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["Glgy 329 Lab 3 "]
["Question 1"]
["grossular + quartz = anorthite + 2wollastonite"]
[]
["Question 2"]
["Given"]
[null,"anorthite","quartz","grossular","wollastonite"]
["Hf1,298 (J)",-4243040,-910700,-6656700,-1635220,null,"DH=",53920]
["S1,298(J/molK)",199.3,41.46,255.5,82.01,null,"DS=",66.36000000000007]
["a",516.83,44.603,1529.3,111.25,null,"Da=",-834.5729999999999]
["b",-0.092492,0.037754,-0.699,0.014373,null,"Db=",0.5974999999999999]
["c",4.1883000000000005e-05,0,0.00025299999999999997,0,null,"Dc=",-0.00021111699999999998]
["d",-4588.5,0,-18943,16.936,null,"Dd=",14388.372]
["e",-1408600,-1001800,7442600,-2777900,null,"De=",-13405200]
["V1,298",100.79,22.688,125.3,39.93,null,"DV=",32.66200000000002,3.2662,"J/bar"]
[]
[]
["2(a) at 1bar, 298K"]
["DG= DH(1,298) - T*DS(1,298) + (P-1)*DV(1,298)"]
["DG= ",34144.71999999998,"J"]
["Gros + Qtz stable"]
[]
["(b) "]
["DV =3.2662J/Bar"]
["Gross + Qtz Favoured by increasing P"]
[]
["c)"]
["DS= 66.36J/molK"]
["An + 2Wo favoured by increasing T"]
[]
["(d)"]
["slope=DS/DV",null,20.317188169738557]
[]
["e)at P=1bar, T=DH/DS"]
[null,812.5376732971661,"K"]
[]
["f) Plot (see figure)you can either use the slope determined in (d)"]
["and the temperature and pressure in (e) or solve the simplified"]
["DG=0=DH - T*DH + DVs(P-1) expression for various pressures."]
["Temp","P={[-DH+T*DS]/DV} -1"]
[800,-255.73026758924743]
[900,1775.988549384606]
[1000,3807.7073663584592]
[1100,5839.426183332313]
[1200,7871.145000306166]
[]
["Question 3"]
["Rearrange DG=0=DH(1,298) + integral of DCp*dT + DVs(P-1 -T[DS(1,298) + "]
["integral of DCp/T*dP] to solve for P"]
["DVs(P-1)=-DH(1,298) - integral of DCp*dT + T[DS(1,298) + integral of DCp/T*dT]"]
["             =-DG(1,T)"]
["Therefore…"]
["P=(-DG(1,T)/DVs)-1   "]
["where   -DG=-DH(1,298) - integral of DCp*dT + T*[ DS(1,298) + integral of DCp/T]"]
["Find DCp and DCp/T"]
["Integral of DCp*dT=Da*(T-298) + Db/2(T^2-298^2) + Dc/3(T^3-298^3) + 2*Dd*(T^0.5-298^0.5) -De*(T^-1-298^-1)"]
["Integral of DCp/T*dT= Da*LN(T/298) + Db*(T-298) + Dc/2*(T^2-298^2) -2Dd*(T^-0.5 -298^-0.5) - De/2*(T^-2-298^-2)"]
[null,"DCp",{"$nulls":6},"DCp/T"]
["Temp","Da term","Db term","Dc term","Dd term","De term","DCp",null,"Da term","Db term","Dc term","Dd term","De term","DCp/T","-DG(1,T)","Pressure(bars)"]
[800,-418954.14,164669.80500000002,-34168.81548565334,317157.4104059993,28226.97147651007,477.28844383589967,null,-824.1530985270067,299.945,-58.18424976000001,-649.5636860843807,65.00254756964776,2.1687902277261344,425.7437383450015,129.34833701089997]
[900,-502411.14,215457.30500000002,-49439.82881898667,366529.2209890356,30088.777032065624,46.7801379833727,null,-922.4512865947547,359.695,-76.12944976000001,-707.7489228555853,67.20051246162306,1.6626740392074026,7253.626497303296,2219.8151666472645]
[1000,-585868.14,272219.805,-68511.00215232001,413226.24047908845,31578.221476510065,-511.31814974163717,null,-1010.3820121473068,419.44500000000005,-96.18584976000001,-756.9719033655325,68.77270381964776,1.0763376385780106,14027.655788319651,4293.7938853467795]
[1100,-669325.14,334957.305,-91804.57548565335,457641.17066170555,32796.857840146426,-1328.0976640942608,null,-1089.9250289066022,479.19500000000005,-118.35344976000002,-799.3198776986128,69.93594762130066,0.3004514107100391,20734.59421587531,6347.231650197572]
[1200,-752782.14,403669.805,-119742.7888189867,500079.10176107596,33812.38814317673,-2588.4102010874703,null,-1162.5421138008376,538.945,-142.63224976,-836.257355545552,70.82068993075887,-0.7926979460444841,27349.172665834085,8372.391912875539]
[]
["Plot Temp and Pressure (see Figure)"]
[]
["Question 4","The curves are very similar, the most noticable difference is in the P, T differences for each curve."]
[null,"Less noticeable is the fact that the curve with defined heat capacity is very slightly curving. "]
[null,"Is it justified? What do you think? If we fix T, the Pressure difference between the two graphs is "]
[null,"approximately 0.5kbar which corresponds to approximately 2km depth. At 30km this is not necessarily a "]
[null,"large change (<10%) but close to the surface at say 4km depth this in a change of 50%! At a fixed "]
[null,"pressure, the temperature difference is approximately 40K. This may not look like much but if we are "]
[null,"looking at geothermometers, the object is to be as accurate as possible, so why not use the heat capacity "]
[null,"expression. Using the geothermal gradient found for this exercise a difference of 40K gives a change in depth of "]
[null,"approximately 800m. On a regional scale this may not seem significant but since thes rocks probably "]
[null,"formed in a contact aureole, a change in depth of 800m would be quite significant since contact aureoles "]
[null,"form in a relatively small space (on a regional scale). "]
[null,"Which method you use depends on what you want the data for."]
[]
["Question 5a"]
["Find the pressure that corresponds to 950K on the graph using the Cp defined curve."]
["T=950K","P=3200bar","P=3.1582*108Pa"]
["r=2600kg/m3"]
["P=rgh"]
["h=P/rg","h=12.4km"]
[]
["Question 5b"]
["Geothermal gradient"]
["from surface Temperature (298K) to 12.4km depth (950K)"]
["(950K-298K)/12.4km"]
[52.58064516129032,"K/km"]
["This is much higher than average geothermal gradient (25K/km), therefore the environment in which the "]
["assemblage formed must be different than simple burial metamorphism. The increased gradient could "]
["be due to proximity to an intrusion, resulting in a contact aureole and an increased geothermal gradient"]
["producing the high temperature assemblage of grossular, quartz, anorthite and wollastonite."]

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[]
["Glgy 329 - Lab 4 - Q2"]
[]
["Balanced reaction:      Ms + Qtz = Kfs + Sil + H2O"]
[]
["Set up equations:"]
[]
["For Q2b:"]
["deltaG = 0 = delta-H - T*delta-S + (P-1)delta-Vs + RTlnP",{"$nulls":4},"ideal gas, meaning f = P",null,"P + RT/dV lnP = -(dH-TdS)/dV + 1"]
[{"$nulls":5},"(fugacity coefficient = 1)"]
["For Q2c:"]
["deltaG = 0 = delta-H - T*delta-S + (P-1)delta-Vs + RTln #P",{"$nulls":4},"where # = fugacity coefficient",null,"P + RT/dV ln#P = -(dH-TdS)/dV + 1"]
[]
["So, the idea is to vary P such that the two sides of the above equations (in bold) equal each other"]
["This has to be done by trial and error because P is in two places in the above equation. "]
[]
["Data from Philpotts:"]
[]
[null,"H kJ/mol","S J/K-mol","V m3/mol * 103"]
["Ms",-5972.275,287.86,0.14071]
["Qtz",-910.648,41.34,0.022688]
["Kfs",-3971.403,213.93,0.10887]
["Sil",-2573.574,96.78,0.0499]
["H2O",-241.818,188.72,"    n/a because deal with fluid phases separately"]
[]
[null,"delta-H kJ","delta-S J/K","delta-Vs m3/mol * 103",null,"V in J/bar = V in cc/10"]
[null,96.1279999999997,170.23000000000002,-0.004628000000000021,null,"V in cc = V in m3 * 1000000"]
[{"$nulls":5},"Thus V in J/bar = V in m3 * 100000 / 10"]
["Normalize units",{"$nulls":4},"e.g. (-0.004628 /1000) * 1000000 / 10 =  -0.4628"]
[null,"delta-H  J","delta-S J/K","delta-Vs J/bar"]
[null,96127.99999999971,170.23000000000002,-0.4628000000000021]
[]
["Q2b: ideal gas"]
[null,null,"---RHS of above equations----",null,"   ---------------------------LHS of above equations------------------------",null,null,null,"Comparison of LHS and RHS"]
[null,null,null,"Col4",null,null,"#","Col8","Col4 - Col8 "]
["T °C","T (K)","dH - TS","-(dH-TdS)/dV +1","P bars","RT/dV","Fugacity coeff.","P + RT/dV ln#P","Difference should be close to 0"]
[300,573.15,-1439.3245000002935,-3109.0356525503175,1.4,-10297.25649308552,1,-3463.340923290792,354.3052707404745]
[350,623.15,-9950.824500000293,-21500.349394987574,6.8,-11195.560296024149,1,-21454.22268749638,-46.126707491192064]
[400,673.15,-18462.324500000293,-39891.663137424825,27.1,-12093.864098962778,1,-39877.01249499315,-14.650642431675806]
[450,723.15,-26973.824500000293,-58282.97687986208,89.4,-12992.16790190141,1,-58285.97830638146,3.001426519382221]
[500,773.15,-35485.32450000031,-76674.29062229936,254,-13890.47170484004,1,-76662.18495626215,-12.105666037212359]
[550,823.15,-43996.82450000031,-95065.60436473662,647,-14788.775507778671,1,-95071.07635797697,5.471993240353186]
[600,873.15,-52508.32450000031,-113456.91810717387,1525,-15687.079310717301,1,-113457.36469969968,0.4465925258118659]
[650,923.15,-61019.82450000031,-131848.23184961113,3501,-16585.38311365593,1,-131849.0595444586,0.8276948474813253]
[700,973.15,-69531.32450000031,-150239.54559204838,9054,-17483.68691659456,1,-150239.20585809313,-0.3397339552466292]
[]
["Q2c: real gas using fugacity coefficients"]
[null,null,"---RHS of above equations----",null,"   ---------------------------LHS of above equations------------------------",null,null,null,"Comparison of LHS and RHS"]
[null,null,null,"Col4",null,null,"#","Col8"]
[null,null,"dH - TS","-(dH-TdS)/dV +1","P bars","RT/dV","Fugacity coeff.","P + RT/dV ln#P","Dif (Col4 - Col8) should be close to 0"]
[300,573.15,-1439.3245000002935,-3109.0356525503175,1.4,-10297.25649308552,1,-3463.340923290792,354.3052707404745]
[350,623.15,-9950.824500000293,-21500.349394987574,7.2,-11195.560296024149,0.95,-21519.485985969084,19.13659098151038]
[400,673.15,-18462.324500000293,-39891.663137424825,28.6,-12093.864098962778,0.95,-39906.71098242301,15.047844998181972]
[450,723.15,-26973.824500000293,-58282.97687986208,99.2,-12992.16790190141,0.9,-58258.727440222676,-24.249439639403136]
[500,773.15,-35485.32450000031,-76674.29062229936,366,-13890.47170484004,0.7,-76669.97808505094,-4.312537248420995]
[550,823.15,-43996.82450000031,-95065.60436473662,1300,-14788.775507778671,0.52,-95066.51656694115,0.912202204533969]
[600,873.15,-52508.32450000031,-113456.91810717387,3280,-15687.079310717301,0.52,-113458.10253867411,1.1844315002381336]
[650,923.15,-61019.82450000031,-131848.23184961113,5367,-16585.38311365593,0.73,-131849.0761288167,0.8442792055720929]
[700,973.15,-69531.32450000031,-150239.54559204838,7806,-17483.68691659456,1.08,-150239.69340995394,0.14781790555571206]
[]
[]
["Summary of two curves"]
[null,"Ideal","Non-ideal"]
[300,1.4,1.4]
[350,6.8,7.2]
[400,27.1,28.6]
[450,89.4,99.2]
[500,254,366]
[550,647,1300]
[600,1525,3280]
[650,3501,5367]
[700,9054,7806]

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["Simple Phase Diagram"]
[]
["Species","H","S","V","V J/bar"]
["kyanite",-2581097,83.68,4.409e-05,4.409]
["andalusite",-2576783,92.88,5.153e-05,5.1530000000000005]
["sillimanite",-2573574,96.78,4.99e-05,4.99]
[]
[null,"dH","dS","dV","convert V to J/bar"]
["k-a reaction",4314,9.199999999999989,7.440000000000003e-06,0.7440000000000003,null,"P=-dH/dV+T*ds/dV+1"]
["k-s reaction",7523,13.099999999999994,5.81e-06,0.5810000000000001]
["a-s reaction",3209,3.9000000000000057,-1.630000000000003e-06,-0.1630000000000003]
[]
["T","P k-a","P k-s","Pa-s"]
[473.15,53.39247311827148,-2279.094664371778,8367.349693251503]
[573.15,1289.9516129032154,-24.361445783137242,5974.711656441688]
[673.15,2526.51075268816,2230.3717728055,3582.073619631872]
[773.15,3763.069892473104,4485.104991394141,1189.4355828220578]
[873.15,4999.629032258048,6739.838209982778,-1203.2024539877566]
[973.15,6236.188172042992,8994.571428571418,-3595.840490797571]
[1073.15,7472.747311827938,11249.30464716006,-5988.478527607389]
[]
[]
["Most stable mineral",{"$nulls":6},"dG=dH-TS+V(P-1)"]
["T","P1","P2","P3","P4","T2","P5","P6"]
[600,1,1000,3000,7000,800,1000,4500]
["dG kyanite",-2631305,-2626900.409,-2618082.409,-2600446.409,null,-2643636.409,-2628204.909]
["dG andalusite",-2632511,-2627363.153,-2617057.153,-2596445.153,null,-2645939.153,-2627903.653]
["dG sillimanite",-2631642,-2626656.99,-2616676.99,-2596716.99,null,-2646012.99,-2628547.99]
[]
[null,"A<S<K","A<K<S","K<A<S","K<S<A",null,"S<A<K","S<K<A"]
[]
["G-P Plot","G Kyanite","G Andalusite","G Sillimanite"]
["P"]
[0,-2643694.049,-2646262.393,-2645970.43]
[8000,-2608422.049,-2605038.393,-2606050.43]
[]
["G-T Plot"]
["T"]
[500,-2607509.909,-2605192.653,-2604503.99]
[1000,-2649349.909,-2651632.653,-2652893.99]