Thermochemistry of Geological Processes
Thermochemistry of Geological Processes GEOL 555
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Date Created: 10/23/15
TOPIC 8 PHASE EQUILIBRIA EQUILIBRIUIVI BOUNDARIES INVOLVING FLUIDS To calculate the position of an equilibrium boundary for a reaction involving uids We start With the expression AG 46 P 71ASV RT an 0 For example consider the equilibrium CaCOKcalcite SlOZquartz lt gt Casi03wollaswnite COZg For this reaction We have assuming pure solids K co X cqxco chT X Cq chPT So if We have the proper thermodynamic data We can calculate the phase boundary as anisopleth on aPT diagram or as an isobar on a TX diagram etc 2 THE GENERALIZED DEVOLATILIZATION REACTION Alt gtB ucoZ va where u v may be positive negative or zero AVG AGE P1AV Whitman P magi 0 Assuming ideal mixing ofideal gases We can Write BTW E u 7 v BX F As Xm XE c0 vuv Zero reaction The above equation tells us that a TX COZ diagram aximum at XCOZ uu v XHZo The gas phase will be buf long as A andB coexist No maximum can exist if either u or v or both are The above equation allows us to predict the geneml the TX boundary for a devolatilization fered at this maximum as A Rn x 8T mt V 0 ple ZMgO sio2 lt gt MgZSiOA RxT I n x M57Si20220H2 lt gt Mgsio3 sio2 H20 AeBucozugtov0 I 67m RT 11 ugt0v0 xm Example Caco3 sio2 lt gt CaSiOZ co2 AeBuCOZvHZOugt0vgt0 I anew 7 T 7 J u gt 0 V gt 0 exec Ax Xa Xaa Example C zMgsSi20220m2 i0 lt gt 5CaMgSi206 3co2 H20 Maximum occurs atxcoz 33 1 0 75 XHZO 131025 H9 075 002 X002 AVHGOQBUCOGL1gt0Vlt0 I saw 7 u 7 V j u gt 0 V lt 0 am Ax Xa X Exam 1e CazAl silo8 Caco3 H20 lt gt CazAIZSiZOH OH co2 EXPERIMENTAL PETROLOGY SOME PRINCIPLES I The PT mnge over Which a phase is found to be stable cannot be extended by the presence of additional phases I The stability mnge of a given phase may be reduced by the presence of an additional phase Example Reduction in the eld of stability of rnuscovite in the presence of quartz At P kbars muscovite persists to 670 C in the absence of quartz but rnuscovite quartz breakdown at 600 C KAlZAlSiKOmOHZ lt gt KAlsip8 A1203 H Zo SiuZ AlzblUS KAlelsipl UOHZ 102 gt KAlsip8 Alzsio5 H20 0 The central goal of experimental petrology is to determine the stability fields of individual minerals and simple assemblages on PT or TX diagrams MEANING OF A PHASE BOUNDARY REVISITED u AG39 O Aphase boundaryisthe locus of a11 points where AG 0 for the reaction H ofinterest REVERSED PHASE EQUILIBRIUIVI MEA SUREMENT S In this methodyou start 39 h both reactants and H products in every run 4 0 Mam 4 0 o favored at each 39 i with this method the actual equilibrium e is not directly obtained 0 M90 002 but the curve can be bracketed to within the desired or possible ee ofaccuracy 13 ONEWAY REACTIONS Some reactions are 00 o H slow in the reverse 0 direction to be 0 H experimentally observed Such experimental data CaMgCQ change are meaningless cacos M90 002 because ofslow kinetics D SYNTHESIS REACTIONS C 0V CV Sometimes when the reaction A gt B is too slow it A has been common practice to start with a less P G stable reactant C such as a glass gel oxide B 39 ure etc Intheory we should observe C gt A in the stability eld of A and c gt B in the B Agtlt stability eld of B T T However it is o en noted that when using this thod the position of the equilibrian boundary appears to depend on the starting material chematic PT diagram showing chematic GT diagram showing location ofthe phase boundary on e nature of the starting ergy curves of the products and eactants in a synthesis SYNTHESIS REACTIONS Cont d The differences in Gibbs free energies between C an and B are both quite large so kinetic factors determine the reaction products instead of thermodynamics Ostwald step rule There is a tendency to prefer that reaction which involves minimum Ar S er than minim AG in systems controlled by kinetics I Synthesis experiments should be taken with a gmin of salt particularly if they contmdict eld observation reliable thermodynamic data or direct reversed or oneway experiments SOLUB ILITY MEASUREMENTS The idea of this method is to determine the solubilities of various phases in water over a broad PT range We assume the solubilities are low so the solution is dilute and we can approximate it as ideal Thus p1 li RT lnX In a solubility experiment when equilibrium is attained pfquot Kl Now suppose we wish to study the phase boundary for the reaction 1 andalusite lt gt kyanite If we measure the solubilities of kyanite and andalusite e amter under identical conditions we can 39 o P determine the equilibrium b undary Ix ArGl ka x 2 andalusite lt gt corundum SiOZaq ArGZ Gmxl Gsmwfnd Gm 1 3 kyanite lt gt corundurn SiOZaq Arc3 Gmxl 15ngka kaX1 In each set of solubility measurements When equilibrium is attained We have AYGZ AYG3 0 so Gama Gama GS OZaqand Gk Gnu1 GSszltaqgt AYGl Gmxl Gsm mky Gmxl Gsmzwky ArGl p 502 RT mxsmzky p 502 RT mxsmzand n AGl RT ln 3 302 ky1 But at equilibrian Are kyanite z 0 so d I and an ng X302 P Thus the equilibrium X30 Xami urve is the locus of all points in PT space Where the solubilities of kyanite and andalusite are equal T andalusite QUESTIONS TO ASK WHEN EVALUATlNG PHASE EQUILIBRIUM EXPERIMEVTS 1 Was the experiment truly reversed Consider C D gt A B Are C and D identical in all respects IfC and D are in one crystal form as products but another as reactants the experiment is not reversed For example Mamp SiO4 SiOZ gt Mamp SiZO cpx is not the reverse of MgZSiZO opx lt gt MgZSiO4 SiOZ 2 Were the reactants and products suf cieme Well characterized We Want to know Which new phases if any have formed and Which original phases if any have dissappeared Xray diffmction is a typical technique used for this purpose but it alone may be insuf cient May need to use multiple techniques 3 Was the quench mte suf ciently fast to really freeze in the high PT assemblages Reactions involving volatiles have to be quenched very quickly 4 Were enough variables controlled or measured to fully constrain the system according to the phase rule 22 5 Were there any gmdients in pressure tempemture etc Temperature gradients In the hydrothermal technique coldseal vessels eg gradients occur because part ofthe vessel is cold and part is hot rherlmocouple gold capsule thermal gradient Pressure gradients In pistoncylinder equipment NaCl talc or pyrophyllite are o en used to tmnsmit pressure May not tmnsmit pressure accumtely or evenly E SCHREINEMAKERS RULES AND THE PETROGENETIC GRID BASIC FACTS Consider anncomponent system 1 At an invariant point n 2 phases coexist 2 Up ton 2 univariant curves radiate from each invariant point 3 Each univariant curve involves n 1 phases and can be labelled conveniently by the one phase not participating in the reaction 4 Divariant assemblages ofn phases each exist in the sectors between e univariant curves None of the sectors subtends an anglegt 18 5 Some equilibria may be degenerate ie a system with fewer phases than the maximum permitted by the phase rule 25 SCHREIN39EMAKER S GEOMETRICAL METHOD This method allows one to construct a qualitative P T or TX etc net by purely geometric arguments based on the phase rule Experimen s an thermodynamic data Clapeyron constraints can help locate the univariant reac 39ons an 39 39 points in phase diagram space We can use the geometric constraints to 1 Supplement experimental data 2 Evaluate experimental data 3 Provide some idea of phase relations when no experimental data are available 26 M39U39LTISYSTEMS Multisystems Systems in which more phases exist than can be in equilibrium at one time according to the phase rule Let b the total number of phases that exist in a system ofC components Let P the number ofphases we wish to consider for a given degree ofvariance We can use the combinatorial formula to determine the maximum possible number of invariant points univariant curves and 39variant elds in this multisystem THE COMBINATORIAL FORMULA For 1 phases taken P at a time ForaninvariantpointFC2P0 soPC 2 Thus the total number of invariant points in this multisystem are a N 39C2 c 2z 7c2z ForaunivariantcuiveFC2PlsoPC land l C1 7C1 Andforadivariantphase eldFC2P2so N Cl y P C and a N P 39 N c W 1W7 P 27 a C C 2 EXAMPLE The petrogenetic gridin PT space for this 6phase multisystem Consider a 3component system with 6 possible has aninvariantpointP F3 0 5 phases coexist Thus the number of invariant points will be 6 654321 N65 6 39 5m 5432101 The number of univariant curves will be 6 7 654321 4z2z 432r2r The number of divariant elds will be 654321 31404321 3 N54 15 2 a 2 N63 15 3 6 n may look like that below FUNDAMENTAL AXIOM This must be stated at the beginning because in the most geneml case it may not hol Fundamental Axiom When tWo divariant assemblages I and 11 each ofn phases meet along a univariant curve of n 1 phases then on one side of the curve assemblage Iis less metastable than assemblage II and on the other side assemblage II is less metastable than I 0st likel case QUESTION Consider anncomponent system With n 2 possible phases labelledl 2 3 n l n2 A divariant assemblage n 1quot 2 is con ned to a sector in PT space bounded by the tWo intersecting univariant curves labelled n l and n 2 The tWo curves will in general divide P T space into tWo sectors one With an angle greater than 180 and one With an angle less than 180 Which sector will contain the assemblage n MORE Y S CHREINEMAKER RULE A divariant assemblage n l n 2 always occurs in a sector Which makes an angle about the invariant point of no greater than 180 Proof based on Fundamental Axiom n1 n1 n2 Lo icall metastabte WWW g y stabie and inconsistent metastabte n2 mum statute and 1 WIJVZ metastabte 39 Stabie n1 wtg ntn2 Eta e meta tab e Logicallyconsistent n2 1 2 M m I metastabte rttn2 o metastabte Consequences The PT region near each invariant point is divided by univariant curves inton 2 sectors each ofwhich is 80 in angular extent and is occupied by one or more divariant assemblages one of which is unique to that sector n n can exist under identical values ofP and T which assemblage prevails depends on bulk composition 35 OVERLAP RULE Each divariant assemblage that extends across the univariant curve 039 Whether stable or metastable contains the phase j 1 Example Assembiage A ASSEWWEUE A 39 If assemblage A occurs on both sides of 1 Without change then assemblage A must contain phase 1 Any assemblage not containing phase 1 Would have to change 1 2 3 4 Possible assemblages 123 245 235 us the only assemblage that remains unchanged after univariant reaction 1 is 1 2 3 which contains phase 1 This rule can be used to check the consistency of a phase diagram EXAMPLE 1 25 lt gt341 4 5 Possible assemblages 123 234 345 37 CONSTRUCTION RULE 5 Any univariant reaction 3 labelled i always occur on the side of other univariant reactions 4 opposite the side on follows from the Morey Schreinemaker Rule 2 1 If 1 occuned on the same side of 5 as it appears as a participating nha 9 1x assemblage 15 2 3 4 would occur over a sector of more than 0d 3r invariant poin itsminorimage Useful fact Any symmetry existing in the chemography of the phases will be re ected in the symmetry of the arrangement of the univariant curves about the invariant oint Three pieces of information are not available from Schreinemakers analysis 1 The actual location ofthe invariant points in PT space 2 Actual values of the slopes ofthe univariant curves near the 3 The choice between a given sequence of univariant curves and m l a HOW TO ORIENT A NET 1 Use experimental data for the position of some invariant points and univariant reactions 2 Use the Clapeyronequation to constrain slopes of the univariant curves in PT space 3 For dehydmtiondecarbonation reactions the volatiles always will occur on the hi temperature lowpressure side of the univariant reaction oundary 4 In solidsolid reactions the denser phases lower molar volume occur on the highpressure side of me univariant reaction boundary n 2 39 ONECOMPONENT SYSTEM N33 l invariant points N32 3 univariant curves and N3l 3 divariant phase elds This system is typical of polymorphic transitions 1 3 Possible reactions 3 J 19 2 3 19 3 2 2 lt gt 3 l The minor image ofthis topology would also satisfy Schreinemaker s rules u TWOCOMPONENT SYSTEM NON DEGENERATE N44 l invariant points N43 4 univariant curves and N42 6 divariant phase elds Chemogmphy 3 4 Possible reactions 4 1 3lt gt 2 Note symmetry of 1 and 4 relative to 2 and 3 n THREECOMPONENT S YSTEM NONDEGENERATE There are three nondegenerate distinct topologies for three component systems degenerades arise due to co11inearity of3 or more phases or polymorphism 1 Possible reactions 1 2 3 lt gt 4 5 2 3 4 lt gt 1 5 512lt gt34 DEGENERATE SYSTEMS A system is said to be degenemte if a TWo or more phases have the same composition ie are polymorphs b In an n component system Three or more phases are colinear Four or more phases are coplanar TWO COMPONENTS DEGENERATE There are four possible different cases Case 1 2 4 In this case phase 4 cannot participate in an ofthe reactions so phase 4 is called the indifferent phase The other phases are called singular phases mm 1 lt gt 2 Case 2 1 2 34 I l 2 3 12 are relatively indifferent 4 l 1 to 34 and vice versa Possible reactions 41lt gt2 Case 3 12 3 4 Possible reactions 2 4lt gt 3 2 1 4 lt gt 3 3 1 lt gt 2 4 1 lt gt 2 Possible reactions 1 2lt gt3 21 4lt gt3 31 4lt gt2 42lt gt3 CO INCIDENCE RULE When tWo indifferent phases lie on the same side of the singular phases then the univariant curves bearing the label of the tWo indifferent phases coincide stable to metastable When the tWo indifferent phases lie on opposite sides of the singular phases then the univariant curves bearing the labels of the indifferent phases coincide stable to stable THREECOMPONENT c 1 DEGENERATE SYSTEMS Possible reactions Case 2 Possible reactions 1 4 5 lt gt 2 3 2 1 lt gt4 3 1 lt gt4 4 1 5 lt gt 2 3 5 1 lt gt4 1 4 APPLICATION To THE SYSTEM AlZOSSiozHZO Andalusite A pyrophyllite P kaolinite K quartz Q Water W Hp Possible reactions VJAKQlt gtP AKQlt gtPW KAQWlt gtP PAQWlt gtK QAPWlt gtK But this net is not properly oriented Let s assume th MOLAR VOLUIMES molar volumes but 2332211136 Phase V cm3 mall at 25 C 1 bar H20 1807 quartz 2269 DVmDthlite 12660 kaolinite 9931 andalusite 5154 55 m ORIENTED NET PT SLOPES Reaction Av cmzmol39l As dPdT KA3QWlt gtP 111lto lt0 gt0 PAQ2wlt gtK 1101lt0 lt0 gt0 A 2QKlt gt WP 002gt0 gt0 gt0 Q 3Klt gtP5W2A 221gt0 gt0 gt0 WA5QKlt gt2P 111 7 7 n 3 M39ULTISYSTEMS Such systems contain one more phase than is required for a single invariant point As a result the petrogenetic grid for such a system contains more than one invariant point It is possible to link the invariant points together 39 univariant curves in avariety of different topologies In general in anondegenemte n 3 multisystem there are n 2n 3 2 alternate base nets possible REE FLUOROCARB ONATES Minerals in the LnCOKFCaCOKFZCOKVl system 7 calcite cc Caco3 7 basm site ha anogF 7 pan39site pa ancmcogzr2 7 synchysite syLnCa COX 7 riintgenite LnZCa2C035F3 If We neglect rontgenite because of its mrity We have ann 3 multisystem This is a three component system With amaximum of 6 possible invariant points 6n WHY ARE FLUOROCARB ONATES IMPORTANT They are the main ores of LREE and are important accessory minerals in avariety ofrocks Carbonatites Mountain Pass CA st Honore and oka Quebec Karonge Burundi W Mountains C0 Wigu Hill Tanzania etc Skarns Bastnas Sweden Granitessyenites Strange Lake and 39lhor Lake CanadaNarsarsuk nd Rodeo de Los Molles Argentina etc Stratabound hydrothermal Bayan Obo China Olympic Dam Australia Veinsbreccias Olympic Dam Australia GallinasMountains New 39 uzo Columbia Snowbird Montana Rock Canyon Creek BC etc m The maximum number of distinct possible base topologies for this multisystem is n 2n 3 2 3 23 3 2 32 In addition We have the mirror images and trivial co 39 Trivial conj1gate a trivial conjugate is formed by changing the stability status ie metastable or stable of all the invariant points ofthe base topologies While maintaining the order of the univariant reactions parity about each invariant point If one base topology has been determined the others an be derived from the rst by systematically tmnsposing each of the invariant points one at a time until all possibilities have been exhausted a TRANSPOSITION A ternary diagram for a ortion ofthe Ln CO F L n C C F P 3 ba 39 cang phase relationships amon 39te cc uocente Pa c uonte c bastnasite ba p t 5y a and synchysite sy portion ofthe diagram is physically inaccessible cc CaCO3 FZCOS4 In this case systematic determination of the possible topologies leads to only 23 different base topologies this is because the system under discussion is degenerate Some possibilities can be eliminated based on volume and entropy constraints For example any topology that contains a triangle in Which the highvolume or highentropy assemblages are all on the inside or all on the outside is physically impossible We can also use Clapeyron slopes and What little experimental phase equilibrian or eld evidence We have as Example of a topology that is impossible to orient to satisfy all volume or entropy constraints ESTIMATING ENTROFIES um mum mm myws A h m Amarme may A m cm Lmabym ML 9 mm m 5mm 5 ELWLMMWOWA m mm comm no Am coio m w WHEELER h MIMIquot Wm Mum 439 X2 rm 7 an wlrme M Ca Mgssz Mn Cd Fe Zn mm k M qulzcmfuz um REE ma zngxrgwe chum xrgwedsoubum 428t561mnl K NW Mahmud muvpyumsmumom e memhrm SL2ltDF mam W m Ema VOLUMES AND ENTROHES OF ELLvoRocAREoNATEs AND RELATED MINERALS ENTROFY AND VOLUME CHANGES REACTION L m 139 r munemuy nun m Mm Mint mu m Mum um nm N AWE n in m r 7ymuyn01p mu m 7 W m m r mu m r mmamwm mmnm n dgmfunlum momma mun Ami mmcmwhcm mun m MAMAMMWWWW n anmmm mmcmw mu mm m WHICH NET IS THE CORRECT omzv 3914 m mm mm Thiasserfbbgl 4m cemrdyocuus mam FIXEDVSLOPE DIAGRAMS m mamas gramme Phase dugzmwe e g mummy mm mm dkmnw moms 0m 01quot womwbgms mm dznvedfmmdrlz m sme Ybyfmung sub Wamn ymms W mm 3911 m Annanew uh Phase smug THE FsruquIAGRAM up 5 3911 mm mum 01wsz m hummg mam nhcmhdds as Am Kain cmbe Bran 3914 snhds an my datean Wm 3914 m whm mg m m bexm39xmlzs ATmum de may Mom WHY BOTHER WITH sUcH A DIAGRAM7 AFFLICA TIONS We Wm my mm m 3914 Mom systans CzorZroisxofxlio 21d ziormisxoirnp zmym shmme mlwdmm mm M mks Nchas mum syemms humus cmde Perugzm Th 5 m p at ma Pgemn ycmhk dmcdy Mm annexed Mcmw mlmdkmm Em ls swam KW mums Iiihue szdszun B squucusdz was mm mm New quotAw em C50 5er3 Zr FHASES IN THE 32075102720pr SYSTEM u 5 Vin n AM A mum 11m A a manna um x mm mzmni m an m 1117 Q My 30 1w 1 m mummy Wahgus m P M m mm mm mmmumm quot Walngtsu hl a W WadJun m AVU n m 1ynmunumu um 1ycno HID a w xyncwu mm 1 am tyne 130 1 am mm 3 am 3 mm mm numu 1 m vycvga as am A mm A n yynon1 mm M Wm no um m mmmm Wmme v a mi ms mm m mam m mm Wm x h P am mm smmPNgmm Homogeneous uids are normally divided into two classes liquids and gases vapors A phase that can be condensed by a reduction 4 of temperature at constant pressure Liguid A phase that can be vaporized by a reduction of pressure at constant temperature NONIDEAL FLUID BEHAVIOR The distinction cannot always be made unambiguously and the two phases become indistinguishable at the critical point l Supercritical a T 1 I My D THE CRITICAL POINT Critical point The maximum pressure and temperature where a pure material can eXist in vaporliquid equilibrium Beyond Tc and PC the designation of gas vs liquid is arbitrary rre m m Bin waan At the critical point the meniscus between phases slowly fades and dissappears J quotEhrllmlinn zip If one moves around the critical point it is 5 possible to get from the liquid to the vapor field 391 I without crossing a phase boundary Iti39mi m39 3 PT phase diagram for a pure material 4 PV phase diagram for a pure l material C critical point nRT V At high T we expect the Gas isotherms to conform to the ideal gas law ie P is inversely proportional to V PV phase diagram for pure 5 l I I P P39 I Hi Liquid THE PV DIAGRAM We can use the lever rule on a PV diagram to determine the proportion of vapor vs liquid at any given pressure The bending of the isotherms in the vapor eld from the ideal hyperbolic shape as the critical point is approached indicates nonideality The PV diagram illustrates the dif culty in developing an equation of state for all regions for a pure substance However this can be done for the vapor phase Schematic isotherms in the twophase field for a pure uid For uid of density A the proportion of vapor is YXY and the is XXY vapor is PPQ and the proportion of liquid is QPQ MOST GENERAL EQUATION OF STATE dV 01ml dP 5T 6P d7VadT pdp Two special cases a Incompressible uid aBO dVV 0 no equation of state eXists V constant b 0L and B are temperature and pressureindependent VIRIAL EQUATION OF STATE The most generally applicable EOS PVabPcP2 a b and c are constants for a given temperature and substance In principle an in nite series is required but in practice a nite number of terms suf ce At low P PV z a bP The number of terms necessary to accurately describe the PVT properties of gases increases with increasing pressure The limit of PV as P gt 0 is independent of the gas T 273 16 K triple point of water PV1cm3atm1g mo i 9 a 3 lim PVTP 0 PV 22 414 cm3 atrn g mol l a P So a is the same for all gases It is in fact RT 11 I THE COMPRESSIBILITY FACTOR proportion of liquid For uid of density B the proportion of THE COWRESSIEILITY FACTOR mm 1 my cvpmmx mcmm kkaovmmm39km ewz mmnh byd zeq z m f D Drzncou39 W n my m mama dun mmtnuncnxs mug mixuks em 1 cmlmm many mum at minus SOME AFFROXIMATIONS 11 g n V V Only a m c m gamma 1mm mm mm wax0 vs a mud H EQUATIONS OF STATE THE OBJECTIVE IN THE SEARCH FOR AN EOS VAN DER WAALs EQUATIoNum p quot m V 5m 2 mm xdahvzlyfzw parameters 2 thatcmbe Emily Amman 4 am can be Tamara xmxuxes Mums 1mm forts mdznllzs 51mm yam15m AHDN Pressma39s 4ng measmxmcumwess um 44 gases A l39 glruynsswes mm mm mm m comps m z Hm n1qu rum x THE VAN DER WAALs PARAMETERS We cmdzknmmlww mama k azrd hymn m m m y m m m Ar an m um mm x as m m m n m m 332 m U ml m m m m 335 m um nm 1 s m c 761 m um mm m sat 2 CRITICAL CoNsTANTs OF CASES 71 m cm 727 M 3m n275 m m 553 76 mm u 715 mm mm m m w m mm m m 2m nzm cm m m ans ms cm as 1m 5m um mmaawmmammmm New mumgm ammmmsmpmmm sues We can rearrange the previous equations to get the der Waal parameters in terms of the critical parameters a 311V b V where V Note that the actual measured value of VB is not used to calculate a and b VAN DER WAALS CONSTANTS FOR GASES W m Ne 02107 1709 cm 3592 4267 1345 3219 H20 5464 3049 2318 3978 NH 4170 3707 Xe 4194 5105 c114 2253 4278 H2 02444 2661 CZHA 4471 5714 02 1360 3183 Csz 5489 6380 N2 1390 3913 C H 1800 1154 a am6 annmol392b 1072 am molquot SCth OTHER EOSS Bertholet 1899 PI7 7b RT The higher the tempemture the less likely particles will come close enough to attract one another signi cantly a and b are different from VdW Dieterici 1899 Keyes 1917 P 37 01 3 A and l are correction factors 27 AAa 7aV BBn17bI7 s a 17 CAD BU are constants We usually don t know V but we know P so an iterative approach is required calculateA B and a with an assumed V value and compute P IfPEalE Pexp then adjust V accordingly and recalculate P Rearmngement of the BeattieBridgeman equation gi 7RT I7 I72 I7 I74 R Where ange1ti RBI y7RTB baA i 5712 5 This shows the BB equation to be simply a truncated form of the vuial equation AAU17al39l BBn17b139I 5 perfect gas volume P a b andc are the same as for the BB EOS One can use the Beattie equation to obtain a rst guess for the BeattieBridgeman equation which is more accurate because it allows for the variation ofA B and a with volume EMORE EOS39s rm W m asemmmhmm ma gm mm 1 maxn4qrpmma JUMP Lula Mailers my 120 5 4m Ammmm macs m u mum u nassm 1 E e 3 m use 3 mum Wm rampmm Wining a ma 7 mm m mm mmmnmmmmm ksmanna E if hnbxszcunsmzrd a an mmm whip mi 1 u nzmmmxfmhmdcdkdk anemfo Carnahan and Starling 1969 haIdsphere model Kerrick and Jacobs 1981 HardSphere Modi ed RedlichKwo HSMRK aPT E M 7 by T217075 aPT an emp icallydelived polynomial aPTcTdTeT zTzzzTaT3 wherezcdor2 37 b 7b Tquotb T39Zib T393 c 70 Tquotc T393 Z1MZIJ HAIMZHI V LEE AND KESLER 1975 V v Mi3em4jew V r V V J 1 bu Abnr t zr 7114773 cn four mgr K V d Tf c y 39i 732 T3ZAVge 1 39 DUAN MOLLER AND WEARE 7 72 73 7 72 73 BialazTy ajTy C414 a5Ty a Ty D a7 JrLIETy39ZJragT3 E am aHTy392 alzTy393 4 F 706T Vy y B y T This isjust a modi ed form the the Lee and Kesler EOS CALCULATING FUGACITY COEFFICIENTS BY INTEGRATING AN EOS Using the van der Waals equation Using the oIiginal Redlich Kwong equation P7 RT 7 a 11771 TZVVb Using the HSMRK EOS ofKenickand Jacobs 1981 h1 8y79y23y37h1P177 c 17y RT MEG17 d e I RT7VI7 b I RT7V207 b C Inii d Rsz Vb RTEVb V d mm 2 W17 V RTZZbZ e b 2 WW 7 Ammsmc Hummsmdememm may dun n mmu om cummmonvc m a mum mst cummdmkksn Human 2n mam m xmxzmwvf x nhmndc xmwm mm mum man Emmusamuum m mmmmmmsm Wmmdmatsamdm AMEwwus mumuwz nmnmmmm h ammuwmsul m AMEwwus mxgmuwzw III CORRESPONDING STATES ad cmmsea m dam pvr Mums D m r gm ow141de 120 5 zvadahlz WLE Caminude 51mmquot mm m A sum Cdekhzzmxrgdunducechlrnnmd mm wk 44m law wedcom mummy Comsymdxrg mm x mugom 5K z m p zmmmwums 12m annnmv zquot quotviltnnxnmmmrxmm v ar S Lm lily KELID 39 u I H MI I p I um 1 I3 an 15 mm um 5 MI I a a 39 Lln 3 ma in LII r E IIJUT iun UniI llllnlialulvulu I 0 10 In IHJ 5 EA T Li Lil Hediml mum Pr Measured compressibility factors for H20 vs those obtained from corresponding state theory I B ll R l Min 4 IF no El 1 1 f lf b n 139 III EEPI39I inn I39L39ICI qul 1 Measured compressibility factors for CO2 vs those obtained from corresponding state theory 10 03 rc15o Igt E 06 39 a n H 739 120 N 04 co2 0 02 TC l00 I I I I o o 100 200 300 400 500 Pt F Generalized density correlation for liquids pr ppc 58 PITZER S ACENTRIC FACTOR The acentric factor of a material is de ned with reference to its vapor pressure The vapor pressure of a subtance may be expressed as vat b logPr a T but the LV curve terminates at the critical point where Tr Pr So a b and If the principle of corresponding states were exact all materials would have the same reducedvapor pressure curve and the slopea would be the same for all materials However the value of a varies 59 The linear relation is only approximate a is not de ned with enough precision to be used as a third parameter in generalized correlations Pitzer noted that Ar Kr and Xe all lie on the same reducedvapor pressure curve and this passes through log Pf l at Tr 07 We can then characterize the location of curves for other gases in terms of their position relative to that for Ar Kr and Xe The acentric factor is a logPf H7 1000 03 can be determined from T 0 Pc and a single vapor pressure measurement at T r 07 10 10 12 14 16 18 20 0 1 I r F a 1 a Slopea ZB 3 Ar Kr Xe E 2 r l I I I 1 139 1 u c Slope m 3 2 Tr 0397 noctane Approximate temperaturedependence of reduced vapor pressure 61 ACENTRIC FACTORS FOR GASES Gas 0 Gas 0 Gas 0 Ne 0 C12 0 073 methane 0 011 Ar 0 004 Br2 0 132 ethylene 0 087 Kr 0 002 C02 0 223 ethane 0 100 Xe 0 002 CO 0 049 benzene 0 212 H2 0 22 0 250 toluene 0 257 O2 0 021 HCl 0 12 nheptane 0 350 N2 0 037 HZS 0 100 propane 0 153 F2 0 048 S 02 0 251 mxvlene 0 331 PRINCIPLE OF CORRESPONDING STATES REVISITED Restatement of principle of corresponding states All uids having the same value of 0 have the same value of Z when compared at the same Tr and Pr The simplest correlation is for the second virial coefficients 2 2 1 3 2 1 5 RT W T The quantity in brackets is the reduced 2nd virial coefficient j 2 13 9131 0 422 0172 B0 0 083 T16 B 20139 T42 The range Where this correlation can be used safely is shown on the chart on the next slide For the range Where the generalized 2nd virial coefficient cannot be used the generalized Z charts may be used ZZO 0Z1 These correlations provide reliable results for nonpolar or only slightly polar gases The accuracy is 3 For highly polar gases the accuracy is 510 For gases that associate even larger errors are possible The generalized correlations are not intended to be substitutes for reliable experimental data Generalized correlation for Z0 Based on data for Ar Kr and Xe from Pitzer s correlation Generalized correlation for Z1 based on Pitzer s correlation Z EXAMPLE 1 What is the volume of SO2 at P 500 atm and T 500 C According to the ideal gas law 1E 0 080256atmLmol 1773K P 500atm Using the acentric factor 0 0273 Tr 7734308 179 Pr 500778 643 From the charts Z0 097Z1 031 Z 097 0273031 1055 500atmv 0 080256atmLmol 1773K I7 0 124Lmol 1 1055 V20131Lmol391 67 4 i T l 1 Use generalized virial coefficients Eq336 and Eqs337 and 338 Z 3 T 39 2 Use generalized compressibility factors 1 1 Pg 339 and Figs 312 and 313 aturation o 1 2 3 4 s 6 7 z i Line defining the region where generalized second virial coefficients may be used The line is based on Vr 2 2 EXAMPLE 2 What is the volume of SO2 at P 150 atm and T 500 C According to the ideal gas law 7 1E 0 080256atmLmol 1773K P 150atm Using the acentric factor 0 0273 Tr 7734308 179 Pr 150778 193 0 414Lmol 1 300083 042126 0 083 179 310139 017422 0124 179 BP ch 30 wBl 0083 02730124 41049 3P P 193 Z1 c r1 0049 0947 kRchTr 179 Z 0947 150mm 0 080256atmLmol 1773K V 0392L mol39l Vc 0122 L mol391 Vr 03920122 325 CORRESPONDENCE PRINCIPLE FOR FUGACITY Correspondence principles and generalized charts exist for fugacity and other thermodynamic properties For fugacity both two and threeparameter generalized charts have been developed Again these are to be used only in the absence of reliable experimental data ln p f de 0 Pr 1 We can use this equation together with the generalized Z charts 1 Look up Pc and T0 of gas 2 Calculate Pr and Tr values for desired TS and PS 3 Make a Table of Z from the generalized charts at various values of Tr and Pr Of course we must have Pr values from 0 to the pressure of interest at each temperature 4 Graph Z1Pr vs Pr for each Tr 5 Determine the area under the the graph from Pr 0 to Pr Pr to get In p 11 Used generalized fugacity charts 12 USE OF TWOVFARAMETER GENERALIZED FUGACITY CHARTS EXAMPLE Cdzuhtn 3914 Elgamty coi A mu c 27mm mum Em hm f112lmm mob TK 3mm 7mm E 227 m EXAMPLEZ Whns w lgmtyuthmd chase ma mu m7 uvzwtmssu DrcE n25 c 5 7 A Fun Wymcoems rgwnhllqmd K 171 am THREEVFARAMETER CORRELA TIONS FOR FUGACITY ETC 1 mm lag 0 ng mt a fxnmmuquynm my fat 3E commmsm E 1995 11le mmams comm IV GASEOUS MIXTURES IDEAL GAS MIXTURES I Mixture as a Whole obeys PI7 RT I TWo such mixtures are in equilibrium With each other through a semipermeable membrane When the partial of each component is the same on each side of the membmne I There is no heat ofmixing The gas mixture must therefore consist of freely moving particles With negli 39ble volumes and having negligible forces of interaction DALTON S LAW VS AMAGAT S LAW I Dalton s Law P PT I Amagat s Law V XVT These tWo laws are mutually exclusive at a given pressure and temperature xn THERMOD YNAMICS OF IDEAL MIXING REVISITED We have previously shown that AG RT 2X 1 lnXx xdealmxx using Dalton sLaW We can derive AthaaIVnx RTZX mi Pr and for entropy We have AStdtzaImx RZX mi Pr x NONIDEAL MIXTURES OF NON DEAL GASES For a perfect gas mixture 11 RTlnIf 1f RTlnPT RTlnX For an ideal mixture of real gases 11 RTlnf pf RTlnfjquot RTlnX f Xfquot X4121 Lewis Fugacity Rule For areal mixture ofreal gases 11 0 RTlnf f mmquot X Qf onection for x DALTON S LAW AND GENERALIZED CHARTS Calculate reduced pressure according to RFi 39 1 PAVT nAZART 133V nBZBRT PCV nchRT PA 133 PCVT nAZA Jrquotst quotCZCRT nXAZA XBZB XCZCRT quotrszT AMAGAT S LAW AND GENERALIZED CHARTS Calculate reduced pressure according to p m P PTVA quotAZART Fri3 quotBZBRT PTVC quotchRT PVA VB VC nAZA 4325 nCZC RT nXAZA XBZB XCZCRT nTZmRT PSEUDOCRITICAL CONSTANTS LVV cuve for A 7V curve for B KAY S METHOD Assumes a linear critical curve between the critical points for A and B PC 39 Z X Pc r r C Z X r When answers are near the critical point for the mixture We cannot be certain that We are not dealing With a liquidvapor mixture JAFFE S METHOD For binary mixtures only am BaXB BJB a T i m X Bxi w iXaXB MIXING CONSTANTS IN EQUATIONS OF STATE Van der Waals and simple Redlichinong EOS a 7 aa x Use ifno mixture data Jk J k are available BeattjeiBridgemanEOS zw v1 v1 am ZXJaJ hm Xij 11 11 4 n 3W ZXJBUJ Cm 2 X16 11 1 BenedicLWebbiRubin E08 1 n AW Bumx EXJBM swam MW F 3 J W 3 2 aggw quotml Virial Equation of State Z1BVCV2 DV3 BM XXBJ 11 11 mex ZZZXIXJXICQJIC 11 11 k1 PREDICTION OF CRITICAL CONSTANTS Critical Temperature compounds with Tbml 1 atm lt 235 K and all elements TE 17 Tb 200 11 All compounds With Tbm11atmgt 235 K A Containing halogens or sul ir TE141Tb 66 11F F No of uorine atoms B Aromatics and napthenes TE141Tb 66 r0388Tb 93 r ratio of noncyclic carbon atoms to total carbon atoms y C All other compounds TE 1027Tb 159 Critical Pressure I K 7 9 Where TB is in K and VB is in cm3 gquot CIiticalVolume V 037717 110 Where F is a pammeter called the Sugten Pamchor SUGTEN PARACHOR VALUES FOR ATOMS AND STRUCTURAL UNITS C 48 S 482 triple bond 466 H 171 F 257 A 167 N 125 CI 543 CI 116 P 377 Br 680 0 85 O 200 I 910 0 61 O esters 600 double bond 232 Pcampmd Z quot1P
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