Exploring Thermodynamics. Table of Contents.

Transcription

1 Exloing hemoynmis eoge K 4 ble o ontents. II. hemo-hemisty. Intoution... Stte Vibles, Stte Funtions, n Stn Sttes.... Stn mol enthly o omtion....4 ble I 4 4. Stn Enthly hnge o Retion emetue eenene o Enthly hnge o Retion...6 Exmle 7 6. Stn hnge in ibbs Fee Enegy o Retion Exmle. 7. emetue eenene o ibbs Enegy hnge o Retion Exmle 8. essue eenene o ibbs Enegy hemil Retions.4. hnge in ibbs Fee Enegy o the enel Retion.. 6. hemil Equilibium o Retion.7. K, K, n moe Ks 8 Exmle Review o ibbs Fee Enegy.. 4. hse Equilibium nsomtion emetue.. Exmle 5 5. Vition o hse Equilibium emetue with essue.5 Exmle 6.6 Exmle 7.7. K, 6/7/4

2 II. hemo-hemisty. Intoution. In hte I. hemo-hysis, the bsi onets o het, wok, n intenl enegy wee exline long with thei eltionshi to eh othe. si evesible oesses, lie to n iel gs wee lso exmine in some etil. he stte vible entoy ws eine, n the stte untions o Enthly, elmholtz enegy, n ibbs enegy wee eive. Sine vitully ll o the bove onets e unmili to most iniviuls, genel isussion with the intent o nsweing questions o mening n useulness ws ovie. In hte II. hemo-hemisty, we tun ou ttentions to some es o hemisty o whih the onets isusse in I. hemo-hysis beome useul. Questions suh s: Wht het is evolve in etion?, Will this etion oee sontneously?, s this system ehe hemil equilibium?, will be nswee. Most o the situtions enountee in hemisty involve etions ie out une onstnt essue. his is beuse the etion vessel is usully oen to the tmoshee. Une the onition o onstnt essue, the stte untion Enthly is most useul. Rell tht the hnge in Enthly, une onstnt essue onitions, my be inteete s het, S + V S U + ( V Q, sine une onstnt essue. hemists oten ee to the hnge in enthly o etion, s the het o the etion o this eson. he stte untion ibbs ee enegy is lso vey useul in hemisty. Rell this untion ws eine, in hte I. hemo-hysis, s: ( S U + ( V ( S S + n o onstnt temetue oess, eesents [negtive net wok o the oess. When system is not in equilibium, it hs vilble enegy (o wht is oten lle FREE enegy to o wok. In moving tow equilibium, the system eleses this enegy in the om o wok s it lowes its enegy stte. In hemil systems oen to the tmoshee, this is usully V wok. hus, o onstnt essue hemil etion, i <, the etion is sontneous; n i, the etion is in equilibium. et V. K, 6/7/4

3 . Stte vibles, Stte Funtions, in Stn Stte. o oee, bsis o inomtion tbultion is ote. With the unestning tht bsolute numeil vlues o stte untions e imossible to obtin, N, sine we e only onene with hnges in these untions, the iel o hemo- hemisty hs eie to ssign vlues to these stte untions in n eot to ete bsis o inomtion tbultion n oblem omuttion. symboli nottion hs lso been ote by the iel o themo-hemisty. zeo suesit is le on the themoynmi quntity to inite tht the essue is t stn stte, X O. b is le ove the quntity to inite tht it is e mole o mol in vlue. subsit is inlue to inite the Kelvin temetue t whih the themoynmi quntity is vli. Fo exmle, 98 ees to mol enthly t tmoshee, n 98 K. We thus eine the Stn Stte. he stn vlue o the envionmentl stte vibles, essue n temetue, e (by hoie: Stn essue is eine s tmoshee. Stn temetue is eine s 98.5 K (5. Entoy, lso stte vible, ws elie intoue s mesue o isoe. It is gue tht (bse on mixing, onigution, n veloity entoy minimum stte o isoe o mteil woul exist une the onitions: ue, well oee ystl, t K. hus, mteil une these onitions is ssigne n bsolute entoy vlue o ZERO. SK. Entoy vlues t othe temetues e omute om themoynmi t (see I. hemo-hysis, se 4. Fo exmle: S98 S S S 98K K q ev q 98K n S98 K SK (o tm onstnt essue K S98K 98K K n ev S whee is untion o temetue. Note tht i hse hnges ou ove the temetue nge, the hnge in entoy o those hse 98K. K, 6/7/4

4 hnges must lso be inlue ( Q nl hse hse hses hse hse n be oun in tbles o themoynmi t.. Vlues o o S 98 K Intenl enegy, Enthly, elmholtz enegy, n ibbs ee enegy o evey element in its most stble stte o ggegtion, t tmoshee n 98.5 K is ssigne the vlue o zeo. ue iel gs, ue liqui n ue soli, exists in stn stte t 98K, tmoshee. issolve o solute seies, inluing eletolytes, non-eletolytes, n iniviul ions, e in thei stn sttes i thei tivities (o eetive onenttions e unity in mol quntities, 98K, tmoshee. e vise, Enthly, Entoy, n ibbs ee enegy o etions t stn stte e bse on the esumtion tht the etion will go to % omletion. Sine ll hemil etions e evesible to some extent, % omletion is neve elize! Rel etions neve ou t stn stte, howeve it is the stn stte etion whih will ovie bsis o the themoynmis o the non-stn stte etion, isusse lte.. Stn mol enthly o omtion. With the stn stte eeene bsis, it is ossible to tbulte vlues o the enthly o omtion o omouns. onsie the omtion etion: ( g + O ( g O( l In inile, we eom this etion une stn stte onitions. he etion is eome in tmoshee onstnt essue vessel. We uthe extt (o het, the het o omtion, to mintin onstnt temetue within the vessel. Sine the stn enthly o the elements in thei most stble stte e ssigne the vlue o zeo, they nee not be uthe onsiee. + 98K O( l ( g O ( g O( l Kl/mole Stn enthly o omtion vlues e oten obtine om loimeti exeiment o t, lthough one ew key vlues e etemine, mny moe stn enthly o omtion vlues my be omute o othe omouns. he ollowing tble is tyil o the tyes o themo hemil t tbulte in vious soues suh s the nbook o hysis n hemisty ublishe by R.. K, 6/7/4 4

5 ble I. hemoynmi vlues o vious mteils. ++, -5 K, s l/mole K o 98 o 98 S o 98 kl/mole kl/mole l/molek yogen (g E E-7. Nitogen (g N E- -6.9E Oxygen (g O E- -.5E bon vo bon monoxie(g O 6.5.8E E Wte (g O E- 4.59E bon ioxie(g O 6.9.4E- -.45E Ethne (g 6..84E- -.97E Methne (g E E Ethene (g 4.9.8E E mmoni (g N E E i (g E E-7 l/molek 9K 7 K Liqui o 98 Kl/mole o 98 Kl/mole S o 98 l/molk Wte(l O.5 -.9E-.7E o t 5.5. l/gk L o 79.7 l/g t L v o59 l/g t 4. Stn Enthly hnge o Retion. In mno simil to ining the enthly o omtion, the hnge in enthly o etion n now be obtine ietly om the blne etion eqution, n the stn mol enthlies o omtion o the etnts n outs. he oeue is quite simle in inile, we obtin the enthly s o omtion o ll etnts n outs, n subtt the sum o the enthly s o the etnts om the sum o the enthly s o the outs; ll being obtine t stn stte. (outs (etnts onsie the etion: O (g + ½ O (g O (g It is esie to obtin the hnge in enthly o this etion une stn onitions. he oeue stts by quiing the stn mol enthly o omtion o ll omonents ( tm, 98K, in Kl/mole om ble I. he lultion is onlue by subtting the hnge in enthly o omtion o the etnts, om tht o the outs. Notie tht sine O is in its most stble stte t stn onitions, its enthly o omtion is zeo. { mole O (-94.5 Kl/mole} - (mole O (-6.4 Kl/mole +/ mole O ( Kl/mole} Kl 98. K, 6/7/4 5

6 onsie the etion: (ghite + O (g O(g + ½ O (g It is esie to obtin the hnge in enthly o this etion une stn onitions. ue to the stit ontol o the oxygen equie, this etion is iiult to eom. oweve, i we ell tht enthly is untion o stte, then the hnge in enthly must be ineenent o the th o the etion. hus, we evise n ltente set o etions whih n be ie out (o o whih the etion enthly s e known. Fo the etions shown below, +. his is ess s Lw. (s + O (g O (g + ½ O (g O (g (s + O (g O (g O (g O (g + ½ O (g Kl Kl (s + O (g O (g + ½ O (g Kl ess s lw my be use with ny Stte untion t stn stte. 5. emetue eenene o the hnge in Enthly o Retion. In genel, will hnge with temetue. o etemine etion s hnge in enthly t temetue othe thn 98K, we hve but to emembe tht is untion o stte. We exloit this t by inventing seon etion th onneting the initil n inl sttes o the etnts n outs though the stn temetue s seen in the shemti igm below. % etnts % outs emetue 98K o 98. K, 6/7/4 6

7 Fist, the temetue o the etnts e eue om to 98K. Seon, the etion is ie out t stn stte (98K. Finlly, we elevte the outs om 98K to. he sum o these enthlies is equivlent to the hnge in enthly o the etion t. his is Kihho s lw; lible to ny stte untion. 98 O n, e. + 98K + 98 n, o. I the seii hets,, e not onstnt ove the temetue nge, we hve little hoie but to eom the integl s. o o so equies the vilbility o s untion o. Mny soues o themo-hemil t ontin these untions. I the temetue nge is smll enough so tht we my onsie ll s onstnt, we my eue the bove eqution by swing the limits o integtion on the ist integl, n emoving the mount n seii het to give, ˆ 98K + ( 98 whee ˆ ( n i, o. ( n j,re. Exmle. One mole o methne is to be ete une stoihiometi onitions with i t 98 K, t onstnt essue o tmoshee, in n ibti n isobi ontine. etemine the enthly o ombustion, O ombustion 98K, n the inl temetue o the outs. We stt by witing the blne etion o methne n oxygen: 4 ( g + O ( g O ( g + O( l We next look u the stn mol enthly o omtion o the etnts n outs om the ble I. he stn stte hnge in enthly o the etion is: (outs (etnts 98 O 98 { mole(-94.5 kl/mole + mole (-68. kl/mole} - { mole(-7.89 kl/mole + mole ( kl/mole} -.8 kl/mole o methne his vlue is lso the stn mol enthly o ombustion t 98 K! 98. K, 6/7/4 7

8 o etemine the inl temetue, we will use Kihho s lw. Fo this exmle, we nee to ise the temetue o the outs to inl. his will lso inlue hse hnge o wte liqui to wte vo. Finlly, sine i is 77% nitogen( by weight, we must lso inlue the oe mount o nitogen n its hnge in temetue. We etemine the numbe o moles o N, whih omny the moles o O. i is mixtue o % oxygen n 77% nitogen by weight (te gses will not be onsiee. Seleting s bsis, gm o i,. gms O +.77 gms N gm i so in one gm o i, the moles o O n N e. g ( mole/ gms.79moles O.77 g ( mole/8 gms.75 moles N n the mole tio beomes, moles O (.75 moles N /.79 moles O 7.65 moles N O : 7.65 N he inl stes will onsume the enthly o ombustion to ise the outs, n the setto gs (N, to thei inl temetue. Notie tht we must inlue the enthly o voiztion o the liqui wte. Fo ese o lultion, we will omute this tnsition in two stges: 98K 7K, n 7K inl. Wte: 98K 7K, stge : hee e moles, o 6 gms o wte. Fom ble I,, liq l/g K, n L v 59 l/g K ( liq m 6 gms *l / gk * (7K 98 K 7l. 7Kl v m * Lv 6 gms *59l / gm 944l 9. 44Kl 98, liq 7, v Kl O : 98K 7K, stge : Fo the gs hse mteils, we will use uve it o sine thee is exete to be wie nge in temetue. Fom the bove tble, +*+ in l/mole K Sine Q o the onstnt essue oess, 98 7K n 7K 98 n 7 98 ( + + n + + 7K 98K. K, 6/7/4 8

9 K moles 6.9( l.76 Kl N : 98K 7K, stge : he oeue o O bove is ollowe K moles 968 l.968 Kl Stge, summy: ( ( ( ( (7 98.Kl +.76 Kl Kl 6.77 Kl o het must ente the etion outs to onvet the liqui wte to vo n ise thee temetue to 7 K. his het omes om the etion. ny itionl het (Stge will ise the temetue o the outs om 7K, to thei inl temetue. O + Stge + Stge ibti Kl Kl Kl Stge Stge Stge ll omonents e now in gs hse t 7K. We will use the integl om s ws use o the gses in stge to etemine the inl temetue. Note tht to o l/kl will be inlue to onvet om l, to Kl ( ( ( moles 7 l / Kl + mole ( 7 + ( 7 ( 7 l / Kl 7.65moles ( 7 + ( 7 ( 7 l / Kl 7 s n be seen, the bove eqution is ubi in. Ou solution will be by til n eo. he bove lultion ws entee in n Exel sesheet, n ws juste until the eqution equle zeo.. K, 6/7/4 9

10 inl 96 K eul obsevtion o the t in ble I shows tht the t is vli o the nge: K to 5K. he inl temetue, s omute, is beyon tht nge. We must theeoe onsie the inl temetue s only oximte lthough the metho is soun. n oximte solution oul lso hve been obtine by seleting eesenttive vlues o o the nge o temetues exete, n using Q n the thn the integl. inl s obtine is oten lle the ibti Flme emetue. his temetue is selom elize in tie; howeve it is use s n ue esign limit o mteil seletion when onstuting onstnt essue etion vessel. 6. Stn hnge In ibbs Fee Enegy o Retion. In mno simil to ining the enthly hnge o etion, the hnge in ibbs Fee Enegy o etion n be obtine ietly om the blne etion eqution, n the stn mol Fee Enegy o omtion o the etnts n outs. he oeue is quite simle in inile. We obtin the Fee Enegy s o omtion o ll etnts n outs, then subtt the sum o the Fee Enegy s o the etnts om the sum o the Fee Enegy s o the outs; ll being obtine t stn stte. (outs (etnts (Note tht othe methos to omute the hnge in ee enegy o etion exist n e isusse in the next setion. Exmle. etemine the hnge in ibbs Fee Enegy o the ombustion etion, 4 ( g + O ( g O ( g + O( l, une stn stte onitions. Will this etion oee sontneously s witten? Fom ble I, the omtion ibbs ee enegy vlues e ollete. Fo the outs: ( mole O gs * ( Kl/mole + ( moles O liq * (-56.7 Kl/mole -7.7 Kl Fo the etnts: ( mole 4 gs * (-.4 Kl/mole + ( mole O gs * (. Kl/mole -.4 Kl 98K [-7.7 [ Kl Sine the hnge in ibbs Fee Enegy is less thn zeo, this etion shoul be ble to oee sontneously une stn stte onitions.. K, 6/7/4

11 7. emetue eenene o ibbs Enegy hnge o Retion. In the onsietion o how the ibbs ee enegy o etion vies with temetue (essue emining onstnt, we ell tht the ibbs enegy n be exesse in tems o enthly n entoy s ollows: (S lying the bove to etion t stn stte onitions gives: ( 98 S 98 o etemine the hnge in ibbs ee enegy o etion t temetue othe thn 98, we ell tht both the enthly hnge n entoy hnge o the etion e untions o temetue. Ou metho is simly just these quntities o new temetue n utilize the bove eqution. he bsi inile o ou temetue justment lies in Kiho s Lw, lthough the etils een on whethe we n onsie the seii hets o the etnts n outs onstnt ove the temetue nge 98K, n whethe ny o the mteils unego hse tnsomtion ove tht temetue nge. Setion o this hte esibes the metho to just the entoy to new temetue, n setion 5 esibes the simil justment o the enthly. he justments e (ssuming no hse tnsomtions: n, 98 K ( n i, o ( n j,re o i the seii het vlues my be onsiee onstnt, ˆ 98 ( 98, whee ˆ ( n i, o. ( n j,re. n n S 98K 98 i, o 98 o, o onstnt seii het, ˆ S 98 ln 98 j,re he hnge in ibbs ee enegy o the etion t the new temetue, stn essue is then, ( + ( S + S 98 98K 98 98K. K, 6/7/4

12 Exmle. Fo the etion to oue mmoni gs om elementl yogen n Nitogen, etemine the hnge in ibbs ee enegy o the etion t both 98K n t 5K, tmoshee. ow oes the ibbs ee enegy hnge with temetue? Use seii het vlues t 4K n ssume the seii het my be onsiee onstnt ove the seiie temetue nge? We wite the blne etion: N ( g + ( g N ( n ollet the equie inomtion om ble I: g 98 Kl/mole 98 Kl/mole S 98 l/k/mole,4k l/k/mole N (g N (g (g Next, the etion ee enegy, enthly, n entoy hnges e omute t stn stte. ( mole N (-.4 Kl/mole- ( ½ mole N ( Kl/mole-( / mole ( kl/mole -.4 Kl 98 S 98 ( mole N (46. l/k/mole- ( ½ mole N (45.8 l/k/mole-( / mole (. l/k/mole -.69 l/k 98 ( mole N (-.976 Kl/mole- ( ½ mole N ( Kl/mole-( / mole ( kl/mole Kl We veiy the hnge in ibbs ee enegy o the etion using the eqution, S Kl 98K (-.69 Kl/K Kl! Sine the seii het vlues will be onsiee onstnt, we omute the hnge in het ity, n, o the etion: Ĉ ( mole N (9. l/k/mole- ( ½ mole N (7. l/k/mole-( / mole (6.96 l/k/mole -4.7 l/k Finlly, we ssemble the eqution o the hnge in ibbs ee enegy t the new temetue:. K, 6/7/4

13 ( + ( S + S 98 98K 98 98K n evlute it t 5K. ( + ˆ ( 98 S + ln K.4 ( ln Kl elow is lot o s untion o temetue o the etion: N ( g + ( g N ( g. / N (g + / (g N (g 4..., Kl emetue, K Notie tht 46K. is ositive o temetue vlues in exess o oximtely. K, 6/7/4

14 8. essue eenene o ibbs Enegy. he essue eenene o ibbs ee enegy o ue substne (mixtues e isusse in setion., une the onstint o onstnt temetue, my be obtine in the genel se om the below eqution, U + ( V ( S S + o whih. he esult, lible o onstnt temetue, is, V Futhe einement o the bove esult will usully equie knowlege o how mteils volume eens on essue. Soues o themo-hemil t oten tbulte untions o the isotheml omessibility o vious mteils. Fo hse tnsomtions om liqui to vo, the mol volume o the liqui my oten be onsiee s insigniint. Fo iel gses, the iel gs lw my be use s substitution o volume lthough one must onsie the ossibility tht the el gs my not behve s iel t highe essues n some oetion to will nee to be inoote. V Fo n iel gs, nr V n on substitution, nr nr ln he bove esult will be use in setion. to ollow. 9. hemil Retions. hemil etion is thought to stt, in mosoi sense, when the etnts e mixe. he tul etion howeve equies oess, uing whih oe bons between toms e boken, n o ome, o both. In miosoi sense, the ow te o the hemil etion eens on the numbe o ollisions between the etnts e unit o time, n the tion o those ollisions whih (ue to enegy, moleul oienttion, n numbe o othe tos e tully uitul in ouing out. One ny out hs been ome, new etion is stte; the evese etion. his evese etion lso hs te o etion whih is govene by extly the sme iniles isusse in the evious gh. t the onset o. K, 6/7/4 4

15 the etion, the te o the ow etion (ue to high onenttion is gete thn the te o the evese etion (low onenttion n etion oees in the ow ietion. s moe out is ome, n moe etnt is onsume, the ow etion te is eue n the evese etion te is inese. he etion ehes stte o hemil equilibium when both the ow n evese etion tes e ientil. It must be unestoo tht the ow n evese etions o not sto when equilibium is hieve, they ontinue to oee in both ietions t the sme te. he equilibium is thus evesible ynmi equilibium. ll hemil etions e evesible to some extent. When ll othe onitions e hel onstnt, the te o the etion shoul be exessible s some untion o onenttion; the big question being wht is this untion? o nswe this question, onsie etion o whih the etnts n outs e mixtues o eetly iel gses. y einition, the volume ontining these gses is vey lge when ome to the volume o the gs moleules themselves. hee e no intenl oes ting between these gs moleules. he gs moleules behve s tiny mehnil tiles, eh emove om the othe, only eete by ollisions with thei ontine, n n osionl ollision with nothe eting moleule. Une these onitions, lton s lw o til essues o mixtue o iel gses sttes tht eh gs omonent in mixtue o gses ts s i it is ineenent o the othe gses esent; eh gs omonent exeting til essue on the ontine. Fo the ni omonent i, i R. he onenttion o eh omonent is ietly V ootionl to the til essue o tht omonent. It is extly the eetly iel ntue o the gses tht uses lton s lw to be tue. Une iel onitions, the til essue (o tul onenttion o the se o ilute solution is ootionl to the etion te. With mixtue o el gses, intemoleul oes o exist. hese oes t to hnge the eetive numbe o tiles, n thus the eetive onenttions o the gses involve. his non-iel onition is uthe exebte when the size o the moleule is signiint when ome to the ontine size, when the gses e o high onenttion, n when the moleules e o high olity to list only ew situtions. With non-iel gses ( o solutions, the til essue ( o tul onenttion is oo mesue o the etion te. It is the eetive onenttion, o tivity whih is ootionl to the etion te. In evious setions, the themoynmis o the Stn Stte Retion wee esente. Rell tht the stn stte etion is esume to go to % omletion; to hve NO evese etion! Ws ou time wste on the. K, 6/7/4 5

16 stn stte etion? No. We hve but to just the stn stte themoynmis to the tul hemil tivities o the etnts n outs.. hnge in ibbs Fee Enegy o the enel Retion. We e now ey to exten the esults o setion 8 to the genel etion. lton s lw o til essues o mixtue o iel gses sttes tht eh gs omonent in mixtue o gses ts s i it is ineenent o the othe gses esent; eh gs exeting til essue on the ontine. he sum o these til essues equling the totl essue on the ontine. hus, o n i moles o the gs omonent i, we evisit the eivtion o setion 8, elling tht the temetue is hel onstnt. V, whee we substitute n i i nir tm i n R i V o the i th omonent, i i ni ( i i ni R ln tm n ( n R ln Next, onsie the ollowing etion o iel gses: + b + i i i I ie out une stn stte onitions, the hnge in ibbs ee enegy o the etion is: ( + ( ( b( I howeve, the etion is ie out une non-stn stte onitions, we ive t: + ( ( b( i ( When we subtt one eqution om the othe, n enge, we ive t: i i ( + ( ( b( n n ientiy eh o the ou tems on the ight s o the om, ni ( i i nir ln i, eviously oun. On substitution n engement, we ive t: R ln + R ln R ln br ln. K, 6/7/4 6

17 + R ln b whee the til essues must be in tmoshee units, n the exonents in moles. hus, we hve eive the hnge in ibbs Fee Enegy t non-stn stte o etion involving mixtue i iel gses.. hemil Equilibium o Retion. s isusse in the Intoution o this hte ( n in etil in hte I. hemo-hysis, setion 6., hemil system will be in equilibium when it hs no vilble enegy (o wht is oten lle FREE enegy to o wok. Une onstnt temetue n essue onitions, the ibbs Fee Enegy stte untion is inteete s the vilble enegy ( - Wok net n s suh, will eit hemil equilibium. When < o the etion, the etion n o net wok n shoul oee sontneously s witten. When, the etion n o no net wok n is theeoe in equilibium. hus, t equilibium, + R ln b R ln b he quntity ontine in the logithm will be nme the Equilibium onstnt n is ssigne the vible, b K. K R ln K e R he equilibium onstnt, K, is oten intoue to stuents by mens o kineti (etion te guments. ee, we hve esente uely themoynmi gument. oth e vli n instutive. Notie tht the vlue o the equilibium onstnt is untion o temetue, but is NO untion o onenttion sine (unlike is bse on the ibbs ee enegies o the outs n etnts in thei stn sttes. When etion is not t equilibium, it is ustomy to wite the ibbs ee enegy eltionshi o setion. s:. K, 6/7/4 7

18 + R ln Q whee Q, the etion quotient, hs the om o the equilibium onstnt K. Remembe howeve tht Q is bse on the tul onenttions o the outs n etnts; NO on thei stn sttes.. K, K, n moe Ks. In the e o hemisty, it woul seem tht the ouning thes h the seil le in thei hets o the lette K (s oes this utho. get mny onstnts titionlly e ssigne to the lette K with subsit tthe to sot out one om the next. In this setion, we look t K, K x, K, n K tivity n the eltionshi between these K s. Wht ll hve in ommon, is to exess the hemil tivity o the outs n etnts in some esonble n onvenient om. (he K vlues to ollow my lso be ele with Q i the etion is not in equilibium. til essue. s ws eviously shown, the til essues e useul when the etion involves ll gs omonents, K b, t lest om mthemtil oint o view. oweve, in til lultions involving gshse systems, it is oten moe onvenient to exess quntities o gses in units othe thn tmoshees o essue. Mole tion. he mole tion o omonent, is the tio o the numbe o moles o tht omonent, to the totl numbe o moles o mteil (both ni i out n etnt in the etion, x i. Fo n iel gs, this is ntotl totl lso equl to the tio o the til essue o omonent to the totl essue. Solving o the til essue, i xi, we ele eh til essue in K s ollows: b ( X ( X X X + b + b X X K b b ( X ( X X X K X, whee K x b. X X K x is oten esie to omute thn K n n lso be use o solutions whih behve s iel. onenttion. he onenttion o omonent, is the tio o the numbe o moles o tht omonent, to the totl numbe o lites o volume vilble to be ouie (both out n etnt in the etion. he sque bes, [, oun the omonent is shot hn o the onenttion o tht ni omonent. Fo n iel gs, i R V. We ele eh til essue in K s ollows:. K, 6/7/4 8

19 K b b R R R R ([ ([ ([ ([ b [ [ [ [ ( b R + K b R + (, whee K b [ [ [ [ K is most useul o solutions whih behve s iel. tivity. When hemil system evites om iel, the hemil tivity o eh omonent must be use. he hemil tivity o omonent, is ootionl to the omonent s onenttion. [ i i i tivity, whee the ootionlity onstnt is lle the tivity oeiient,. We my ele the onenttions in K s ollows: K b [ [ [ [ b b b b tivity K, whee K tivity b hemil tivities, n tivity oeiients e tble in vious hemil t hnbooks. Exmle 4. One gin onsie the etion: ( ( ( g N g g N + o whih we stt with ½ mole o N (g n / mole o (g ombine in tmoshee onstnt essue ontine hel t 98K. Let equl the tion o omletion o this etion. We seek to etemine how the vlue o the hnge in ibbs ee enegy o this etion eens on. We ssume the gses to behve iel. ( We stt by omuting the numbe o moles o eh omonent n the totl numbe o moles t ny stge in the etion. N (g: mole * N (g: ½ mole * (- (g: / mole * (- otl moles + ½*(- + /*(- (- moles. Next, the mole tions o eh omonent e omute. ( X N ( ( / X N ( (( / X. K, 6/7/4 9

20 Q is now omute using Q Q x (-/-/. Note tht o essue o tmoshee, Q Q x. ( Q.5.5 ( (.5.5 ( ( n inlly, using the etion beomes: + + R ln Q Kl.976 Kl (.986l molk 98 / l he bove untion is ghe using Exel to give:, the hnge in ibbs ee enegy o 98 K ( mole ln [ Q / N (g + / (g <---> N (g, kl , tion to omletion t 98K he bove gh inites tht when.97,. hus the etion is eite to go to 97% omletion. s tehnil note, the etion kinetis t 98 K, tm e vey vey slow. Fo this eson, the etion is not esible in til sense.. K, 6/7/4

21 he hnge in ibbs ee enegy o the etion t temetue o 7K ( tm is lso omute (justing ll tems o the new seii het vlues t this temetue, n ghe below s untion o etion omletion,. / N (g + / (g <---> N (g, kl , tion to omletion t 7K t this elevte temetue, when the omletion is oximtely 8%. Finlly, the hnge in ibbs ee enegy o the etion t temetue o 77K, tm, is omute (justing ll tems o the new seii het vlues t this temetue, n the new essue, n ghe below s untion o etion omletion,. / N (g + / (g <---> N (g, kl , tion to omletion t 77K, tm. K, 6/7/4

22 t 77K, tmoshees, t omletion o oximtely 5%. t the elevte temetue n essue, the kinetis (etion te e muh moe voble o the oution o mmoni. oweve, une bth oessing, the tion to omletion is vey low n woul be onsiee ineiient. I oue by ontinuous oessing howeve, the yiel eiieny is suiient n eonomil.. Review o ibbs Fee Enegy. elow, we hve omute the bsolute ibbs Fee Enegy,, o the etion: N ( g + ( g N ( g by numei integtion o the evious gh (77K, tm. he vlues o on the vetil xis e bse on the bity ssignment tht the ibbs Fee enegy o the ue etnts equl zeo. he esult o this numei integtion is the tm uve seen below. he sloe o this uve is 77K o the etion. Note tht t.5 (5% omletion, the sloe is zeo! his is the omletion tion t equilibium. It n be seen tht o vlues o <.5, the sloe o the uve is negtive, initing sontneous etion t these onenttions. bove 5%, the sloe is ositive initing tht the etion is sontneous in the oosite ietion (ight to let s witten. bsolute ibbs Enegy 8 6, kl , tion o omletion t 77K, tm. lso shown in the gh is line onneting the oint (, to the oint (,+8.94kl. Fo this etion, 77K kl, whih is the sloe o this line. onsieing only 77K whih is gete thn zeo, we my be temte to (wongly onlue tht the etion is not sontneous s witten t 77K.. K, 6/7/4

23 4. hse Equilibium nsomtion emetue. Suose we hve ue substne,, existing in two hses (hse n hse tht e in equilibium with eh othe, suh s soli n liqui o liqui n vo. hse hse Sine the two hses e in equilibium, the hnge in mol ee enegy must be zeo. his imlies tht t equilibium, the mol ee enegy o hse must be equl to the mol ee enegy o hse. Futhe, i some smll hnge in the mol ee enegy o hse wee to tke le, n ientil hnge in the mol ee enegy o hse must tke le to mintin the equilibium. hse hse hse hse We set out to etemine the temetue t whih n equilibium hse tnsomtion will ou, t stn essue o one tmoshee. Fom Setion 6. we ell (o mole o mteil, ( hse ( hse n t equilibium, the etion ee enegy must equl zeo. hus, ( hse ( hse We now emloy the esults o Setion 7. to eh o the omtion ee enegy s bove. ( ( + S + S hse ( + ( S + S hse (sine the mthemtil tetment o hse will be ientil to hse, we sve some se We now ele the temetue eenent vlues bove with the oite integls. + mole S + mole (hse (hse s n be seen, one we ele with its oite untion o temetue, n eom the equie integtions, solving the esult o temetue will be next to imossible. hus, we will emloy ieent (n inomtive tti; we will ele n eom the integtions o both hse n hse. hen we will lot the esulting o eh hse s untion o temetue. It is exete tht these two uves will inteset, n the intesetion temetue will be the equilibium hse tnsomtion temetue. Exmle 5. We selet the mteil O, n the tnsomtion om liqui to vo. he neessy t is oun in ble I. he gol is to etemine the equilibium liqui to vo hse tnsition temetue. he oh is outline bove.. K, 6/7/4

24 he esults o the substitution n integtion is, kl 98 + { ( 98 + ( 98 + ( 98 } l kl S 98 + ln + ( 98 + ( l he to o / is equie beuse,,, S om ble I. e in loies. he esult o the omuttion will be in Kl. We show the esults o ou omuttions in the gh below. Mol Fee Enegy o Fomtion o vo liqui emetue, K elow 7K, it is seen tht the Fee Enegy o the liqui is moe negtive thn tht o the vo. hus, liqui is the voe hse t these temetues. bove 7K, vo is the voe hse sine the ee enegy o tht hse is moe negtive. t 7K, both liqui n vo hses hve the sme ee enegy e mole, thus thei ieene is zeo; sine, he hse etion is in equilibium. lthough we hve suessully eite the equilibium temetue, one must emembe tht the outome is only s goo s the t use. Mesuement o s untion o temetue is teious exeise, n the subsequent uve it will ontin some eo; oten too muh. he iet mesuement o the tnsomtion temetue is esie n ute.. K, 6/7/4 4

25 5. Vition o hse Equilibium emetue with essue. Review the ist two ghs o Setion. We hve shown tht o the tnsomtion,, tht the mol ee enegy eltionshis t equilibium must be: hse hse hse hse hse hse Relling tht V S, we ele the ieentils in the bove s ollows: hse hse V S V S S V V ( ( S S V Fo n equilibium hse tnsomtion t onstnt essue, the hnge in hse entoy hnge is hses n on substitution gives, hse hse hse hse V his is the leyon eqution. One o its mny uses is to esibe how the equilibium hse tnsomtion temetue is eete by essue. n ltente om o the eqution is: V Exmle 6. ow oes the equilibium eezing temetue o ue wte hnge with essue? Ou etion is: O( sol O( liq he bove esult will be use to solve this oblem. Fist, the equie t is ollete (om the nbook o hemisty n hysis, R. ensity o ie t.97 g/ml ensity o wte t.9999 g/ml moleul weight o O 8 g/mole he mol volume o eh is now omute, ml 8g lite Fo Ie,.96 lite/mole.97g mole ml ml 8g lite Fo wte,.8 lite/mole.9999g mole ml. K, 6/7/4 5

26 he hnge in mol volume is, V V liq V soli.8 lite/mole -.96 lite/mole -.6 lite/mole Fom ble I., L 79.7 l/gm. hus, 79.7l 8g.85litetm sol liq 59. lite tm/mole g mole.986l V.6lite / mole Finlly, we omute 7.5K -.77 K/tm 59.litetm / mole We hve omute the hnge in the equilibium eezing temetue with essue o wte. Note tht this vlue is only ute in the viinity o 7K. s l n s l V will vy with temetue. Using the oint-sloe om o stight line bout the oint ( tm, 7.5K we ive t the eqution, (, tm ( 7.5, I one o the hses is n iel gs, we n exten the leyon eqution into moe onvenient lthough oximte om. onsie liqui to vo tnsomtion, V V V V. his oximtion is bse on the vo liqui vo t tht the volume o mole o gs is usully times lge thn tht o the liqui une the sme onitions. ssuming n iel behvio n mole, R V Vvo Vliqui Vvo On substitution into the leyon eqution, we hve V R R R Now, s inl oximtion, let be onsiee onstnt s well. We integte both sies. R R i he bove is the lusius-leyon eqution i i ln i We now set out to onstut hse equilibium igm using the bove esult, long with the esult o Exmle 6.. K, 6/7/4 6

27 Exmle 7. ow oes the equilibium voiztion temetue o ue wte vy s untion o essue? Ou etion is: O( liq O( v he lusius-leyon eqution will be use to solve this oblem. Fist, the equie t is ollete. Fo one mole o O t tm, 7K, om ble I., 59l 8g.85litetm l v 4.8 lite tmoshees/mole gm mole.986l We hoose the initil oint t 7.5K, tm. Ou eqution beomes, 4.8ltm / mole ln.85ltm / molek 7.5K tm ex K bove, we hve n oximte eltionshi, in the viinity o 7K, o the vition o boiling temetue with essue. elow is the equilibium hse igm o the O system. he uves e omute om the esults o Exmle 6 n Exmle 7, n e wn to sle..5 W te hse igm liq-v soli-liq Li, tm.5 Soli Liqui.5 Vo em K. K, 6/7/4 7

28 t the sle esente, it is imossible to esolve the soli-vo line, s well s the negtive sloe o the soli-liqui line. Most texts on the subjet esent sketh o this hse igm using highly istote (tisti sle to onvey these ies. bove is n exmle o n tisti O hse igm. Obseve tht both sles e istote. he istne om.6 to tmoshee es ientil to the istne om to 8 tm. Likewise, the istne om K (-7 to 7K ( is hl o the istne om 7K ( to 7K (. hse igms, in wht eve om, o onvey muh inomtion. bove, one n lely see tht s the essue is inese, the liqui egion is wiene o ; the eezing temetue eeses while the boiling temetue ineses. It shoul be unestoo tht the negtive sloe o the soli-liqui uve is highly unusul. Most mteils soli-liqui uve hve ositive sloe. ht is, most mteils elevte both the eezing n boiling temetues with inese essue. Wte is unusul in this eset.. K, 6/7/4 8