A bref vervew f the prncples f thermbarmetry Cn-y Lee (2009) Hmgeneus tem Fr a ne phase tem (hmgeneus tem) characterzed by a fxed cmpstn, the state varable knwn as Gbbs Free energy G s defned as fllws: G H S () where H s enthalpy, s temperature, and S s entrpy are all state varables. akng the ttal dfferental f eq. and substtutng n the fundamental equatn fr nternal energy ( du ds PdV ), ne can shw that dg Sd VdP (2) where V s vlume and P s pressure. hs means that G s a functn f the state varables and P, the state varables f mst nterest t a gelgst, hence G (, P) (3) G s essentally ne way t express energy, just lke nternal energy U, enthalpy H, and Helmhltz Free energy A are dfferent ways t express energy. Fr example, U s the energy n a tem capable f dng pressure-vlume wrk (PdV) r transferrng heat. H s the amunt f energy avalable fr transferrng heat at cnstant pressure. A s the amunt f pressure-vlume wrk avalable at a cnstant pressure. G s the amunt f energy asscated wth changng temperature and pressure, whch culd ptentally be used t drve ther frms f wrk. Except fr and P, all f the energy state varables (G, H, A, U) are extensve quanttes, that s, the amunt f G fr a tem depends n hw bg yur tem s. If, hwever, we allw ur tem t grw n sze r change cmpstn, then G wll change. In ther wrds, G s actually a functn f, P, and the number f n mles f varus cmpnents n the tem G (, P, n ) (4) Fr example, f we were nterested n the G f Mg 2 SO 4, yu wuld have t specfy, P and hw many mles f Mg 2 SO 4 yu have. We can take the ttal dfferental f Eq. 4 t gve G G G dg d dp dn P n P n n (5),,, P, n j n cmparsn t Eq. 2 shws that G G S V (6) P, n P, n As fr the thrd term n Eq. 5, we make the fllwng defntn G n (7), P, n j n a quantty, whch we call the chemcal ptental and Eq. 5 becmes dg Sd VdP dn (8) Eq. 8 shws that the change n G f a tem depends n the change n, P, and sze (n mass) f the tem. An mprtant qualty f G at any gven and P s that G (9) n
Eq. 9 can be dfferentated t yeld And substtutng Eq. 0 nt 8 yelds dg n d dn (0) Sd VdP n d 0 () hs equatn s knwn as the Gbbs-Duhem equatn and shws hw changes n, P and chemcal ptental are related. Fr a tem cmpsed f n cmpnents, there are n+2 ntensve varables and n+ varables can vary ndependently. Frm Eq. 9, we can als defne the partal mlar Gbbs Free energy G f a gven cmpnent G G (2) n Whch s smply equal t the chemcal ptental f the pure-endmember cmpnent. What des chemcal ptental mean? If yu lk at Eq. 7 carefully, yu wll see that the chemcal ptental s a measure f hw much Gbbs Free energy wuld change n yur tem (fr nw, ur tem s a sngle phase) f yu were t ncrease the mass f the tem. Fr example, f yu have mle f Mg 2 SO 4, the Gbbs free energy f yur tem wuld change by 2 f we added 2 mre mles f Mg 2 SO 4 t the tem. Nw, let s cnsder tw phases, A and B, each cntanng tw cmpnents, and 2. hs, fr example, culd be the exchange f Mg 2 S 2 O 6 () and Fe 2 S 2 O 6 (2) between clnpyrxene (A) and rthpyrxene (B), px cpx cpx 2S2O6 Fe2S2O6 Mg2S2O6 Mg Fe2S2O6 (3) Because we have tw phases n ur tem nw, the ttal change n Gbbs Free Energy f the tem s the sum f the Gbbs Free energy changes f the tw phases dg dg dg (4) And because A dn dn and A A A B B B dn dn dn dn dg 2 2 2 2 B A B dn2 dn2, Eq. 5 smplfes t (5) dg dn 2 2 dn2 (6) At equlbrum, G s mnmzed and therefre dg =0. In general, settng Eq. 6 t zer requres that 2 2 px (7) hs means that at equlbrum, the chemcal ptental f a cmpnent s equal n all phases! When the tem s nt at equlbrum, dg<0, that s the tem seeks t decrease G untl t reaches a mnmum n G. If, fr example,, then cmpnent wll tend t flw ut f phase A and nt phase B untl the chemcal ptentals f cmpnent n phases A and B are equal. Chemcal ptental s thus analgus t and P. If s the same n tw phases, there wll be n flw f heat. If ne phase has a hgher than anther adjacent phase, heat wll flw frm the htter phase t the cler phase untl s equal between the tw phases (equlbrum).
Smlarly, gases nt cmpartments held at dfferent pressure but allwed t flw freely between the tw cmpartments wll eventually equalze ther pressures. Chemcal ptental s thus lke a chemcal pressure that drves chemcal reactns. Hetergeneus tem We nw cnsder a hetergeneus tem, that s, a tem cmpsed f mre than ne type f phase. Fr example, assume we have a tem that can be descrbed by the fllwng phase cmpnents, A, B, C, and D. Let s assume that these cmpnents can be related t each ther stchmetrcally by the fllwng reactn (ths reactn s nt unque): aa bb cc dd (8) We can defne a prgress varable, whch s a measure f hw many mles cmpnent dnc dnd dna dnb d (9) c d a b Or mre generally dn d (9a) v Where v represent the stchmetrc ceffcents fr each cmpnent n the reactn, e.g., a, b, c, and d wth the prperty that t s pstve f the cmpnent s n the rght sde f the reactn and negatve f t s n the left sde. Pluggng Eq. 9a nt Eq. 8, yelds the general equatn f dg fr the entre tem n the frm f the prgress varable d G S d V dp d (20) d G Sd VdP d (2) Whch fr the example at hand dg Sd VdP cc dd a A bb d (22) At cnstant and P and assumng equlbrum (dg =0) 0 cc dd a A bb d (23) hs mples that at equlbrum 0 cc dd aa bb (24) Or mre generally, at equlbrum, 0 (25) whch means that fr any reactn, the sum f the chemcal ptentals f the reactants s equal t the chemcal ptentals f the prducts, hence there s n net chemcal pressure drvng the reactn frward r backward. Equlbrum cnstant In mpure tems, fr example, a phase cnsstng f a sld slutn f tw cmpnents - lke lvne beng a sld slutn f frsterte and fayalte endmember cmpnents, e.g., (Mg,Fe) 2 SO 4 ) - the chemcal ptental f ne cmpnent s decreased frm ts pure state by a factr related t the entrpy f mxng. he chemcal ptental n general s gven by
R ln a (26) Where the frst term represents the chemcal ptental f the pure endmember cmpnent and the secnd term accunts fr mpurtes n the frm f dlutn. he value a represents the actvty f a cmpnent n a gven phase. Fr an deal slutn, the actvty s smply equal t the mle fractn f that cmpnent n the phase, that s, a =X. Actvty s n sme sense a measure f the cncentratn f the cmpnent n a phase. In the case f a gas speces wthn a mxture f gases, actvty s equal t the partal pressure f the gas speces under deal cndtns. By deal, we mean the case n whch the mlecules f nterest (gas speces f mlecular cmpnents) d nt nteract wth each ther, e.g., n electrstatc nteractns and n lss f energy durng cllsns (any cllsns are perfectly elastc). In reallty, when ne mxes tw cmpnents tgether, there are electrstatc nteractns. w mmscble lquds wll repel each ther, drvng the effectve pressure r actvty f each lqud cmpnent dwn. In anther case, the tw cmpnents that mx actually attract each ther electrstatcally, ncreasng the effectve actvty. In these nn-deal scenars, a X, where under deal cndtns. At equlbrum, reactn between varus cmpnents requres that And upn re-arrangng, Where the equlbrum cnstant s gven as 0 R ln a (27) v v R a (28) v a ln K (29) and ( P, ) R ln K (30) Nte that the chemcal ptentals n the abve equatns refer t that f the pure endmembers (standard state) at the and P f nterest. Eq. 30 shws that the actvtes f varus cmpnents n the phases asscated wth a gven reactn must be related t each ther accrdng t the equlbrum cnstant. Because the left hand sde f the equatn depends n P and, K shuld depend n P and, that s, there s a unque K fr any gven P and. hs s the bass fr thermbarmetry. If we can measure K, then we can nfer P and. hermbarmetry In rder t apply Eq. 30 as a thermbarmeter, we need t calculate hw vares as a functn f and P. hs requres that we calculate the chemcal ptental f each cmpnent nvlved n the reactn at dfferent and P. d s, we calculate the Gbbs Free Energy f each cmpnent, startng frm SP cndtns, K and bar. Nte that chemcal ptental s smply the partal mlar free energy; everythng belw refers t the partal mlar free energy. he ndvdual Gbbs Free energes f the pure endmember cmpnents are dented by ther Gbbs Free energy f frmatn frm pure cmpunds. At SP cndtns, we have
G f, K,bar H f,k,bar S K, bar (3) calculate the varatn wth and P, we can ntegrate ver any P- path because G s a state varable and s hence path-ndependent. Hwever, sme ntegratn paths are easer than thers. We wll frst ntegrate up n temperature at cnstant pressure ( bar), and then we wll ntegrate up n P at the temperature f nterest. Integratng frst wth temperature yelds cp G f,,bar H f,,bar K bar cpd S, d (32) ntegrate up n pressure, we make use f the fllwng relatnshp. P G P G f,, P G f,,bar dp G f,,bar bar V bar dp (33) P Nte that V represents the mlar vlume at the temperature f nterest, whch means that n thery, we shuld have calculated V frm V K,bar. hs s dne by accuntng fr thermal expansn (nt shwn). We can cmbne the abve equatns t yeld: cp P G f,, P H f,,bar cpd S K,bar d V bar dp (34) where the heat capacty s gven by c d c p a b (35) 2 / 2 Fr a reactn, Eq. 34 can be mdfed accrdngly. cp P G,, P H,,bar,,, 0 K bar cpd S d V dp (35) bar Eq. 35 descrbes the P- curve fr a reactn nvlvng pure phases. If the phases are nt pure, then ne needs t accunt fr mdfcatns t the chemcal ptental and Eq. 35 becmes cp P G,, P H,,bar cpd S,K,bar d V, dp R ln K bar (36) Eq. 36 descrbes the P- curve f a reactn fr gven equlbrum cnstants K. Eq. 36 can be appled drectly t a barmeter by ntegratng upwards n and P frm K and bar. Hwever, n practce, a smplfed versn f Eq. 36 s used: G,, P H,, P S,, P ( P bar) V R ln K (37) Here, ne assumes that the enthalpy f reactn H,, P, entrpy f reactn S,, P, and mlar vlume change f reactn V are rughly cnstant ver the and P range f nterest. Nte that ths s f curse an apprxmatn and these values fr enthalpy, entrpy and mlar vlume are based n emprcal calbratns n the and P f nterest usng labratry experments. hs means that ths apprxmate equatn (37) shuld never be appled utsde f the -P range ver whch the equatn was expermentally calbrated.
Recgnzng the apprxmatns that went nt Eq. 37, we can re-arrange 37 t gve lnk as a functn f P and S H ( P bar) V ln K (38) R r expressed as a thermmeter ( P bar) V H (39) S R ln K r as barmeter S R ln K H P (40) V Applcatn f thermbarmeters and what makes a gd thermmeter r barmeter Applcatn f Eq. 38 as a thermbarmeter requres emprcal calbratn f the entrpy, enthalpy and vlume f reactn cnstants. hese cnstant can als be theretcally calculated by ntegratng up n and P frm the standard states f each phase partcpatng n the reactn. Once the calbratn cnstants are knwn, Eq. 37 s appled as fllws. One frst has t determne the actvtes f the cmpnents n each phase n rder t calculate K. hs s dne by measurng the cmpstn f each phase partcpatng n the reactn and then calculatng mle fractns f the cmpnents f nterest accrdngly. If the tem s deal, then the mle fractns are all that are needed. If the tem s nt deal, actvty ceffcents must be estmated. Unfrtunately, n many cases, the nn-deal terms are nt knwn very well. Fr these reasns, many emprcal thermbarmeters d nt treat nn-deal terms explctly, but rather add n extra cmpstnal terms n ther thermbarmeterc equatns (39, 40) n rder t apprxmate nndeal effects. Once K s knwn, the equatn can be slved t generate a unvarant lne n P- space. here are unfrtunately tw unknwns, P and. hs means that ne needs at least tw equatns as cnstrants, r essentally tw reactns. he best bet s t get tw reactns wth dfferent senstvtes t and P, that s, dfferent dp/d s that the tw equatns can be slved smultaneusly fr and P wth mnmal errr. he cmmn slutn essentally represents the ntersectn f the tw unvarant lnes n P- space. A thermbarmeter that s senstve t temperature requres that the term n Eq. 38 be large, and hence a large S,, P s preferred. A thermbarmeter that s mre senstve t pressure s ne that requres the P term t be large, hence a large V s preferred. he -P dependency f a unvarant lne descrbng a reactn between pure phases s gven by the Clapeyrn slpe dp S (4) d V If the phases are nt pure, there s stll a Clapeyrn slpe fr a gven equlbrum cnstant K.
dp d S R ln K V here are bascally tw types f reactns that are wrth examnng n detal. he frst s called an exchange reactn, wheren a certan catn s exchanged between tw dfferent phases. Fr example, we have the exchange f Fe and Mg between lvne and rthpyrxene: Fe2S2O6 ( px) Mg2SO4 ( l) Fe2SO4 ( l) Mg2S2O6 ( px) (43) he vlume change f ths reactn s lkely t be very small because n new phases are generated n ths reactn. Instead, the entrpy change f reactn s lkely t be mre mprtant. As a cnsequence, exchange reactns are typcally temperature senstve and pressurensenstve. Net transfer reactns are mre lkely t be pressure-dependent because the phases n ether sde f the reactn are nt the same. An example s as fllws: Mg2S2O6 ( px) Mg2SO4 ( l) SO2 ( melt) (44) Whch represents the reactn f SO 2 cmpnent n a melt wth the frsterte cmpnent n lvne t generate enstatte cmpnent n pyrxene. Because lvne and melt have very dfferent denstes (nverse f mlar vlume) than rthpyrxene, the vlume change f reactn s lkely t be nn-neglgble. Such a reactn wll thus have a pressure-senstvty. (42) Prblem Set. Yu are gven a sute f perdttes, ultramafc rcks cnsstng f garnet, clnpyrxene, rthpyrxene, and lvne as the dmnant phases. Each f these phases can be descrbed by a set f phase cmpnents. Olvne Mg 2 SO 4 Frsterte Fe 2 SO 4 Fayalte Ca 2 SO 4 Mntcellte Pyrxene CaMgS 2 O 6 Dpsde Fe 2 S 2 O 6 Ferrslte Ca 2 S 2 O 6 Wllastnte Mg 2 S 2 O 6 Enstatte CaSO 6 Ca-schermakte MgSO 6 Mg-schermakte Garnet Mg 3 2 S 3 O 2 Pyrpe Fe 3 2 S 3 O 2 mandne Ca 3 2 S 3 O 2 Grssular Wrte ut at least fve reactns that nvlve these dfferent cmpnents. We wll gve yu tw examples t start wth:
Fe2S2O6 ( px) Mg2SO4 ( l) Fe2SO4 ( l) Mg2S2O6 ( px) () CaSO6 ( px) Ca2S2O6 ( px) Ca32S3O2( gt) (2) Usng able 5 n Hlland and Pwell (998), calculate S,, P and V (at K, bar). Use these values t estmate the Clapeyrn slpe. Nte whch reactns are pressure-senstve and temperature-senstve. 2. hermmetry. Read Ells and Green 979. Answer the fllwng questns. a. What reactn s ths thermmeter based n? What mneral phases are requred fr ths thermmeter t wrk? b. Explan hw they gt the generalzed thermbarmeter equatn (Eq. 3) and the meanng f each term; what assumptns have gne nt makng ths equatn. c. Derve Eq. 4 d. Explan the meanng f Eq. 5 e. able 2. Shw that yu can reprduce ther values f K D frm ther clnpyrxene and garnet cmpstns. f. Shw that the frm f Eq. 9 s the same as Eq. 3 except fr the extra cmpstnal term t accunt fr Ca. Explan the mtvatn fr ths extra emprcal term why Ca? 3. Barmetry. Read Harley and Green (982). Answer the fllwng questns. a. What reactn s the barmeter based n? What mneral phases are requred fr ths barmeter t wrk? b. Eq 2 s the generalzed smple frm f a thermbarmeter equatn. Explan hw ths dffers (r nt) frm that used n Ells and Green 979. c. he authrs have started ff n a smple tem MAS (Mg--S). On page 698, px p M M M M they shw that aen amgs X ( X ) Mg. Explan hw ths was derved frm actvty slutn mdels. d. In mvng frm the MAS tem t mre realstc natural tems, whch have ther cmpnents, such as Ca, the authrs have been frced t deal wth nn-deal effects. Explan qualtatvely hw they accunted fr these nn-deal effects. What nn-deal effects were they mstly wrred abut? 4. Applcatn f thermbarmeters. Cntnents are underlan by thck thermal bundary layers, typcally thcker than what s seen beneath cean basns. hese thck thermal bundary layers have a number f gedynamc mplcatns fr heat flw thrugh the cntnents as well as the lng-term stablty f cntnents. hus, t s mprtant fr us t knw hw thck cntnents truly are. If we culd map ut the thermal state f the cntnents wth depth, we wuld gan valuable nsght nt the structure f cntnents. One way t d ths s t nvestgate mantle xenlths carred up as fragments n magmas
f deep rgn, such as kmberltes, the same magmas that brng up damnds. hese xenlths are thught t have been transprted t the surface very rapdly and are thught t have quenched, freezng n whatever gechemcal sgnatures they at the tme they were resdng n the mantle just prr t entranment n the hst kmberlte. he assumptn s that the dstrbutn f elements between mneral phases n these xenlths reflects equlbrum cndtns and that these mneral chemstres are frzen n at eruptn. Kpylva et al. (999) present electrn mcrprbe data n mnerals frm mantle xenlths frm the Canadan cratn. hese xenlths cntan garnet, clnpyrxene, rthpyrxene, and lvne. hey thus have the apprprate mneral assemblage fr estmatng P and f equlbratn usng the thermbarmeters just dscussed. d ths, yu wll have t set up a spreadsheet where yu cnvert majr element xdes n wt. % fr each mneral nt catn percents. Cnvert nt catns fr the apprprate number f xygens per frmula unt: Olvne = 4 xygens Pyrxene = 6 xygens Garnet = 2 xygens Fr a few samples n Kpylva et al. (999), calculate the actvtes f varus cmpnents n garnet and pyrxenes, whch are apprprate fr the Ells and Green (979) and Harley and Green (982) thermbarmeters. Ells and Green (979) gt-cpx Mg-Fe exchange thermmeter 304X ( K) gt Ca 3030 0.86P( kbar) ln K.9034 D And the Harley and Green (982) barmeter s P( kbar) ( R ln K 2.93) 5650 557( X Vr where s n Kelvn K M X ( X ( X M gt 3 Ca) ) M M V 83.3 78.98( X ( X )) (cal/kbar) M )( 2X M ) X px Fe 6300( X 5. hermbarmetry recap. Dscuss the results f yur calculatns n the cntext f the thermal state f the Earth r cntnental lthsphere. What assumptns have gne nt yur calculatns. gt Ca X gt Fe ( X gt Ca ) 2