracted from Eq. (4-4) tself, yeldng G d R V R H R = d dt (4-4) V R As a consequence, = T H R (G R /) (G R ) and = T T (4-43) (4-44) Equaton (4-43) provdes a drect lnk to VT correlatons through the compressblty factor Z as gven by Eq. (4-36). Thus, wth V = Z/, Z V R V V g = = (Z ) Ths equaton n combnaton wth a rearrangement of Eq. (4-43) yelds G d R V R d = d = (Z ) (constant T) Integraton from = to arbtrary pressure gves G R = d (Z ) (constant T) (4-45) Smth, Van Ness, and Abbott [Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., pp., McGraw-Hll, New York (5)] show that t s permssble here to set the lower lmt of ntegraton (G R /) = equal to zero. Note also that the ntegrand (Z )/ remans fnte as. Dfferentaton of Eq. (4-45) wth respect to T n accord wth Eq. (4-44) gves H R = T Z d (constant T) (4-46) T Because G = H TS and G g = H g TS g, then by dfference, G R = H R TS R, and = (4-47) Equatons (4-45) through (4-47) provde the bass for calculaton of resdual propertes from VT correlatons. They may be put nto generalzed form by substtuton of the relatonshps = c r T = T c T r d = c d r dt = T c dt r The resultng equatons are G R H R c S R R = r H R = T r r d (Z ) r (4-48) Z Tr r d r (4-49) r The terms on the rght sdes of these equatons depend only on the upper lmt r of the ntegrals and on the reduced temperature at whch they are evaluated. Thus, values of G R / and H R / c may be determned once and for all at any reduced temperature and pressure from generalzed compressblty factor data. G R r ROEY CALCULATIONS FOR GASES AND VAORS The most sfactory calculaton procedure for the thermodynamc propertes of gases and vapors s based on deal gas state heat capactes and resdual propertes. Of prmary nterest are the enthalpy and entropy; these are gven by rearrangement of the resdual property defntons: H = H g + H R and S = S g + S R These are smple sums of the deal gas and resdual propertes, evaluated separately. EVALUATION OF ENTHALY AND ENTROY IN THE IDEAL GAS STATE For the deal gas state at constant composton: dh g = C g dt and ds g g dt d = C R T Integraton from an ntal deal gas reference state at condtons T and to the deal gas state at T and gves H g = H g + T T C g dt S g = S g + T g dt C R ln T T Substtuton nto the equatons for H and S yelds H = H g + T T C g dt + H R (4-5) S = S g + T g dt C R ln + S R (4-5) T T The reference state at T and s arbtrarly selected, and the values assgned to H g and S g are also arbtrary. In practce, only changes n H and S are of nterest, and fxed reference state values ultmately cancel n ther calculaton. The deal gas state heat capacty C g s a functon of T but not of. For a mxture the heat capacty s smply the molar average y C g. Emprcal equatons relatng C g to T are avalable for many pure gases; a common form s g C = A + BT + CT + DT (4-5) R where A, B, C, and D are constants characterstc of the partcular gas, and ether C or D s zero. The rato C g /R s dmensonless; thus the unts of C g are those of R. Data for deal gas state heat capactes are gven for many substances n Table -55. Evaluaton of the ntegrals C g dt and (C g /T) dt s accomplshed by substtuton for C g, followed by ntegraton. For temperature lmts of T and T and wth τ T/T, the equatons that follow from Eq. (4-5) are T C g B C D dt = AT (τ ) + T(τ ) + T(τ 3 3 ) + T R τ 3 T τ (4-53) T T R CT τ+ g dt = A ln τ+ BT D + CT + τ T (τ ) (4-54)

12 ROEY CALCULATIONS FOR GASES AND VAORS 4-9 Equatons (4-5) and (4-5) may sometmes be advantageously expressed n alternatve form through use of mean heat capactes: H = H g + C g H (T T ) + H R (4-55) T S = S g + C g S ln R ln + S R (4-56) where C g H and C g S are mean heat capactes specfc, respectvely, for enthalpy and entropy calculatons. They are gven by the followng equatons: C g H R C g S R B C D = A + T (τ+) + T (τ +τ+) + (4-57) 3 τt = A + BT + CT + (4-58) RESIDUAL ENTHALY AND ENTROY FROM VT CORRELATIONS The resdual propertes of gases and vapors depend on ther VT behavor. Ths s often expressed through correlatons for the compressblty factor Z, defned by Eq. (4-36). Analytcal expressons for Z as functons of T and or T and V are known as equatons of state. They may also be reformulated to gve as a functon of T and V or V as a functon of T and. Vral Equatons of State The vral equaton n densty s an nfnte seres expanson of the compressblty factor Z n powers of molar densty ρ (or recprocal molar volume V ) about the real gas state at zero densty (zero pressure): Z = + Bρ+Cρ + Dρ 3 + (4-59) The densty seres vral coeffcents B, C, D,... depend on temperature and composton only. In practce, truncaton s to two or three terms. The composton dependences of B and C are gven by the exact mxng rules B = y y j B j (4-6) j C = j y y j y k C jk (4-6) k where y, y j, and y k are mole fractons for a gas mxture and, j, and k dentfy speces. The coeffcent B j characterzes a bmolecular nteracton between molecules and j, and therefore B j = B j. Two knds of second vral coeffcent arse: B and B jj, wheren the subscrpts are the same ( = j), and B j, wheren they are dfferent ( j ). The frst s a vral coeffcent for a pure speces; the second s a mxture property, called a cross coeffcent. Smlarly for the thrd vral coeffcents: C, C jjj, and C kkk are for the pure speces, and C j = C j = C j,... are cross coeffcents. Although the vral equaton tself s easly ratonalzed on emprcal grounds, the mxng rules of Eqs. (4-6) and (4-6) follow rgorously from the methods of statstcal mechancs. The temperature dervatves of B and C are gven exactly by dc dt db dt = = j j T k D τ T y y j (4-6) dt dc jk y y j y k (4-63) dt An alternatve form of the vral equaton expresses Z as an expanson n powers of pressure about the real gas state at zero pressure (zero densty): Z = + B + C + D (4-64) Equaton (4-64) s the vral equaton n pressure, and B, C, D,... are the pressure seres vral coeffcents. Agan, truncaton s to two db j τ+ τ ln τ or three terms, wth B and C dependng on temperature and composton only. Moreover, the two sets of coeffcents are related: B =B (4-65) C =(C B )() (4-66) Values can often be found for B, but not so often for C. Generalzed correlatons for both B and C are gven by Meng, Duan, and L [Flud hase Equlbra 6: 9 (4)]. For pressures up to several bars, the two-term expanson n pressure, wth B gven by Eq. (4-65), s usually preferred: Z = + B = + B (4-67) For supercrtcal temperatures, t s sfactory to ever hgher pressures as the temperature ncreases. For pressures above the range where Eq. (4-67) s useful, but below the crtcal pressure, the vral expanson n densty truncated to three terms s usually sutable: Z = + Bρ+Cρ (4-68) Equatons for resdual enthalpy and entropy may be developed from each of these expressons. Consder frst Eq. (4-67), whch s explct n volume. Equatons (4-45) and (4-46) are therefore applcable. Drect substtuton for Z n Eq. (4-45) gves R G B = (4-69) R T R T Dfferentaton of Eq. (4-67) yelds By Eq. (4-46), and by Eq. (4-47), Z = db B = (4-7) db = (4-7) R dt An extensve set of three-parameter correspondng-states correlatons has been developed by tzer and coworkers [tzer, Thermodynamcs, 3d ed., App. 3, McGraw-Hll, New York (995)]. artcularly useful s the one for the second vral coeffcent. The basc equaton s B c = B +ωb (4-7) R Tc wth the acentrc factor defned by Eq. (-7). For pure chemcal speces B and B are functons of reduced temperture only. Substtuton for B n Eq. (4-67) by ths expresson gves Z = + (B +ωb r ) (4-73) T r By dfferentaton, Z db = r r dt B +ω db r B r dt r T Tr Tr r Tr Tr Upon substtuton of these equatons nto Eqs. (4-48) and (4-49), ntegraton yelds G R = (B +ωb r ) (4-74) Tr H R db db = r B T r +ω T r (4-75) c dtr B dtr The resdual entropy follows from Eq. (4-47): S R R T H R S R R R dt B T db dtr T db dt db dtr = r +ω (4-76)

13 4- THERMODYNAMICS In these equatons, B and B and ther dervatves are well represented by Abbott s correlatons [Smth and Van Ness, Introducton to Chemcal Engneerng Thermodynamcs, 3d ed., p. 87, McGraw-Hll, New York (975)]:.4 B =.83 (4-77).7 B =.39 (4-78).675 = (4-79).7 = (4-8) Although lmted to pressures where the two-term vral equaton n pressure has approxmate valdty, these correlatons are applcable for most chemcal processng condtons. As wth all generalzed correlatons, they are least accurate for polar and assocatng molecules. Although developed for pure materals, these correlatons can be extended to gas or vapor mxtures. Basc to ths extenson are the mxng rules for the second vral coeffcent and ts temperature dervatve as gven by Eqs. (4-6) and (4-6). Values for the cross coeffcents B j, wth j, and ther dervatves are provded by Eq. (4-7) wrtten n extended form: B j = (B +ω j B ) (4-8) where B, B, db /dt r, and db /dt r are the same functons of T r as gven by Eqs. (4-77) through (4-8). Dfferentaton produces db j dt db j dt db dtr db dtr cj cj db = db cj +ω j cj R db = +ω j (4-8) cj dt dtrj Tr.6 Tr 5. where T rj = T/T cj. The followng combnng rules for ω j, T cj, and cj are gven by rausntz, Lchtenthaler, and de Azevedo [Molecular Thermodynamcs of Flud-hase Equlbra, d ed., pp. 3 and 6, rentce-hall, Englewood Clffs, N.J. (986)]: ω +ω j ω j = (4-83) T cj = (T c T cj ) ( k j ) (4-84) Z cj = cj cj (4-85) Vcj Tr.6 Tr 4. db dtrj Z c + Z cj wth Z cj = (4-86) and V cj = 3 3 V c + V cj (4-87) 3 In Eq. (4-84), k j s an emprcal nteracton parameter specfc to an j molecular par. When = j and for chemcally smlar speces, k j =. Otherwse, t s a small (usually) postve number evaluated from mnmal VT data or, absence data, set equal to zero. When = j, all equatons reduce to the approprate values for a pure speces. When j, these equatons defne a set of nteracton parameters wthout physcal sgnfcance. For a mxture, values of B j and db j/dt from Eqs. (4-8) and (4-8) are substtuted nto Eqs. (4-6) and (4-6) to provde values of the mxture second vral coeffcent dt B and ts temperature dervatve. Values of H R and S R are then gven by Eqs. (4-7) and (4-7). A prmary vrtue of Abbott s correlatons for second vral coeffcents s smplcty. More complex correlatons of somewhat wder applcablty nclude those by Tsonopoulos [AIChE J. : 63 7 (974); bd., : (975); bd., 4: 5 (978); Adv. n Chemstry Seres 8, pp (979)] and Hayden and O Connell [Ind. Eng. Chem. roc. Des. Dev. 4: 9 6 (975)]. For aqueous systems see Bshop and O Connell [Ind. Eng. Chem. Res., 44: (5)]. Because Eq. (4-68) s explct n, t s ncompatble wth Eqs. (4-45) and (4-46), and they must be transformed to make V (or molar densty ρ) the varable of ntegraton. The resultng equatons are gven by Smth, Van Ness, and Abbott [Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., pp. 6 7, McGraw-Hll, New York (5)]: R G = Z ln Z + R ρ (Z ) d ρ (4-88) T ρ R H = Z R T ρ Z T d ρ (4-89) T ρ ρ By dfferentaton of Eq. (4-68), Z db dc = ρ+ ρ ρ T dt Substtutng n Eqs. (4-88) and (4-89) for Z by Eq. (4-68) and n Eq. (4-89) for the dervatve yelds, upon ntegraton and reducton, G R 3 = Bρ+ Cρ ln Z (4-9) H R = B T db ρ+ C T dc ρ (4-9) dt dt The resdual entropy s gven by Eq. (4-47). In a process calculaton, T and, rather than T and ρ (or T and V), are usually the favored ndependent varables. Applcatons of Eqs. (4-9) and (4-9) therefore requre pror soluton of Eq. (4-68) for Z or ρ. Wth Z = /ρ, Eq. (4-68) may be wrtten n two equvalent forms: B C Z 3 Z Z = (4-9) () ρ 3 + B + ρ ρ = (4-93) C C C In the event that three real roots obtan for these equatons, only the largest Z (smallest ρ), approprate for the vapor phase, has physcal sgnfcance, because the vral equatons are sutable only for vapors and gases. Data for thrd vral coeffcents are often lackng, but generalzed correlatons are avalable. Equaton (4-68) may be rewrtten n reduced form as r (4-94) Tr Z Tr Z where Bˆ s the reduced second vral coeffcent gven by Eq. (4-7). Thus by defnton, B c Bˆ = B +ωb (4-95) c The reduced thrd vral coeffcent Ĉ s defned as Z = + Bˆ + Ĉ C c Ĉ (4-96) R T c A tzer-type correlaton for Ĉ s then wrtten as Ĉ = C +ωc (4-97) dt r

14 ROEY CALCULATIONS FOR GASES AND VAORS 4- Correlatons for C and C wth reduced temperature are C =.47 + (4-98) C = (4-99) The frst s gven by, and the second s nspred by, Orbey and Vera [AIChE J. 9: 7 3 (983)]. Equaton (4-94) s cubc n Z; wth T r and r specfed, soluton for Z s by teraton. An ntal guess of Z = on the rght sde usually leads to rapd convergence. Another class of equatons, known as extended vral equatons, was ntroduced by Benedct, Webb, and Rubn [J. Chem. hys. 8: (94); : (94)]. Ths equaton contans eght parameters, all functons of composton. It and ts modfcatons, despte ther complexty, fnd applcaton n the petroleum and natural gas ndustres for lght hydrocarbons and a few other commonly encountered gases [see Lee and Kesler, AIChE J., : 5 57 (975)]. Cubc Equatons of State The modern development of cubc equatons of state started n 949 wth publcaton of the Redlch- Kwong (RK) equaton [Chem. Rev., 44: (949)], and many others have snce been proposed. An extensve revew s gven by Valderrama [Ind. Eng. Chem. Res. 4: (3)]. Of the equatons publshed more recently, the two most popular are the Soave-Redlch-Kwong (SRK) equaton, a modfcaton of the RK equaton [Chem. Eng. Sc. 7: 97 3 (97)] and the eng- Robnson (R) equaton [Ind. Eng. Chem. Fundam. 5: (976)]. All are encompased by a generc cubc equaton of state, wrtten as a( T) = (4-) V b (V + b )( V + σb) For a specfc form of ths equaton, and σ are pure numbers, the same for all substances, whereas parameters a(t) and b are substancedependent. Sutable estmates of the parameters n cubc equatons of state are usually found from values for the crtcal constants T c and c. The procedure s dscussed by Smth, Van Ness, and Abbott [Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., pp , McGraw-Hll, New York (5)], and for Eq. (4-) the approprate equatons are gven as α(t a(t) =ψ r )R T c (4-) b =Ω c (4-) Functon α(t r ) s an emprcal expresson, specfc to a partcular form of the equaton of state. In these equatons ψ and Ω are pure numbers, ndependent of substance and determned for a partcular equaton of state from the values assgned to and σ. As an equaton cubc n V, Eq. (4-) has three volume roots, of whch two may be complex. hyscally meanngful values of V are always real, postve, and greater than parameter b. When T > T c, soluton for V at any postve value of yelds only one real postve root. When T = T c, ths s also true, except at the crtcal pressure, where three roots exst, all equal to V c. For T < T c, only one real postve (lqudlke) root exsts at hgh pressures, but for a range of lower pressures there are three. Here, the mddle root s of no sgnfcance; the smallest root s a lqud or lqudlke volume, and the largest root s a vapor or vaporlke volume. Equaton (4-) may be rearranged to facltate ts soluton ether for a vapor or vaporlke volume or for a lqud or lqudlke volume. Vapor: Tr V = R T a(t) + b Tr.7 V b (V + b) (V + σb) (4-3a) b V Lqud: V = b + (V + b)(v + σb) (4-3b) a(t) c c Tr.5 Tr.5 Soluton for V s most convenent wth the solve routne of a software package. An ntal estmate for V n Eq. (4-3a) s the deal gas value /; for Eq. (4-3b) t s V = b. In ether case, teraton s ntated by substtutng the estmate on the rght sde. The resultng value of V on the left s returned to the rght sde, and the process contnues untl the change n V s sutably small. Equatons for Z equvalent to Eqs. (4-3) are obtaned by substtutng V = Z/. Vapor: Z β Z = +β qβ (Z + β) (Z + σβ) (4-4a) Lqud: Z =β+(z + b)(z + σb) (4-4b) b where by defnton β (4-5) a(t) and q (4-6) b These dmensonless quanttes provde smplfcaton, and when combned wth Eqs. (4-) and (4-), they yeld β=ω (4-7) Ψα(T r ) q = (4-8) ΩTr In Eq. (4-4a) the ntal estmate s Z = ; n Eq. (4-4b) t s Z =β. Iteraton follows the same pattern as for Eqs. (4-3). The fnal value of Z yelds the volume root through V = Z/. Equatons of state, such as the Redlch-Kwong (RK) equaton, whch expresses Z as a functon of T r and r only, yeld two-parameter correspondng-states correlatons. The SRK equaton and the R equaton, n whch the acentrc factor ω enters through functon α(t r ; ω) as an addtonal parameter, yeld three-parameter correspondng-states correlatons. The numercal assgnments for parameters, σ, Ω, and Ψ are gven n Table 4-. Expressons are also gven for α(t r ; ω) for the SRK and R equatons. As shown by Smth, Van Ness, and Abbott [Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., pp. 8 9, McGraw- Hll, New York (5)], Eqs. (4-4) n conjuncton wth Eqs. (4-88), (4-89), and (4-47) lead to H R G R r Tr +β Z qβ = Z ln(z β) qi (4-9) = Z + d ln α(tr) d ln T qi (4-) TABLE 4- arameter Assgnments for Cubc Equatons of State* For use wth Eqs. (4-4) through (4-6) Eq. of state α(t r ) σ Ω Ψ RK (949) / T r SRK (97) α SRK (T r ; ω) R (976) α R (T r ; ω) *Smth, Van Ness, and Abbott, Introducton to Chemcal Engneerng Thermodynamcs, 7th ed., p. 98, McGraw-Hll, New York (5). α SRK (T r ; ω) = [ + ( ω.76ω / )( T r )] α R (T r ; ω) = [ + ( ω.699ω / )( T r )] r