Thermochemistry of Geological Processes
Thermochemistry of Geological Processes GEOL 555
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Date Created: 10/23/15
TOPIC 6 SOLID SOLUTIONS REVIEW OF IDEAL MIXING Recall that for an ideal mixture we have found the following relationships AQdealmxx RTZXx lnXx Asidealmxx RZXx lnXx Adeeazm AdeEalmxx 0 ANOTHER APPROACH TO THE ENTROPY OF MIXING Recall the expression from Boltzmann S k In W where W the number of microstates corresponding to a single macrostate We can calculate W for the mixing ofN objects some of which are indistinguishable t W J N XN it where X is the fraction of objects of type i 3 EXAMPLE KAISi308 We have 4 tetrahedral sites How do we arrange 3 Si and 1 Al ion over 4 sites with the same energy 1 w i 4 3m 3200 Combining the B oltmann expression with the formula for calculating W we get 1 Skln kn X N J H Now we introduce a mathematical approximation called Stirling s approximation Note that the following equality holds In N lnNN 71N7 21 lnN lnN71lnNr 2ln1 N Zlny H Now if N is suf ciently large we can replace the 39on with an integral so we have N In N Iln ydy I Now we need to integmte by parts Iudvuvi vdu ulny dvdy y N and lan ylnyllNrJ dyNlnNrN1 1 For very large N We can neglect the 1 to get In Ni N lnN7 and noW the entropy becomes 3 7 kln M7iXNz 71 kNlnN 7N7 XN1nXN 7 XN 7 kNlnN 7N7 XN1nXN N 71 7 kN lnN7iX lnXN s kNlnN7 2X lnX 7 km 7 7 7mg X In X 7 If We take N nNU We get n is the number of distinguishable sites 7 7 41622 In X 7 This gives the ideal con gurational entropy 39 sites are energetically equivalent KAlSi3 08x1 ordered 7 KAlSi308xL disordered microcline lt7 sanidine 3mm 7 3mm 7nRZ X In X 7 74Ry41n 5an 47calKquot mol Measurement shows that the actual AS 35 cal K39l mol39l Which tells us that sanidine is not completely disordered Aluminum avoidance principle No tWo adjacent formula units have aluminum in adjacent sites Kerrick and Darken 1975 GCA 39 14311442 Using the avoidance principle the maximum con gumtion entropy should be 29 cal K39l molquot IDEAL IONIC MIXING IN OLIVINE For ideal mixing We have AHrmx 0 AVrmx 0 AS Scun guratmnal max Olivine solid solution XMgZ 104 1XFez sio4 lt7 Mg2X Fem 104 The olivine structure can be represented as M2 M1 T04 and 1 mole of olivine contains tWo moles of octahedral metal sites mix 7 AS Asmax kln W mzl m cov g W mmnl Wmmai wfurstentewfayalite 7 2X1VDJ717 W 7 21xN ZXIV 21xN Wm 2N Asm 7 Icme 7 km 2XN2 17 XN kh12NL1n2XNL1n217 XN Using Stirling s approximation We get ASW k2N1n2N 2N7 2XNln2XN 2XN 7 217 XNln2l7 XN 217 XN u AS 7 2kNln2N17X1n2XN x 17X1n217XN 17 X ASW 2RnZnN 17Xn 2nXnNX 717X1n2ln17XnN17X AS 2Rh12Xh12717XnZ 7XnX717Xn1X 1X 1 X NXl N1XlnN R7X1nX717Xn1X ACTIVITY OF FORSTERITE IN Gxdealxal n quotF116 quotFag ZRT quotFa lnXFa nFa1nXFa 6112ath 6mm mxx Aadmlmxx HF 6 ZRTI39HXF Aim 2RTXmXlt17gt01nlt17gt01 HZNRTIHXEG G tdtzalsol n X51 1 05 Bm recall that 2RTXlnX17X1n17X uFauaRTnaFa F0 mtdaalml n so an X an EMF quot Or alternatively M1 M2 Gxdealml n quotFa quotFacxd2a1al n an XMg 39XMg Ternary diagram showing compositions ofFe Ii oxides ACTIVITY OF Fe Ti OXIDES xFejoA 1XF97Ti04 lt gt Fe32xFe2 ZVXTiLXOA W 7 max 7 nal Mm Ascow g kl mmnl Total number of sites 3N Asz kln Wx W W mg w W W 304w g xN2xN V 217xN17xN W 7 3N FeO FaOt Fezoz 2xN27xN17 xN magneme hematite 5 6 ASW 7R27 x1n2 7x 17x1n17 x xFejO4 1xFeZTiOA lt gt Fe32xFe2 ZVXTi17XOA 2xh1x7217xh12 WagGmgRTlnx227x nmainnm gg WHOM quotm 2quotm 1 o Zileix RT quotmg 2quot V1quotTWTW quotm IHLW VIM um GM RT ln 4 q T an 1niyznmmzj a m mn quotmg rim ulv 4 Where nmag 2nmv number of Fe nu1v number of 24 an 7 3 Note the 4 is a normalization factor to insure that a 1V T1 2nmag 7 number ofFe for pure ulvospinel is unity PYRODGENES General structural formula for pyroxene M2 M1 T2 06 M2 Large cation octahedml site Takes Na Ca Mg M1 Small cation octahedml site Takes Fe Fe Mgzy Ti A13 T Tetrahedral site Takes Si and Al Commonly used end member pyroxene components diopside CaMgSizO hedenbergite CaFeSizO Ca tschermakite CaA1A1SiO jadeite NaAlSiZO and acmite NaFe3SiZO w ACTIVITIES OF PYRODGENES IN THE IDEAL IONIC MIXING MODEL M2 M1 T 7 M2 M1 139 adtXCa 39XMg39XSrZ athXCa 39an 39XS39 7 M2 M1 TZ MZ Ml TZ Jd XNa 39XAI 39st a XNa XE X3 am X g2 X Aff X XSTZ If perfectly ordered am 4 X2 X2quot X2 XS If perfectly disordered A REALLIFE PROBLEM Consider the following reaction relevant to a quartz ec ogite NaAlSiZO sioZ lt gt NaAlSiKOB We may wish to plot this reaction boundary in PT space Quartz is usually pure but jadeite will be a component of a pyroxene solid solution and albite will be a component of plagioclase solid solution We therefore cannot assume that their activities are unity we must calculate them For plagioclase we can assume ideal molecular mixing ie nan m Kalb For pyroxene we use an ideal ionic mixing model Suppose we are given the following analysis of the pyroxene determined by electron microprobe in terms of atoms per 6 oxygens S1 1788 Mn 0005 A1 0246 Mg 0710 T1 0088 Ca 0873 Fe 0240 Na 0050 We must determine which cations go into which sites This is done in a stepwise hio 1 First apportion all Si on to the T sites 2 Next ll the rest of the T sites with Al and put the rest ofthe A1 into M1 3 Put Ca and Na into M2 4 Put suf cient Mg into M2 to ll it up 5 PutTi Fe Mn and the remaining Mg into M1 This results in the following T 1788 Si 0212 A1 200 M1 0034 A1 0088 Ti 0240 Fe Mn 0005 0633 Mg 100 M2 0873 Ca 0050 Na 0077 Mg 100 ajdXAArIzZIXVm39XZ M1 M2 1 Z ajd0050 0034 1788 000152 2 1 1 1 L 23 onsider the simpli ed ypothetical cpx shown I I I I Molecular mixing assumes shortrange order and that molecules of diopside and jadeite actually exist in the solid solution can X f 07 Ionic mixng Assumes no shortrange order Considers the system to be a random mixture of Mg and Al on M1 and Na and Ca on M2 L13 Xg f X 0707 049 Unless there is evidence for shortrange order it is best to use mixing on sites ionic mode For example chargebalance considerations may force shortrange ordering This occurs in the plagioclase feldspars NaAl SiKOE CaAlZSiZOB 115 Xazb an Xan Thus we need to know the structure of minerals before we can apply activity models IONIC MIXING ACTIVITY MODEL R MICAS General structqu formula for micas AM1M2Z TA om vZ A alkali interlayer site takes Na K Rb Ca Ba M1 octahedml site takes Lit Felt Mg and vacancies 39 M2 octahedml site takes FezFe3 Cr Mg Al Tiquot Mn T tetmhedral site takes Si Al V volatile site takes OH39 F39 Cl39 0239 COMMONLY RECOGNIZED MICA END MEMBERS Dioctahedral Micas Muscovite K A12AISizOmOHZ Pamgonite Na AlZAlSi3OuOHz Margarite Ca AlzAleizOmOHz MgAl Celadonite K MgAlSiAOmOHZ FeAl Celadonite K F tAlSiAOmOHZ Fuchsite K CIZAISi3OmOHZ Ferrimuscovite K Fe3ZAlSi3Ol UOHZ Ferriceladonite K Fethe3Si40mOHZ 27 Trioctahedral Micas Phlogopite KMg3AlSi3OmOHZ Annite KFe3Alsi3OmOHZ all Fe Fluorophlogopite KMg3AlSi3OmFZ Oxyannite KFe Fe ZAlSiKOlZ Zinnwaldite KLiFeA1AlSi3OmOHZ Eastonite KMg2 A1AlZ SiZOmOHZ Wonesite Naphlogopite NaMnglAlZ SiZOmOHZ Lepidolite KZLiKAl4 Si7OZIOH3 Suppose microprobe analysis of a mica yielded the following Nan mKn 97Mg1 sFeu 8A11 sSiz BOIUOHI 7Fn 3 We need to rst assign the cations to various sites 1 All Si goes onto the T sites leaving 4 29 11 vacancies to be lled by Al 2 Put the remainder ofAl 04 onto the M4 site 3 Put all the Na and K onto the A site this just lls the A site with none le over 4 Put all OH andF onvolatile site 5 The Fe and Mg go onto the M1 and the 2 MZ sites The total cations on all M sites is 04 A1 165 Mg 08 Fe 285 This is 015 short suggesting vacancies oan n So now we have XKA 097NaA 003 X M 15A1MZ 04202 XAlT 114 0275 XS 2 4 0725 XOHV 172 085 and XFV 032 015 However we need to make some assumption about how Fe and Mg partition among M1 and M2 If we assume that Fe and Mg do not show an preference between M1 and M2 then we can write M1 M2 mm X72 X7 E 0485 XMg XMg XMg 165 we alsohave X ijg1Xm1 M2 M2 M1 7 And XFEXMEXA171 Solving these three equations results in XFEMl 0277 XFEW 0261 XMgW 0572 XMgM 0539 Now suppose We need to calculate the ideal activity of muscovite in this mica We Would Write Z W cX2X01 Xi Xi ngXgHY qmscm l for pure end member muscovite A normalization factor different from unity arises When you have mixing on the same site eg here AlSi mixing on the T site in the end member So to calculate the normalization factor assume We have pure muscovite T en amuscm 1 clt1gtlt1gtlt1gtzlt14gtlt34gt3lt1gtz 1 C27256 C 948 Now the activity of muscovite in the actual mica for Which We have an anal sis is amuscm 94809701502Z027507253085Z quotquot53 000418 To calculate the activity of annite component in the mica We Write d7 CX 4 X ZZ ltX ngXgHY mm To calculate the normalization factor again assume We have pure annite T en amm 1 clt1gtlt1gtlt1gtzlt14gtlt34gt3lt1z 1 C27256 C 948 For the activity of annitein the actual micaWe Write gamma 9480970277026 1Z027507253085Z ammm 00131 IONIC MIXING ACTIVITY MODEL FOR AMPHIBOLES Geneml structqu formula for hornblende A M4z M3 M2z NH2T14T24 022 V2 1 Al can only go on the T1 but not the T2 tetmhedral sites 2 M2 is the smallest octahedml site and is preferred by trivalent cations 3 M4 is the largest octahedml site and is preferred by alkali and alkaline earth ions 4 A contains Na K I COMMONLY RECOGNIZED AMPHIBOLE END MEMB ER S Tremolite I CazMgS SiEOZZOHZ Tschermakite I CaZAlZMg3AlZSi OZZXOHZ Gedme MgzltA12Mg3gtltAlzsi o gtltOHgtz Actinolite I Ca2FeSSiBOZZOHZ Cummingtonite I MgZMgSSiEOZZOHZ Edenite NaCaZMg5AlSi7OZZOHZ Pargasite NaCazltA1Mg0ltAlz Si gtOZZltOHgtz Glaucophane I NaZAlZMg3SiXOZZOHZ Richterite NaNaCaMgSSiEOZZOHZ IDEAL ACTIVITY OF PARGASITE We Write 0355 CXz laXW X392 39Xn f 39X fxz 39XL S X 2 X 1 X504 am assuming pure end member pargasite We Write argamph 1 311Z1z1212112Z12z1 1Z amph Cl64 aparg REVIEW OF IDEAL MIXJNG m m rum 44 mmwe m rum MW nh nxsl y Em ARTZX MX admms an MW mm Z in Tum TRULY NONVIDEAL SOLUTIONS AQMM m 39Gm m RTEXJM animalgm Ehxy my arm mmmmms m smmnmx mmdsam msmunwn mem hmArdAsm mmuwz Ehxy myadegmsfmm mmmmms m m zysm xwmhnnhgmmymm dysm xwm p 5 mm PmmArdAma uwuwzw mm WWWavg qu m Hm mma mm mm sum m m 54mm mi rahm quot nymtmxndsanum Wzmmmms m I Mpgumkv hsmhm u quot 3 3923 35 quot 733 1quot A m Hymn mung Tawnum yquot m w mama r rrzy zxrxgzmz a Graar2mnmx GIEE 5 FREE ENERGY OF A REAL SOLUTION w I mm G G im Hafiz 1Mquot 5 RELATIOSHIFEETWEENEXCESS PROPERTIES AND THE ACTIVITY COEFFICIENT This leads to the relationship C7 RT In 9 The excess partial molar entropy enthalpy and volume can now be written EX S 6G ARTlnxiRT 6T 1 6T P X RX 6 gm T aln referred to as the oneparameter Margu es H EX 7T M iRTZ equation This equation is symmetrical about the BT RX 6T PX 11 composition ie XA XB 05 This yields a 651m 61 y symmetrical solvus on a TX diagram VIEX RT7 The pammeter WG has units of energy it is TVX f n independent of X but dependent on T and P m MARGULES EQUATIONS SYMMETRICAL REGULAR SOLUTIONS The main equation for a binary solution is GEX WGXAXB WGXAUXA waxi xii This Margules equation gives the total excess free energy per mole as aparabolic function o concentration of the two components It is o en The Margules parameter WG can be thought of as the energy required to interchange a mole of A with 39 mixture without changing composition If WG gt 0 then molecules A and B prefer to be with molecules ofthe same type if WG lt 0 then they prefer to associate with each other Other excess functions vEX waAxB HEX WHXAXB SEX WSXAXB THE ACTIVITY COEFFICIENT IN TERMS OF WG We can write ILA 6 1 XAXdGdX4 But Grant Guiml GEX XAG2 XEG2RTXAln XAXEln XE WGXAXE Now inserting XB l XA and G HA we write uA ujRT1nXA WGX Now we know that for nonideal solutions u u RTlnXyx 5 RTIH74 n Combining the three previous relations we can write 51 RThm WGX 6g RThuB WGXi Note that these equations have the form of a cated virial equation They have the same parabolic form as the original expression GEX WGXAXB The above equation has avery simple convenient form but relatively few solid solutions can be expected to behave so simp y RELATIONSHIP OF WG TO THE HENRY S LAW CONSTANT Recall that Henry s Law states that f Km39X Ll NA 722 f 722 f X f eWGRT3 Km f 5WGRT AS YMIWETRIC SOLUTIONS Most real nonideal solutions will be assymetric One possible solution would be to extend the polynomial in a form such as RT lny aXZbX3cX4 This approach is called the RedlichKisfer expamion and Works quite Well for many nonideal solutions Another approach is the twoparameter Margules equation GEX X1Wch1Xz XZWGIXIXZ or GEX szXz WGI 39 2WczXzZ wcz 39 V Iv123 The latter has the form of a truncated virial equation once again 5 ACTIVITY COEFFICIENTS IN THE TWO PARAMETER MARGU39LES MODEL GT RT 1n n lt2Wm 7 WmgtX 2Wm 7 WGZgtX 5 RThwz 2W5 WGZW 2W52 7 W51X13 Note that when x 1 we get RTI i z W52 andthat whenx2 1 we get RTln 11 W51 MARGULES EQUATIONS FOR TERNARY AND HIGHER ORDER SYSTEMS For atemary symmetric solution We Write for the excess Gibbs free energy GEX WGIZXIXZ WGZKXZXK WGIKXIXK and for the activity coef cients RTlni t W512X22 Wszz W512 Watr azQXzXz RTlniz WGan Waquz W512 Wazr W513X1X3 RTlniz W513X12 Waquz Wm Wazr 512X1X2 57 For a quaternary symmetrical solution We Write GEX WGIZXIXZ WGIKXIXK WG14X1X4 WGZKZXK WGZ4XZX4 WG34X3X4 1 RTlnyl Waiin W513X32W514X42 W512 W513 T WGZQXZXK W512 Wm W524X2X4 WGIK W514 T GEAX3X4 For a ternary asymmetric system We have GEX WGZKXZZXK WGKZXKZXZ WcuX12X chxzzxt WGIZXIZXZ WGZIXZZX1 sx SOME PRACTICAL SIMPLIFICATIONS 0 It is found experimentally that in many minerals WG for Fe mixing is small 0 Thus when we have CaMgFe mixing we can make the following simplifications WFe Mg m 0 Wotang w WC are THE MOLECULAR PICTURE FOR A BINARY SYMMETRIC SOLUTION Assume that molecules are located on a lattice of Na Nb sites that have a coordination number Z When a and b are in sepamte phases We have lZZNa nearestneighbor interactions for a and lZZNb nearestneighbor interactions for b Random mixing yields a probability Xa that any site contains a molecule of a ande that any site contains a molecule ofb Let 23 23 and 23b re nt the energy of aa bb and ab interactions respectively an We cmmwm mZNoN zMM2N mum mz o in w mzunaz z a mw mmm wymmsamawm mm msysymummam m mmu 2m mg nmmsmW5 mmmmw 2m 1m mm I an um m m walk mm m 12 mmxmym mlmxmmnx mmmm mmmmmuwm m m mummy mwn cumudsammmmmxmmmdngmnmum Wmmzmmmgmaslwmmmmym mwmmhdnmymenmkmwmlm 77 Nm 39 n x a I iquot mm hum mi mm mm man mm mailmanswan xmm mak m I m m mm I smmm an quot 7 mumw rubsz ESTIMATING MARGULES PARAMETERS ERoM SYM METRIC SOLVI Con nwa PIN ofwmwsmanb ma c waning wmk sulvusz EEIEIK macaw W m mam stmz mm And mu39mvhlwu zmpzmm nd mumvhmd mmas rm i Because ms Ammmmman 1m m Mm mmth m4 mam Rom m W a gt rm W TuguWLzrdW ESTIMATING MARGULES PARAMETERS ERoM ASYM METRIC SOLVI mm K AMIgram mum m m WSW mi 1 1 I1 m I I1 RTlnX 39 0W WEIJXI 2er WEAK ma I1 I1 oRTlnX1 39 0W WEIJXI 2er Weaxe We noW have tWo equations relating WGl and WGZ RT mX1 2Wcz WclXz Z 2Wm WczXz 3 RT mxl ZWGZ WclXz Z 2Wm WGZXXZ 3 RT hixz ZWm WczX1 Z 2woz Wc1X1 3 RT 111X2 ZWm WczX1 Z 2wcz WGIXXFV These equations are not nearly as convenient to solve as in the symmetrical case