CHEMISTRY 383 LECTURE NOTES

Transcription

1 CHEMISRY 383 LECURE NOES Chate III R. H. Schwendeman Deatment f Chemisty Michigan State Univesity East Lansing, MI 4884 Cyight 996 by R. H. Schwendeman All ights eseved. N at f this text may be educed, sted in a etieval system, tansmitted, in any fm by any means, electnic, mechanical, htcying, ecding, thewise, withut i emissin f the auth.

2 III - III. hemdynamics: the secnd and thid laws A. Intductin; sntaneus cesses he fist law f themdynamics, the idea that the enegy f the univese emains cnstant, was and is acceted by almst eveyne. he ntin that enegy cannt be ceated destyed in any cess is cnsistent with exeience. he secnd law f themdynamics, which has t d with the edictin f the diectin f sntaneus cesses, is me subtle and was nt always acceted. A sntaneus cess is, as its name imlies, a cess that takes lace seemingly by itself. We ecgnize many such cesses. A cld bject bught int a wam m becmes wam. A unctued balln lses its ai. Zinc metal ded int HCl slutin in wate eacts t fm H, etc., etc. By cntast, many systems ae at equilibium, whee even if cmeting cesses ae ccuing cntinuusly, they ccu at ates that balance ne anthe (dynamic equilibium). F examle, afte the cld bject eaches m temeatue, thee is n futhe aaent change. If the balln is nt unctued and is tightly sealed, it etains its inflated shae. Afte all f the Zn is disslved, the cmbined ZnCl, HCl slutin emains unchanged. hus, we have tw kinds f cesses: sntaneus cesses and equilibium cesses (called nnsntaneus cesses by Atkins). Althugh equilibium nnsntaneus cesses ae aaently null cesses, they ae f inteest, because it is ssible t envisin an actual cess that is nly infinitesimally diffeent fm an equilibium cess: the s-called evesible cess that we discussed ealie. If the m with the bject in it is cled slightly, the bject will be cled als; if it is heated slightly, the bject will be heated. Again the name is at; a evesible cess is ne whse diectin can be evesed by means f a slight change in the system the suundings. Althugh it is imaily cncened with the diectin f sntaneus cesses, the fmulatin f the secnd law f themdynamics equies cnsideatin f evesible cesses. B. he diectin f sntaneus cesses What detemines the diectin f sntaneus cesses? We knw that it can nt be, as was nce thught, the diectin f enegy change. he Fist Law says that the enegy f the univese is cnstant; yet thee is cntinuus sntaneus change. We can cite simle examles that shw that this is nt just the esult f cmensating cesses in diffeent ats f the univese. Cnside the fllwing system in in which tw halves f a glass bulb imbedded in an insulated cntaine ae seaated by a valve. V L V R Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

3 III - 3 he system is eaed by filling ne half f the bulb with an ideal gas at sme essue while the valve between the tw halves is clsed. If the valve is then ened, the gas escaes fm the filled t the emty side. hee is n wk dne and n heat exhange with the suundings. heefe, U = 0. Yet a cess has ccued. Equilibium is ested when g (left) = g (ight). Nw, cnside a secnd system in which tw metal bas ae insulated fm each the and fm the utside wld. One metal ba is at a highe temeatue than the the. We emve the insulating themal baie between the tw bas. We knw that heat will flw fm ne ba t the the until the temeatues f the tw bas ae the same. N wk is dne and n heat is exchanged with the suundings; s U = 0. Yet a cess has ccued. Equilibium is ested when =. C. he enty and the Secnd Law f hemdynamics: statement hee was cnsideable discussin in the ealy histy f themdynamics abut the existence and natue f a quantity that detemines the diectin f sntaneus cesses. he Secnd Law f hemdynamics is the esult f that discussin. he essential featue f the Secnd Law is the intductin f a new state functin f a system, the enty. he enty f the univese is stulated t incease when any sntanus cess ccus. We use the lette S t designate the enty and then state that S > 0 f any sntaneus cess ds = 0 f any infinitesimal cess away fm equilibium. he S hee is the enty change f the univese. S = S system + S suundings > 0 f any sntaneus cess. Since a evesible cess is a sequence f infinitesimal stes away fm equilbium, S = 0 f a evesible cess. D. he enty and the Secnd Law: futhe cnsideatins he statement f the Secnd Law f hemdynamics has thee imlied ats: Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

4 III - 4. he existence f the enty hee exists a quantity called the enty that is a state functin f the system.. Calculatin f changes in the enty he enty change f a clsed system is defined such that ds = dqev S = S S = dqev In these equatins, dq ev is the heat inut t the system f an infinitesimal ste in a evesible ath at temeatue. 3. Statement f the law F any sntaneus cess in an islated system, S > 0. E. Altenate statements f the Secnd Law f hemdynamics Many distinguished scientists cntibuted t the undestanding f the Secnd Law and each f them had thei wn unique way f stating the law. hee f these (tw by Clausius) ae as fllws: R. Clausius Heat cannt f itself, withut the inteventin f any extenal agency, ass fm a clde t a htte bdy. W. hmsn It is imssible t cnstuct a machine functining in cycles which can cnvet heat cmletely int the equivalent amunt f wk withut ducing changes elsewhee. R. Clausius Die Enegie de Welt ist knstant; die Entie de Welt stebt einem maximum zu. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

5 III - 5 F. Calculatin f the enty f simle cesses calculate the enty f any cess, we make use f the integal ve the ati f the heat inut t the temeatue f a evesible cess. We can use this cedue f any cess, evesible nt, by making use f the fact that the enty is a state functin (we just stulated this, but it can be demnstated t be tue f many cesses). If the enty is a state functin, S is the same n matte hw we get fm ne state t anthe. heefe, if we need t calculate the the enty change f an ievesible (i.e., sntaneus) cess that takes a system fm state t state, we simly invent a cnvenient evesible cess that takes the system fm state t state and use the integal ule t calculate the enty change. his enty change will be the cect value f any cess, including the ne that we ae inteested in.. Isthemal exansin f an ideal gas calculate the enty change f the isthemal exansin f an ideal gas, we assume a evesible exansin. F any isthemal exansin f an ideal gas, we have aleady seen that U = q + w = 0. Als, V w dv nr dv V = = = nr ln V V V Because q = w f this cess, q = nr ln(v /V ). hen, since the cess is isthemal, V V S dq q = = = nr ln V V. Cnstant vlume heating F this cess, we imagine heating a system fm t while keeing the vlume cnstant. Again, we invent a evesible ath t take us fm t. Als, at cnstant vlume, dq = dq V = du = C V d. If we assume that C V is cnstant ve the temeatue ange f inteest, we find dq S = ds = = d S = CV = C V CVd ln Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

6 III Cnstant essue heating Cnstant essue heating fllws the same agument as cnstant vlume heating excet that at cnstant essue, dq = dq = dh = C d. he esult f cnstant essue heating is S d = C = C ln 4. Chemical eactin Since the enty is a state functin, a system cnsisting f a ue substance has a definite enty, just as it has a definite enegy and a definite enthaly. We fund that because f u inability t establish a cnsistent ze f enegy, it is nt geneally ssible t detemine unambiguusly the abslute enegy enthaly f a ue substance. We shall see shtly that this is nt the case f the enty. hee exists a well-defined ze f enty f a ue substance at any temeatue and thee exist well-defined ways t measue this quantity exeimentally. heefe, abslute enties can be and ae tabulated f many substances. A few f these ae tabulated in Atkins in able 3. and many me ae tabulated in Aendix A. hee als exist the extensive tabulatins f abslute enties. We ecall that f a eactin invlving ue substances, such as H (g) + O (g) = H O(l), the net esult is that tw mles f hydgen gas and ne mle f xygen gas disaea, while tw mles f liquid wate aea. If the eactin ccus at cnstant temeatue and cnstant essue unde cnditins such that the atial mla enties f all secies in the eactin vessel stay the same, the enty change f the cess is ( H O( )) ( O ( )) ( H ( )) S = S l S g S g If the eactants and ducts ae in thei standad states, the enties e mle ae standad enties e mle, which ae the quantities usually tabulated. If the substances ae nt in thei standad states, then in many cases the enties e mle can be calculated by ne f the equatins given f ats -3 f this sectin. Late, we will give additinal methds f calculatin f the enties e mle f use in this equatin. Revesible, cnstant temeatue, cnstant essue hase change A hase change f a substance can be eesented by a "chemical eactin", as fllws: H O(l) = H O(g) Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

7 III - 7 his cess ccus at cnstant temeatue, s S = q/. It als ccus at cnstant essue, s q = q = H. (In this aticula case, H is witten H va and is called the enthaly f vaizatin.) heefe, the enty f vaizatin is S va = H va Similaly, f the cess H O(s) = H O(l) S fus = H fus he cess f melting a slid is called fusin, and S fus and H fus ae the enty f fusin and the enthaly f fusin, esectively. hee is an inteesting and imtant emiical finding f the enty f vaizatin that is knwn as utn's ule. his ule states that the enty f vaizatin at the nmal biling int f all "nmal" liquids is abut 85 J K - ml -. A "nmal" liquid is ne that des nt have any unusual stuctual featues that affect the bnding in the liquid. F examle, S va = 85.9 J K - ml - at the nmal biling int (76. C) f cabn tetachlide, and cabn tetachlide wuld be called a nmal liquid. By cntast, S va = 09. J K - ml - at 00 C f wate and wate is nt a nmal liquid. he easn f the diffeence is that extensive hydgen bnding in liquid wate inceases the de in liquid wate cmaed t nmal liquids. We will see that enty is elated t disde, s it equies a geate than nmal enty change t cnvet liquid wate t gaseus wate. he utn ule has sme actical alicatins, because it allws estimatin f the enthaly f vaizatin f a nmal liquid at the biling int by simle measuement f the biling int. We will see late that this allws estimatin f the equilibium va essue f the liquid at temeatues away fm the biling int, which allws estimatin f the change in biling int with change in alied essue. G. Enty, enty change, and disde he tw examles given abve - gas exanding t fill bth halves f a cntaine and heat flwing fm ne side f a metal ba t anthe - aise the suggestin that the Secnd Law may have a statistical basis. F the examle in which ne half f a cntaine filled with gas is seaated by a valve fm the the half, which is emty, we cnfidently stated that the gas wuld fill the emty half as sn as the valve is ened. Why wee we s sue? If, f examle, thee wee nly fu mlecules n the filled side, afte ening the valve we wuld nt be suised t find all fu mlecules n the the side, thee mlecules n ne side and ne n the the, even thugh n the aveage, we wuld exect t find tw mlecules n each side.when thee ae millins f mlecules, hweve, we exect t find a nealy equal numbe n each side. his can be Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

8 III - 8 cnfimed by dinay statistics. It aeas that the easn f the vewhelming cetainty f the edictin f the Secnd Law in this case is a esult f the fact that macscic samles f substances cntain such a lage numbe f mlecules. he same cnclusin esults fm cnsideatin f the examle f the tw heated bas cming int themal cntact when the insulating themal baie is emved. We knw that heat flws fm the ht side t the cld side. his means that enegetic atms n ne side lse enegy while less enegetic atms n the the side gain enegy. But, with the andm distibutin f enegy amng the atms, it seems that it might be ssible f all f the enegetic atms t be n ne side the the, just by chance. he fact that this des nt ccu esults fm the vey lage numbe f atms in any macscic samle. It is ssible t shw that the ssibility f even a small deviatin fm the equilibium distibutin f enegy amng the atms is s small as t be cmletely negligible. he mathematical exessin f the statistical asect f the enty is that the enty is diectly tinal t the lgaithm f the numbe f micscic states cnsistent with a given themdynamic state. If me states ae made available, the system will send sme time in them. When the valve between the full and the emty halves f the gas cell is ened the mlecules that wee cnfined t the full half can nw ccuy any at f the whle cell. Because these exta states ae available, the mlecules ccuy them, and the enty inceases. H. he hid Law f hemdynamics A simle statement f the hid Law f hemdynamics is as fllws: he enty f a efect cystalline mateial is ze at = 0 (Atkins,. 9). A me ecise statement was given in a famus text by Lewis and Randall (G. N. Lewis and M. Randall, "hemdynamics and the Fee Enegy f Chemical Substances," st Ed.,. 448, McGaw-Hill, 93): If the enty f each element in sme cystalline state be taken as ze at the abslute ze f temeatue, evey substance has a finite sitive enty; but at the abslute ze f temeatue, the enty f a substance may becme ze and des s becme in the case f a efect cystalline substance. he hid Law was discveed in the ealy at f the 0th centuy when it became ssible t study the themal eties f eactins at vey lw temeatues. It was fund that f many eactins, S aaches ze as the temeatue ges t ze. F eactins with the same numbes f mles f eactants and ducts, this equies nly that all substances have the same value f the enty at = 0. Hweve, many eactins have diffeent numbes f ttal mles f eactants and ducts. F these eactins t have S = 0 at = 0, it is necessay f all f the eactants and ducts t have S = 0 at = 0. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

9 III - 9 If the abslute enty is ze at = 0 f a ue substance, it is ssible t detemine the enty at any > 0 by the fllwing equatin: S C d H C d H = n C d In this equatin, S is the abslute enty e mle f the substance in its standad state at temeatue ; C is the heat caacity e mle f the substance at cnstant essue; and H is the enthaly e mle f a hase change f the substance at temeatue. hus, measuement f heat caacities and hase-change enthalies allws exeimental deteminatin f abslute enties f ue substances at any temeatue. Many enties have been detemined this way. We will see shtly that abslute enties may als be deduced by measuement f equilibium cnstants f eactins at seveal temeatues. Finally, we nte that f a efect cystalline slid with all f its atms in thei lwest enegy state, thee is nly ne micscic state cnsistent with the given themdynamic state. heefe, if, as we stated abve, we assume the enty t be tinal t the lgaithm f the numbe f micscic states cnsistent with the themdynamic state, then S = klnω whee Ω is the numbe f micscic states cnsistent with the given themdynamic state. Fm this equatin, it is aaent that if Ω =, S = 0. I. Calculatin f S f a simle cess We cnside again the fllwing cess invlving tw metal bas in an insulated cntaine with a themal baie between the bas, as fllws: Befe the themal baie is emved, the tw bas ae at temeatues and with >. Afte the baie is emved, the tw bas cme t the same final temeatue f (heat flws fm the ight-hand, highe temeatue ba t the left-hand, lwe temeatue ba). hee is n lss f enegy t the suundings as heat, and n wk dne, s U = 0 f the whle system. he blem is t detemine the final temeatue f and t shw that S > 0. We assume that the tw halves f the ba have the same heat caacity C V and that the vlume change is negligible (w = 0). Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

10 III - 0 detemine the final temeatue f, we wite U = 0= q+ w = q = q+ q whee q and q ae the heat inut t the left and ight bas, esectively, as they cme t equilibium. Nw, and heefe, f ( ) q = C d C V V f f ( ) q = C d C V V f ( ) ( ) C + C = V f V f 0, afte slving f f, f = + Nw, t calculate S we devise evesible aths f each f the tw metal bas in which the lefthand ba slwly heats u fm t f and the ight-hand ba slwly cls fm t f. he enty change f each f these cesses may be cmuted and then the ttal enty change is the sum f the enty changes f the tw bas. We use the esult btained ealie f the enty change f cnstant vlume heating cling; S CV d C = = V ln f f S CV d C = = V ln f f Afte additin f the tw cntibutins t the ttal enty, shw that S > 0, we must shw that S = CVln f + CVln f = CVln f > whee f f = +. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

11 III - his can be dne as fllws: = = 4 f = + = 4 > 0 As a numeical examle, assume = 300 K, = 400 K, and C V = 0 J K -. hen, and f ( 0 J K ) = = 350K ( 350) ( )( ) S = ln = J K Althugh small, S is clealy sitive f this sntaneus cess. Nte that the units n the enty ae enegy temeatue - ; hee, J K -. I. Gibbs fee enegy. Intductin; elatin t the enty Accding t the Secnd Law, f any infinitesimal cess, ds univ = ds sys + ds su 0 ds univ > 0 ds univ = 0 f any sntaneus cess f any infinitesimal cess away fm equilibium Alicatin f this law equies infmatin abut the suundings as well as abut the system. It wuld be useful t have a ety f the system that edicts the diectin f a eactin withut exlicit cnsideatin f the suundings. his can be dne, vided that we caefully descibe the suundings in advance. J. W. Gibbs agued that if we make the simle assumtin that heat tansfe t the suundings is always evesible, then we can calculate the enty change f the suundings f evey such system. he esult wks ut in such a way that we can incate the enty change f the suundings int a new state functin f the system. We nw call this new state functin the Gibbs fee enegy. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

12 III - see hw this wks, we again assume a cylinde with a tight-fitting istn, but this time we enclse the cylinde in a heat bath f cnstant temeatue. We assume that the bath is s lage that any heat tansfeed t fm the bath is tansfeed evesibly, as fa as the bath is cncened, and that the amunt f heat tansfeed t fm the system is nt able t change the temeatue f the bath. Finally, we assume a cnstant essue cess. It shuld be aaent that this cess is vey simila t a cess caied ut in an en cntaine. he suunding atmshee is s lage that it has an enmus heat caacity. Cnsequently, heat tansfeed t fm the atmshee fm a system f easnable size is an infinitesimal heat tansfe, as fa as the entie atmshee is cncened. We assume a small amunt f heat tansfeed between system and suundings. hen, s that ds su dq = dq = dq sys su dq dq = su = = dh he last f these equalities esults fm the fact that the cess ccus at cnstant essue, and f a cnstant essue cess, the heat inut t the system is equal t the enthaly change f the system. heefe, dh hee is f the system. If we nw use the nmenclatue that S = S system, then accding t the Secnd Law. dh dssys + dssu = ds 0 Nw, we define the Gibbs fee enegy G, as fllws: G H S and ealize that since H,, and S ae all state functins, G is a state functin. see what this accmlishes, cnside a cnstant temeatue, cnstant essue cess. dg = dh d( S) = dh ds heefe, if as just shwn, ds dh 0, is equied by the Secnd Law f an allwed cess at cnstant temeatue and essue, then dh ds 0 Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

13 III - 3 dg 0 summaize: f a cnstant and P cess, dg < 0 dg = 0 dg > 0 f an infinitesimal sntanteus cess f any infinitesimal cess away fm equilibium f a fbidden cess In a simila fashin, it is ssible t define a Helmhltz fee enegy A, such that At cnstant and V, A U S da < 0 da = 0 da > 0 f an infinitesimal sntanteus cess f any infinitesimal cess away fm equilibium f a fbidden cess We emhasize again that G and A ae state functins. hey ae eties f the system that ae indeendent f its evius histy, including the methd f eaatin.. Alicatin f the Gibbs fee enegy t hysical cesses a. Intductin he state f a ne-cmnent, ne-hase system is defined by tw vaiables which we can take t be the temeatue and the essue. heefe, Accding t calculus, G = G(, P) whee G = G e mle dg G G = d + d and this exansin is unique. Nw, we efm sme algeba f an infinitesimal evesible cess at cnstant and. dq ds = dq = ds dw = dv Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

14 III - 4 du = dq + dw = ds dv dh = d( U + V ) = du + dv + Vd = ds dv + dv + Vd dh = ds + Vd dg = d( H S) = dh ds Sd = ds + Vd ds Sd heefe, dg = Vd Sd On a e mle basis Cmae Aaently, dg G = V dg = Vd Sd G G = d + and d G = S b. Vaiatin f the fee enegy with temeatue G H detemine hw G vaies with, we begin with S = fm which we btain G G = H his is the same as ( G / ) H = which is knwn as the Gibbs-Helmhltz equatin. It allws the fee enegy t be calculated at diffeent temeatues vided the enthaly e mle f the substance is knwn. c. Vaiatin f the fee enegy with the essue At cnstant temeatue, d = 0, s dg = Vd. heefe, Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

15 III - 5 G G G = dg = Vd We assumed G = G at = and G = G at =. If the substance is an ideal gas, then G G R G d R = = ln If the substance is a eal gas, it is necessay t mdify this equatin smewhat. his is dne by inventing a quantity called the fugacity f and by witing the equatin in tems f the fugacities at the tw essues, G G = Rln f f define the fugacity cmletely, it is cnventinal t assume that the gas becmes ideal as the essue is lweed t ze. It is then ssible t detemine the fugacity at any essue if the equatin f state f the gas is knwn. If the substance is a liquid a slid, it is almst incmessible. heefe, the mla vlume des nt change much with the essue and a gd aximatin is t assume that it des nt change at all with essue. ( ) G G = Vd = V d = V 3. Relatin between the Gibbs fee enegy and maximum nn-v wk he Gibbs fee enegy can be shwn t be the maximum nn-v wk that can be efmed by a evesible cess at cnstant and. his is an imtant esult, because it is ften the nn-v wk that is useful. he V wk has t be dne in a cnstant essue cess, whethe ne likes it nt. shw this ety f the Gibbs fee enegy, we begin by ecalling that at cnstant, d = 0, s fm G = H S, we find dg = dh ds Sd = dh ds Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

16 III - 6 Nw H = U +V, and at cnstant essue, d = 0, s dh = du + dv + Vd = du + dv But du = dq + dw (w is ttal wk) and dw = dw extdv (wk = nn-v wk + V wk) hus, dg = dq + dw extdv + dv ds F a evesible cess at cnstant and, dq = ds and, s dg = dw Since this hlds f evey ste in the evesible cess, G= w f a evesible cess at cnstant and. We will use this elatin late when we discuss electchemisty. 4. Alicatin f the Gibbs fee enegy t chemical eactins a. Reactants and ducts in thei standad states Cnside the fllwing eactin at sme fixed temeatue, H (g) + O (g) = H O(l) Accding t this eactin, when tw mles f hydgen gas eacts with ne mle f xygen gas, tw mles f liquid wate is duced,, in the wds, tw mles f hydgen gas and ne mle f xygen gas disaea while tw mles f liquid wate aeas. If all f the substances ae in thei standad states, the Gibbs fee enegy change f this cess is ext = ( H O ) ( O ) ( H ) G = G ( l ) G ( g ) G ( g ) In this equatin, f examle, G ( H O l ) ( ) is the standad fee enegy e mle f liquid wate at the temeatue f inteest. Me geneally, G = G ( ) G ( ) hat is, the standad Gibbs fee enegy change f a eactin (the standad eactin fee enegy) is the sum f the Gibbs fee enegies e mle f the ducts minus the Gibbs fee enegies e mle f the eactants. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

17 III - 7 Since G = H S = U+ V S, the deteminatin f the abslute value f a fee enegy e mle is lagued by the same blem f ze f enegy that caused tuble with the deteminatin f the enthaly e mle. We use the same tick that we used befe with the enthaly. We define the standad Gibbs fee enegy f fmatin f a substance t be the Gibbs fee enegy change f the fmatin f ne mle f the substance in its standad state fm its elements in thei standad efeence states. hen, because evey eactin may be viewed as a sequence f evesed fmatin eactins f the eactants fllwed by a sequence f fmatin eactins f the ducts, we can use a vaiatin f Hess's Law f Gibbs fee enegies t wite f G = G ( ) G ( ) hat is: he standad Gibbs fee enegy change f a eactin is the sum f the standad Gibbs fee enegies f fmatin f the ducts minus the sum f the standad Gibbs fee enegies f fmatin f the ducts. As might be exected fm the last equatin, thee exist tables f standad Gibbs fee enegies f fmatin f substances. In the Atkins text thee is a small table as able 3.3 and a lnge table in Aendix A. Anthe way t btain the standad Gibbs fee enegy change f a eactin fllws fm the definitin f the Gibbs fee enegy. Since G = H S, G = H S (cnstant ) at cnstant temeatue. We have aleady seen hw t cmute H and S fm tables f standad enthalies f fmatin and abslute standad enties. Afte cmutatin f these quantities, it is a simle matte t btain G. b. Gas-hase eactants and ducts at essues the than atmshee We have seen hw t calculate G, which is the Gibbs fee enegy change f a eactin in which all f the eactants and ducts ae in thei standad states. We als have inted ut that the sign f the Gibbs fee enegy change detemines whethe the eactin will ceed as witten. hus f a eactin vessel in which all f the eactants and ducts ae esent in thei standad states, we find that if G < 0, eactants eact t fm ducts if G = 0, the system is at equilibium if G > 0, ducts eact t fm eactants f Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

18 III - 8 Since it wuld be a ae situatin in which all f the eactants and ducts ae esent in thei standad states, we need a way t detemine the Gibbs fee enegy change f a eactin mixtue in which the vaius secies ae esent in abitay cncentatins. We will d this hee f a gaseus mixtue and then extend the cedue t slutins and mixtues afte we discuss hase equilibia. We have aleady shwn that f an ideal gas, G G = R ln Nw, we assume that state is the standad state in which G =, the standad Gibbs fee enegy e mle, and =, the essue f the gas in the standad state (usually ba). hen, we assume that state is sme abitay state f the gas (we d the subscit ), in which case, we can wite G G = Rln It is custmay t efe t the ati in the lgaithm as the activity f the samle in the eactin mixtue. We will use the lette a t eesent activity, s that G G G = Rln R ( a) a = ln, i.e., = If the gas is nt sufficiently ideal, it is necessay t use fugacities instead f essues in these exessins (nly ideal gases will be cnsideed in this cuse). If this elatin is alied t a chemical eactin, we btain whee and G = G +( ml) R ( lna) G= G( ) G( ) G = G ( ) G ( ) ( ) ln a = ln a ln a = ln Q Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

19 III - 9 hen, G = G +( ml) RlnQ he " ml" in this equatin esults fm the fact that when we calculate G G 0, the fee enegy e mle f each substance is multilied by the numbe f mles f the substance in the balanced chemical equatin. he Q in this equatin is called the eactin qutient, which is defined in tems f sums ve lgaithms f a, the activity f a duct secies, and lgaithms f a, the activity f a eactant secies. If we ecall that the sum f lgaithms is the lgaithm f the duct, we find that Q = a a (Nte: a = aa 3 a ) hat is, Q is the qutient f the mathematical duct f the activities f the chemical ducts divided by the mathematical duct f the activities f the eactants (nte the tw diffeent meanings f the wd "duct" in this statement). hese mathematical ducts must cntain the activities f all f the eactants and the activities f all f the chemical ducts. F examle, cnside the eactin, F this eactin, CO(g) + O (g) = CO (g) a Q = a CO( g) CO( g) ao ( g) he activities f CO (g) and CO(g) ae squaed in the equatin because they each ccu with a cefficient in the chemical equatin. With the exessin just given, we can calculate G f a eactin in a eactin mixtue in which the eactant and duct gases ae at abitay atial essues. hen, if G < 0, eactants eact t fm ducts if G = 0, the system is at equilibium if G > 0, ducts eact t fm eactants c. Equilibium cnstants If G = 0, the eactin mixtue is at equilibium. hen, if Q = Q eq = K is the equilibium value f the eactin cefficient, we find that G = ( ml) R ln K Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

20 III - 0 Hee, K is called the equilibium cnstant. In tems f the activities f the vaius secies, and K = his exessin f K lks just like the exessin f Q given abve. Bth equatins ae cect. In the equatin f Q, the a's ae the activities f the eactants and ducts unde a secific set f cnditins that may may nt be equilibium, wheeas in the exessin f K, the a's must be activities f eactants and ducts f a aticula equilibium cnditin in the eactin mixtue. a a We emhasize again that evey mle f eactant duct must be eesented by an activity in the equatin f K. F examle, if the ducts f the eactin include tw mles f CO (g), the numeat must include tw facts f the activity f CO (g); i.e., the activity f CO (g) squaed. It shuld be nted that the activity f a gas is a ati f atial essues ( fugacities) f the gas, and is theefe unitless. Since Q and K ae atis f activities, they ae als unitless. A emakable featue f the elatin between G and the equilibium cnstant is that the equilibium cnstant f a eactin in any abitay samle mixtue may be calculated by efeence nly t the Gibbs fee enegy change f the eactin with all f its ducts and eactants in thei standad states. heefe, it aeas that it is nly necessay t set u a table f atial mla standad Gibbs fee enegies f the vaius substances t be able t calculate the equilibium cnstant f a eactin. Such tables exist and ae extemely useful. summaize, we calculate G by ne f the equatins, f G = G ( ) G ( ) G = H S f and then use G = ( ml) R ln K t calculate the equilibium cnstant K. O, it is ssible t wk in the the diectin: measue the equilibium cnstant and use it t detemine G. his cess is descibed in me detail belw. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

21 III - J. Le Chatelie Pincile. Intductin By substitutin f G in the equatin f G, we find G Q R K R Q R ml = + = ln ln ln K With this equatin and the ules f the sign f G, we can geneate a table f the diectin f a eactin by cmaisn f Q with K, as fllws: If Q < K, G < 0, If Q = K, G = 0 If Q > K, G > 0 eactin ceeds t the ight eactin is at equilibium eactin ceeds t the left hese esults ae deduced by examinatin f the lgaithm in the exessin f G. If Q/K < (i.e., Q < K), then the lgaithm is negative and G is negative. If Q/K =, the lgaithm is ze and G is ze. Finally, if Q/K >, the lgaithm is sitive and G is sitive. At equilibium, Q/K =. By examinatin f the effect f a vaiety f changes in eactin cnditins n Q/K, it is ssible t deduce what is knwn as the Le Chatelie Pincile, which eads as fllws: Wheneve a change in eactin cnditins uts a stess n a eactin at equilibium, the cmsitin f the eactin mixtue shifts in such a way as t educe the effect f the stess. We will nw descibe a seies f "stesses" and shw that the esnse f the eactin mixtue is in accd with the Le Chatelie Pincile. Fist, we mentin a change in eactin cnditins that des nt duce a stess and theefe des nt fall within the sce f the Le Chatelie Pincile. he additin f a catalyst t a eactin mixtue changes the ates f the chemical eactins, but the ates f all f the eactins ae changed in such a way that the equilibium cmsitin is nt changed.. Effect f the additin f a eagent F a mixtue f ideal gases, the activities f the cmnents f a eactin ae the numeical values f the atial essues f the individual gases exessed in ba. hat is because we defined the activity t be a = in which is the essue in the standad state f the gas, which is ba. heefe, a = ba (f an ideal gas) Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

22 III - If is in ba, the units cancel, s a is the numeical value f the essue in ba. If we add sme eactant, the denminat f Q inceases, which deceases Q. his, in tun, deceases Q/K, s that it becmes less than, in which case the eactin ceeds t the ight t e-establish equilibium. If we add sme duct, the esult is the evese: the numeat f Q inceases, Q inceases, Q/K becmes geate than, and the eactin ceeds t the left. hus, the geneal ule is If sme eactant duct is added t a eactin mixtue that is at equilibium, the eactin ceeds in the diectin that will emve at f the added cmnent. As an examle, cnside the eactin N (g) + 3 H (g) = NH 3 (g) If sme NH 3 (g) is added t the eactin at equilibium, the eactin will ceed t the left in de t cnsume at f the added NH 3 (g). 3. Effect f incease in essue If we ecall that the atial essue f a substance in a gaseus mixtue is equal t the mle factin f the substance times the ttal essue, we can wite f the eactin at equilibium, K = x x Hee x is the mle factin f a duct mlecule, x is the mle factin f a eactant mlecule, is the ttal essue, and n is the numbe f mles f gaseus ducts minus the numbe f mles f gaseus eactants. If n is sitive and the essue is inceased, n inceases. In this case, the ati f mle factins must decease if K is t emain cnstant. his can nly haen if the mle factins in the numeat decease and/ the mle factins in the denminat incease. But, bth f these things haen if the eactin ceeds t the left. If n is sitive, thee ae me mles f gaseus ducts than mles f gaseus eactants in the equatin f the chemical eactin, s the shift is away fm the side with the geate numbe f mles f gaseus substances in the equatin f the chemical eactin. If n is negative, thee ae me mles f gaseus eactants than gaseus ducts. If the essue is inceased in this case, n deceases, s the ati f mle factins must incease. his haens if the eactin ceeds t the ight. he Le Chatelie Rule is still that the shift is away fm the side with the geate numbe f mles f gaseus substances in the equatin f the chemical eactin. heefe, in eithe case, n Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

23 III - 3 If the essue n a eactin mixtue at equilibium is inceased, the eactin ceeds in the diectin away fm the side f the chemical equatin that has the lagest numbe f gaseus substances. We nte that the effect f a change in the vlume f the eactin vessel at cnstant temeatue is t change the ttal essue f the eactin mixtue. At any given temeatue, a decease in vlume inceases the essue and vice-vesa, and the effect n the eactin mixtue can be edicted fm the change in essue. As an examle, we again cnside the eactin, N (g) + 3 H (g) = NH 3 (g), f which n = 3 =. If we incease the essue n this eactin, the eactin ceeds t the ight t educe the numbe f mles f gas in the cntaine. 4. Effect f a change in temeatue. see the effect f a change in temeatue n a eactin at equilibium equies an analysis that gets a little cmlicated. We begin with the Gibbs-Helmhltz equatin discussed ealie. he equatin that we used ealie shws hw the ati G/ vaies with temeatue. We aly it t G and t a eactin t btain ( G / ) H = Since is necessaily sitive, if H > 0, then G deceases as the temeatue inceases. By cntast, if H < 0, then G inceases as the temeatue inceases. Nw, s that G = ( ml) R lnk K = e G /( ml ) R If G inceases, K deceases, and the eactin ceeds t the left; if G deceases, K inceases, and the eactin ceeds t the ight. hus, we have the fllwing sequences f events, all based n the sign f H : Exthemic eactin H < 0 G K Reactin ges left H < 0 G K Reactin ges ight Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

24 III - 4 Endthemic eactin H > 0 G K Reactin ges ight H > 0 G K Reactin ges left Examinatin f these esults leads t the simle cnclusin that if the eactin gives ff heat as it ges t the ight (is exthemic), the eactin ges left if the temeatue is inceased (t absb heat). If the eactin absbs heat as it ges t the ight (is endthemic), the eactin ges ight if the temeatue is aised (t give ff heat). Again, we cnside the eactin, N (g) + 3 H (g) = NH 3 (g) f which H = 9. kj (twice the standad enthaly f fmatin f ammnia gas). he eactin is, theefe, exthemic. If the temeatue is inceased, the eactin ceeds t the left t absb sme f the heat added t incease the temeatue. K. emeatue deendence f K: deteminatin f H, G, and S he Gibbs-Helmhltz equatin was given ealie in the fm Nw, ( G / ) H = G = ( ml) RlnK Cmbinatin f these tw equatins gives the van't Hff equatin dln K d H H = dln K = ( ml) R ( ml) R d Integatin f the secnd f these tw equatins ve the temeatue ange between, whee K = K, and, whee K = K, leads t the fllwing vey imtant equatin that shws hw the equilibium cnstant at ne temeatue is elated t the equilibium cnstant at anthe temeatue: K ln K H = ( ml) R Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman

25 III - 5 btain this equatin, it is necessay t assume that the enthaly des nt change significantly between the tw temeatues (which is ften a vey gd aximatin). It aeas that measuement f the equilibium cnstant at tw temeatues makes it ssible t detemine H exeimentally. Measuement f the equilibium cnstant at eithe f the tw temeatues makes it ssible t detemine G exeimentally ( G = ( ml ) R lnk). Finally, if H and G ae knwn at the same temeatue, it is ssible t detemine S H G S =. heefe, fm tw measuements f an equilibium cnstant at diffeent temeatues, it is ssible t extact all f the themdynamic eties f the eactin. Actually, it is smewhat bette t make seveal measuements f the equilibium cnstant at seveal temeatues. hen, a lt f ln K against / shuld give a staight line whse sle is H /( ml ) R. Lectue Ntes f Chemisty 383, Chate III 996 R. H. Schwendeman