Chap 4 Fuametal Equatos of hermoyamcs he essetal thermoyamc propertes:. For solate system: (q0) Etropy as a goo crtero for spotaety a equlbrum. For cl

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1 Chap 4 Fuametal Equatos of hermoyamcs able of Cotets 4. Fuametal Equato for the Iteral Eergy for Ope ystems 4.2 Defto of toal hermoyamcs otetals sg Legere rasformatos Legere 4.3 Effect of emperature o the bbs Eergy 4.4 Effect of ressure o the bbs Eergy 4.5 Fugacty a ctvty 4.6 he gfcace of the Chemcal otetal 4.7 tvty of artal Molar ropertes wth pplcatos to Ieal ases 4.8 bbs-duhem Equato pecal opc: pplcatos of Maxwell Relatos Maxwell 2005//04

2 Chap 4 Fuametal Equatos of hermoyamcs he essetal thermoyamc propertes:. For solate system: (q0) Etropy as a goo crtero for spotaety a equlbrum. For close system (costat composto but ot costat q) : Not a goo crtero for costat ( ) or ( ). Nee two more thermoyamc propertes as crtero. sg Legere trasforms to geerate two ew fuctos. elmholtz free eergy () as crtero for costat a. bbs free eergy () as crtero for costat a. For ope system: Chemcal potetals for each speces of a system at equlbrum s the same. potaeous mxg of two partally mscble lqus at specfc a. 2005//04 2

3 Chap 4 Fuametal Equatos of hermoyamcs Combato of st a 2 law for close system st law: ƌq ƌw. 2 law: > ƌq rrev / a ƌq rev /. If oly -work volve ƌw - combe st law a 2 law for close system: - It s the fuametal equato a close system volvg oly work (a equato volves oly state fuctos ). It apples to both reversble a rreversble processes (state fuctos). Just lke - a are cougate varables for work ƌw - ow a are also cougate varables for heat: ƌq rev. ce ( ) s a exact fferetal: π π Τ : teral pressure 2005//04 3

4 2005//04 4 Dervatve of a extesve property wth respect to a extesve property gves a tesve property. ( ) or ( ) ca ot be use to calculate all the thermoyamc propertes of system oly ( ) wll. a are calle the atural varables of If ( ) > t costat (/ ) (/ ) are ot coveet forms. α C π α Chap 4 Fuametal Equatos of hermoyamcs x

5 2005//04 5 Chemcal otetal I 876 bbs trouce the cocept of the chemcal potetal to the fuametal equatos orer to scuss phase equlbrum a reacto equlbrum of of a speces. For : 2 2 If moles of speces are ae to system at costat a there s a chage teral eergy terms of. If a system cotas N s fferet speces s a fucto of ts atural varables a { }: he atural varables of are all extesve: N { } { } N

6 2005//04 6 Fuametal Equato: expresso show how the teral eergy ( ) chages wth atural varable of - where s a exact fferetal. everal mportat expressos ca be erve from ths relato clug the Maxwell relatos. he relatos: show how measurable propertes of a ope system ca be relate to thermoyamc quattes lke. Furthermore Chap 4 Fuametal Equatos of hermoyamcs N { } { } { } { }

7 2005//04 7 Chemcal otetal - ( ).. : { } { } { } { } { } { } N { } { } { }

8 Chemcal otetal If we substtute the seco law the form q/ a : q/ ƌw- ext N N ext he teral eergy remas costat f the ftesmal chage occurs at equlbrum uer costat etropy volume a { }. ( ) { } 0. he crtero for spotaeous chage the system volvg -work a specfe amouts of speces. must be approachg the mmum at costat a { }:. N he tegrate form as: : 2005//04 8

9 2005//04 9 For close system we erve the Maxwell relatos ( ) ( ) ( ) ( ); : Iteral eergy Ethalpy elmholtz free eergy bbs free eergy p p Chap 4 Fuametal Equatos of hermoyamcs

10 2005//04 0 o show how measurable propertes of a close system (costat composto) ca be relate to thermoyamc quattes lke : For ethalpy ( ) s ( ) ethalpy chages wth atural varable of Furthermore as exact fferetal the relatos: p p p p Chap 4 Fuametal Equatos of hermoyamcs

11 2005//04 For elmholtz free eergy - ( ) s free eergy ( ) chages wth atural varable of - Furthermore as exact fferetal we have Now the teral pressure : Chap 4 Fuametal Equatos of hermoyamcs π

12 2005//04 2 For bbs free eergy - ( ) s free eergy ( ) chages wth atural varable of - Furthermore as exact fferetal we have p p p Chap 4 Fuametal Equatos of hermoyamcs

13 Chap 4 Fuametal Equatos of hermoyamcs ( ) ( ) ( ) ( ) : //04 3

14 2005//04 4 Furthermore from ( ) ( ) ( ) ( ) ) we have the four relatos: p p Chap 4 Fuametal Equatos of hermoyamcs

15 2005//04 5 : { } { } { } { } Chap 4 Fuametal Equatos of hermoyamcs { } { } { } { } - -

16 Legere trasforms: lear chage of varables that volves subtractg the prouct of cougate varables from a extesve property of a system. For ay state fucto f (XY) sce f (XY) s exact fferetal; f (f /X) Y X (f /Y) X Y f m X Y wth m (f /X) Y a (f /Y) X ; [m /Y] X ( /X) Y Now we efe a ew state fucto g f -m Xf -(f /X) Y X g f - (mx) (m X Y ) - (m X X m) -X m Y g g (my) f (XY) - (f /X) Y X For the state fucto g (my): g (g /m) Y m (g /Y) m Y -X m Y wth -X (g /m) Y a (g /Y) m (f /Y) X lso g s exact fferetal > [(-X)/Y] m (/m) Y 2005//04 6

17 2005//04 7 : - { } { } { } { } Chap 4 Fuametal Equatos of hermoyamcs N { } { } - N { } { }

18 2005//04 8 : - - { } { } { } Chap 4 Fuametal Equatos of hermoyamcs N { } - N - { } { } { } { } -

19 Chap 4 Fuametal Equatos of hermoyamcs From ( ) ( ) ( ) ( ) we have N N N - N - N //04 9

20 2005//04 20 ( ) : { } { } N { } N { } N N Chap 4 Fuametal Equatos of hermoyamcs { } { } { } { }

21 2005//04 2 Example 4.2 Calculato of molar thermoyamc propertes for a eal gas. ce the molar bbs eergy of a eal gas s gve by Derve the correspog expressos for s: sg equato where Note that the teral eergy a ethalpy of a eal gas are epeet of pressure a volume. R l a. R R - -R l l R R l - -R a Chap 4 Fuametal Equatos of hermoyamcs

22 2005// 22 : { } N { } N { } N N ( ) { } 0 ( ) { } ) ( ext 0 ( ) { } 0 ( ) { } ) ( surr ext 0 Chap 4 Fuametal Equatos of hermoyamcs

23 Chap 4 Fuametal Equatos of hermoyamcs For Irreversble rocesses ( ) {} > 0 ( ) {} < 0 ( ) {} < 0 For Reversble rocesses ( ) {} 0 ( ) {} 0 ( ) {} 0 ( ) {} < 0 ( ) {} 0 ( ) {} < 0 ( ) 0 {} able 4. Crtera for Irreversblty a Reversblty for rocesses Ivolvg No Work or Oly ressure-olume Work. 2005// 23

24 Chap 4 Fuametal Equatos of hermoyamcs Fg 4. Whe a system uergoes spotaeous chage at costat a the bbs eergy ecreases utl equlbrum s reache. 2005// 24

25 Determe partal molar volume bbs. bbs. : exteso surface bbs : - f L γ f L. s. Legere rasform Maxwell : 2005// 25

26 2005// 26 Whe work other tha work occurs the system: { } L f N L γ f { }L γ { } L Chap 4 Fuametal Equatos of hermoyamcs f: the force of exteso. : the surface teso. : the chemcal potetal of speces.

27 Chap 4 Fuametal Equatos of hermoyamcs : ƌq/ surr ƌ q ƌ w ƌ w ƌ w (- ) at costat whch meas ƌ w ( ) where s the elmholtz eergy. he symbol actually comes from arbet the erma wor for work. hus a reversble process at costat temperature the work oe o the system s equal to the crease the elmholtz eergy. - (- ) - ƌ w : the ecrease s a upper bou o the total work oe by the system to the surrougs at costat temperature. 2005// 27

28 Chemcal otetal 参 : ƌ q/ surr or ƌ q ƌ w ƌ w o ext ƌ w o - ext - ƌ w o at costat a ext costat whch meas - ( -) - ƌ w o ( ) ƌ w o 2005// 28

29 Chemcal otetal For a reversble process at costat a the chage bbs eergy s equal to the o- work oe o the system by the surrougs. hus whe work s oe o the system the bbs eergy creases a whe the system oe work o the surrougs the bbs eergy ecreases. I geeral the ecrease s a upper bou o the o- work oe o the surrougs at costat temperature. Whe the system oes work o the surrougs the work oe s less tha the ecrease bbs eergy. 2005// 29

30 Chap 4.3 Effect of o the bbs Eergy he varato of the bbs eergy of a system wth (a) temperature at costat pressure a (b) pressure at costat temperature. he slope of the former s equal to the egatve of the etropy of the system a that of the latter s equal to the volume. 2005// 30

31 Chap 4 Fuametal Equatos of hermoyamcs he temperature varato of the bbs eergy s eterme by the etropy. Because the etropy s largest for the gaseous phase of a substace the bbs eergy chages most steeply the gas phase followe by the lqu phase a by the sol phase. 2005// 3

32 2005// 32 bbs-elmholtz equato { } { } ; - { } { } { } { } 2 { } ( ) ( ) { } ( ) ( ) ( ) { } 2 ( ) ( ) { } { } 2 he arato of the wth emperature { } - { } -

33 2005// 33 he arato of the wth emperature he bbs-elmholtz equato for a chage betwee state a state 2 the equato ca be wrtte as: { } { } p 2 p { } { } p 2 { } p 2 { } 2 { } 2 or

34 bbs-elmholtz Equato hs equato s very useful because:. If we ca eterme for a process or a reacto as a fucto of temperature the for the process or reacto ca be calculate wthout usg calormeter. 2. If a are kow at oe temperature the equato ca be tegrate to calculate at aother temperature assumg that s epeet of temperature. 3. plot of [ ( )/ ] versus (/ ) ca gve a lear le wth a slope of : p { } 2005// 34

35 2005// 35 : 2 2 p p p { } p p 2 p 2 p { } ; p ( ) ( ) R p R o p p o l l he arato of the wth ressure ( ) R o l m m R o l N

36 Chap 4 Fuametal Equatos of hermoyamcs Example 4.2 Calculato of chages thermoyamc propertes the reversble sothermal expaso of a eal gas. s: ce the teral eergy of a eal gas s ot affecte by a chage volume or - 0; q - w J mol ( ) R l J - qrev 5746 J mol K J K J mol K - mol - - mol J K - mol // 36

37 Chap 4 Fuametal Equatos of hermoyamcs Example 4.3 Calculato of chages thermoyamc propertes the rreversble sothermal expaso of a eal gas. s: eal gas expas sothermally at 27 to a evacuate vessel so that the pressure rops from 0 to bar; that s t expas from a vessel of L to a coectg vessel such that total volume s L. Calculate the chage thermoyamc quattes. hs process s sothermal but s ot reversble. w 0; 0; q - w J mol - ll state fuctos a are the same as Example 4.2 because the tal a fal states are the same 2005// 37

38 Chap 4 Fuametal Equatos of hermoyamcs he varato of the bbs eergy wth the pressure s eterme by the molar volume of the bbs eergy chages most steeply for the gas phase followe by the lqu phase a the the sol phase of the substace.. Because the volumes of the sol a lqu phase are smlar they vary by smlar amout as the pressure s chage. 2005// 38

39 Chap 4 Fuametal Equatos of hermoyamcs Example 4.4 Calculato of the bbs eergy of formato of gaseous a lqu methaol as a fucto of pressure. s: (a) he pressure effect o the bbs eergy of a gas s much larger tha the lqu ue to the molar volume fferece. he staar bbs eergy of formato for lqu methaol f º(C 3 O l) at K s kj mol - a that for gaseous C 3 O f º(C 3 O g) s kj mol -. he esty of lqu methaol at K s g cm -3. (a) Calculate f (C 3 O g) at 0 bar at K assumg methaol vapor s a eal gas. (b) Calculate f (C 3 O l) at 0 bar at K. (b) f ( ) o R l ( ) kj mol - ( kj K - mol - ) (298.5 K) l kj mol - f ( ) o ( o - ) f f (32.04 g mol - ) /(0.794 g cm -3 ) (0-2 m cm - ) x0-6 m 3 mol - f (C 3 O l) kj mol - (40.49x0-6 m 3 mol - )(9x0 5 a/ 0 3 J kj - ) kj mol // 39

40 he arato of the wth ressure he fferece bbs eergy for a eal gas at two pressures s equal to the area show below the eal-gas sotherm. 2005// 40

41 he arato of the wth ressure - 0 > for gases a logarthmc epeece of the molar bbs eergy o the pressure: ( ) o R l he evato from ealze behavor stll ca preserve the form of expresso by replacg the true pressure ( ) wth a effectve pressure (f ) calle the fugacty: f o m m R l tegrato betwee ay two pressures to obta: - R l // 4

42 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.2 Depeece of the bbs eergy of formato of a eal gas o the pressure of the gas relatve to the bbs eergy of the gas at the staar pressure of bar. 2005// 42

43 he arato of the wth ressure m he fferece bbs eergy for a sol or lqu at two pressures s equal to the rectagular area show. We have assume that the varato of volume wth pressure s eglgble. pf m p ( ) ( ) p ( ) f m m m 2005// 43

44 Fugacty Fugacty s a fucto of pressure a temperature. lso a characterstcs of a fugtve. he ame comes from the Lat for fleetess the sese of escapg teecy. Fugacty has the same meso as pressure. Defe as: ( ) ( ) o f ( ) m m R l f If we wrte the fugacty as f p s the mesoless fugacty coeffcet relate to the compresso factor Z of the gas betwee 0 a the actual pressure of terest by 2005// 44

45 Fugacty Coeffcet For ay gas at fferet pressures p p' p p' p p' m p m m p f m' R l f' m p m' R l p' ( ) f p p R l l m m f p' l x p f' R p f' ( ) m m p' p p' f m o m R l ' f ( ) l p l φ p' 0 p' m m p R 0 p f p 2005// 45

46 Fugacty Whe attractve forces are omat the molar bbs eergy a the chemcal potetal are lower tha those of a eal gas. t hgh pressure whe repulsos are omat the molar bbs eergy a the chemcal potetal are hgher tha those for the eal gas. 2005// 46

47 Chap 4 Fuametal Equatos of hermoyamcs 0 4 Ntroge gas at 300 K m/m m m f l φ l p R R p 0 RZ R p 0 ( ) m p p 0 m p Z p /bar lm 0 f p 2005// 47

48 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.4 lot of fugacty versus pressure for a real gas. he ashe le s for a eal gas. he staar state s the pure substace at a pressure of bar a hypothetcal state whch t exhbts eal gas behavor. 2005// 48

49 Chap 4 Fuametal Equatos of hermoyamcs rue for all gases For Ieal as: ake fferece: Rearrage: ( ) f mp R l f ( - ) m ( ) p R l f p R l -l f l γ ( ) m - p R 2005//5 49

50 he fugacty of Ntroge 2005//5 50

51 Chap 4 Fuametal Equatos of hermoyamcs Example 4.5 Expresso of the fugacty terms of vral coeffcets. ve the expresso for the compressblty factor Z as a power seres what s the expresso for the fugacty terms of the vral coeffcets? s: l l f f 0 0 Z- Z B' C' 2... C' 2 ( B' C'...) B' //5 5

52 Chap 4 Fuametal Equatos of hermoyamcs Example 4.6 he fugacty of a va er Waals gas. sg the expresso for the compressblty factor Z of a va er Waals gas gve equato what s the expresso for fugacty of a va er Waals gas? s: s a approxmato terms 2 a hgher the seres expaso are omtte. 2 a Z B' C'... b- R R f Z- a l b- 0 0 R R a b- R R a f exp b- R R 2005//5 52

53 Chap 4 Fuametal Equatos of hermoyamcs Example 4.7 Estmatg the fugacty of troge gas at 50 bar a 298 K. ve that the va er Waals costats of troge are a.408 L 2 bar mol -2 a b L mol - estmate the fugacty of troge gas at 50 bar a 298 K. s: f exp b - a R R ( 50 bar ) bar ( )( 298 ) ( )( 298 ) Φ f exp b- a R R //5 53

54 Chap 4 Fuametal Equatos of hermoyamcs he molar bbs eergy of a real gas coces wth the perfect gas value as 0 Whe the attractve force omat f<p a the molecules have a lower escapg teecy. 2005//5 54

55 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.6 Chemcal potetal of a eal gas as a fucto of pressure relatve to the chemcal potetal º of the gas at the staar pressure of bar. o R l p 2005//5 55

56 Chap 4 Fuametal Equatos of hermoyamcs he molar bbs eergy of a perfect gas s proportoal to l p a the staar state s reache at p o. Note that as p-> 0 the molar bbs eergy becomes egatvely fte. 2005//5 56

57 Chap 4 Fuametal Equatos of hermoyamcs Example 4.8 Calculatg the actvty of lqu water at 0 a 00 bar. What s the actvty of lqu water at 0 a 00 bar at 25 assumg that s costat? s: for lqu ( - ) ( ) ( ) R l( a) ( ) ( - ) a exp R t bar a t 0 bar exp.007 t 00 bar a.075 a exp R ( - ) ( kg mol )( 9 bar ) ( bar K mol )( 298 K) 2005//5 57

58 2005// gfcace of Chemcal otetal bbs (chemcal potetal). : (atural varables) { } : : { } { } { } { } { } { } N

59 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.5 wo phases ( ) at the same temperature a pressure. May speces may be preset but we wll focus o speces. ( ) - ( α ) ( β ) [ ( β )- ( α )] t equlbrum ( ) 0 or ( β ) ( α ) 2005//5 59

60 2005//5 60 Determe partal molar volume (partal molar bbs eerges J ) J ( J ) J bbs. J J p B B B B B B w -max o

61 2005//5 6 : { } { } { } { } N { } { } { } Chap 4 Fuametal Equatos of hermoyamcs : the partal molar etropy of speces. : the partal molar volume of speces. : the partal molar bbs eergy of speces.

62 Chap 4 Fuametal Equatos of hermoyamcs p he chemcal potetal of a substace s the slope of the total bbs eergy of a mxture wth respect to the amout of substace of terest. I geeral the chemcal potetal vares wth composto as show for the two values at a a b. I ths case both chemcal potetals are postve. 2005//5 62

63 2005//5 63 { } { } { } x { } N { } Chap 4 Fuametal Equatos of hermoyamcs { } R x R R

64 Chap 4 Fuametal Equatos of hermoyamcs R R l º bar0 5 a. : sce - { } o -R l wth - { } 2005//5 64

65 2005//5 65 ce N N { } N N - - ( ) { } N N - / 2 { } Η Η { } N N Chap 4 Fuametal Equatos of hermoyamcs he partal molar ethalpy of speces

66 2005//5 66 Determe partal molar volume Example 4.9 Dervatos of relatos betwee partal molar propertes ake the ervatves of a (/ ) {} wth respect to to obta the correspog equatos for the partal molar propertes. s: { } { } { } { } { } { } { } { } { } hs last equato s actually a Maxwell relato from. Note that whereas a for a system are always postve a may be egatve.

67 Determe partal molar volume (eal mxture).. y y y R l N R l y R l y R y l y R l tot tot 2005//8 67

68 Determe partal molar volume (eal mxture) { } R l y R l y R y l y R l tot : tot ( ) [ ] y tot tot where 2005//8 68

69 Determe partal molar volume (eal mxture) R { } R 2005//8 69

70 Determe partal molar volume Example 4.0 Calculato of chages thermoyamc propertes o mxg eal gases. If coser the mxg of two eal gases that are tally at the same temperature a pressure but are separate from each other by a partto. Calculate the chages bbs eergy ethalpy etropy a volume whe the partto s wthraw. Note that fal pressure s the same as the tal pressure of each gas. s: he tal values of these quattes are: ( ) t R l 2 2 R l R l ( ) t R l 2 2 R l R l 2 2 t R R ( ) t 2 2 R 2005//8 70

71 Determe partal molar volume Example 4.0 Calculato of chages thermoyamc propertes o mxg eal gases. (cotue) s: he chages upo pullg out the partto are the fferece betwee tal a fal values after mxg. he values of these quattes are: mx t R ( y ly ) R ( y ly y ly ) 0 l 2 2 tot 2 2 < mx t R ( y ly ) R ( y ly y ly ) 0 l tot > mx t 0 mx t //8 7

72 Chap 4 Fuametal Equatos of hermoyamcs he bbs eergy of mxg Let two eal gases two cotaers be a B ; both at the same a p. R l where s the staar chemcal potetal for the pure gas at bar. he tal bbs eergy of the bary system s t B B t o o R l B R l B 2005//8 72

73 Chap 4 Fuametal Equatos of hermoyamcs fter mxg the partal pressure of the gases are p a p B wth p p 2 p. he total bbs eergy chage to o o R l 2 R l 2 fal 2 he fferece of the bbs eergy fal - t s the bbs eergy of mxg mx mx 2 R l 2R l tot R ( y l y y 2 l y 2 ) < //8 73

74 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.7 he bbs free eergy of mxg mx of two eal gases to form a eal mxture. he total amout of gas s represete by t. mx R( l y 2 l y 2 ) he bbs eergy of mxg s egatve for all compostos a temperatures so perfect gases mx spotaeously all proportos. 2005//8 74

75 Chap 4 Fuametal Equatos of hermoyamcs Other thermoyamc mxg fuctos Because (/) p - for a mxture of eal gases the etropy of mxg mx s: mx -( mx /) p B -R (y l y y B l y B ) > 0 for a sothermal sobarc mxg of eal gases sce o teractos betwee the molecules the ethalpy of mxg s zero mx mx mx 0 o etropy of the surroug bee create the rvg force for mxg comes from the crease etropy of the system. 2005//8 75

76 Chap 4 Fuametal Equatos of hermoyamcs Fg 4.7 he etropy of mxg mx for two eal gases to form a eal mxture. he total amout of gas s represete by t. mx -R( l y 2 l y 2 ) he etropy creases for all compostos a temperatures so eal gases mx spotaeously all proportos. 2005//8 76

77 Chap 4 Fuametal Equatos of hermoyamcs Example: Calculate bbs eergy of mxg cotaer s ve to two equal compartmets. Oe cotas 3.0 mol 2 at 25 ; the other cotas.0 mol N 2 at 25. Calculate the bbs eergy of mxg whe the partto s remove. ssume perfect behavor. Metho: We procee by calculatg the tal bbs eergy from the chemcal potetals. o o so we ee the pressure of each gas. Wrte the pressure of troge as p; the the pressure of hyroge as a multple of p ca be fou from the gas laws. Next calculate the bbs eergy for the system whe the partto s remove. he volume of each gas oubles so ts partal pressure falls by a factor of //8 77

78 Chap 4 Fuametal Equatos of hermoyamcs he tal a fal states cosere the calculato of the bbs eergy of mxg of gases at fferet tal pressures. 2005//8 78

79 Chap 4 Fuametal Equatos of hermoyamcs swer: ve that the pressure of troge s p the pressure of hyroge s 3p therefore the tal bbs eergy s ( )( ) ( )( ) o R l 3p R lp 2 2 N2 N 2 o f fter mxg the pressure of troge s p/2 the pressure of hyroge s 3p/2 therefore the bbs eergy chages to N l ( ) ( ) o R o mol l.0 mol R 2005//8 79

80 Chap 4 Fuametal Equatos of hermoyamcs swer: bbs eergy of mxg s the fferece betwee f - mx 3p 2 ( 3.0 mol) R l (.0 mol) R l 3p p ( 3.0 mol) R l2 (.0 mol) R l2 ( 4.0 mol) R l2 6.9 kj p //8 80

81 Chap 4 Fuametal Equatos of hermoyamcs Commet: I ths example the value of mx s the sum of two cotrbutos: the mxg tself a the chages pressure of the two gases to ther fal pressure 2p. Whe 3.0 mol 2 mxes wth.0 mol N 2 at the same pressure the chage of bbs eergy s mx R ( ly ly ) ( ) 3 ( ) 3.0 mol R l.0 mol R l kj a s epeet of the tal commo pressure. B B //8 8

82 Determe partal molar volume bbs-duhem equato - : ; B B - - B B 0 B B - B B 0 B B 0 x x B B x (-x ) B 0 bbs-duhem equato : 0 B B ( ) ( ) //8 82

83 Chap 4 Fuametal Equatos of hermoyamcs bbs-duhem equato s B B J ; B B but we have B B 0 uer costat a p. I geeral bbs-duhem equato as: B 0 B 2005//8 83

84 Chap 4 Fuametal Equatos of hermoyamcs Example : sg bbs-duhem equato he expermetal partal molar volume of K 2 O 4(aq) at 298 K are gve by the expresso: K2O4 /(cm 3 mol - ) m ½ where m s the umercal value of the molalty of K 2 O 4. sg bbs-duhem equato to erve a expresso for the partal molar volume of water the soluto. he molar volume of pure water at 298 K s cm 3 mol //8 84

85 Chap 4 Fuametal Equatos of hermoyamcs Metho: Let eote K 2 O 4 a B eote 2 O the solvet. sg bbs- Duhem equato for partal molar volumes of & B. B B 0 uer costat a p. or B B herefore B ca be fou by tegrato: B B m 0 B where B* s the molar volume of pure B. he frst step s to chage the varable to the molalty m a the to tegrate the rght-ha se betwee m0 (pure B) a the molalty of terest. 2005//8 85

86 Chap 4 Fuametal Equatos of hermoyamcs swer: Wth eote K 2 O 4 a B eote 2 O /(cm 3 mol - ) m ½ (½) 8.26 m -½ m herefore B ca be fou by tegrato: B B B B m / m 0 B m however the molalty m s relate to the amouts of a B by the expresso: m /( B M B ) where M B s the molar mass of water ( kg/mol). herefore B 2 * 9.08 m B MB 3 m * /2 B M B m 0 B /(cm 3 mol - ) b 3/2 ( 9.08 m 3/2 ) 2005//8 86

87 Chap 4 Fuametal Equatos of hermoyamcs For Bary mxture:.? Example: If a bx c x 2 x for molar fracto f? swer: x - 22 x 2 2 ( x ) 2 2 x x x 2x2 ( ) b 2cx x //

88 Chap 4 Fuametal Equatos of hermoyamcs he partal molar volume of ethaol ( E ) as expresse by the polyomal : E /ml mol - ( / m ) p w ( )m 3( )m2 2005//8 88

89 Chap 4 Fuametal Equatos of hermoyamcs Example t 20 the esty of a 20 percet by mass ethaol-water soluto s kg m -3. ve that the partal molar volume of ethaol the soluto s 52.2 cm 3 mol -. Calculate the partal molar volume of the water. swer: Let eote water a B ethaol. he total volume of the soluto s B B We kow B ; we ee to eterme a B orer to solve for. ssume we have 00 cm 3 of soluto; the the mass s m ( g cm 3 ) (00 cm 3 ) g of whch (0.20) (96.87 g) 9.4 g s ethaol a (0.80) (96.87 g) 77.5 g s water g/8.02 g mol mol 2O B g/46.07 g mol mol ethaol ( B B )/ [00 cm 3 - ( mol) (52.2 cm 3 mol - )]/4.3.0 mol 8.5 cm 3 8 cm //8 89

90 Chap 4 Fuametal Equatos of hermoyamcs he partal molar volumes of the compoets of a aqueous soluto of potassum sulfate. 2005//8 90

91 2005//8 9 pecal opc: toal applcatos of Maxwell relatos: he fuametal equatos for a are: For a mole of a substace the Maxwell relatos are:

92 Chap 4 Fuametal Equatos of hermoyamcs Expaso Coeffcet ( ) a compressblty ( ) he expaso coeffcet s a measure of the varato of the volume o chage temperature at costat pressure. (/ ) (/ ) By substtutg R/p to equato we may obta / he sothermal compressblty s a measure of the varato of the volume o chage pressure at costat temperature. -(/ ) (/ ) By substtutg R/p to equato we may obta / whch shows that t s easer to compress a perfect gas at a low pressure tha at a hgh pressure. 2005//8 92

93 2005//8 93 κ α π π κ α he varato of teral eergy wth volume

94 2005//8 94 Example For eal gas 0 for va er Waals gas a/ 2 ( ) 2 (b) - -b R a 0 π R (a) Chap 4 Fuametal Equatos of hermoyamcs ( ) -b R π ( ) ( ) ( ) R - - R - - R a b b b 2 a

95 Determe partal molar volume Example 4. Calculato of the chage molar teral eergy the expaso of propae gas assumg t s a va er Waals gas ropae gas s allowe to expa sothermally from 0 to 30 L. What s the chage molar teral eergy? s: he chage teral eergy for a gve chage volume at costat temperature s gve by: a - a 2 a - 2 ccorg to able.3 a L 2 bar mol -2 but we ee to covert to I ut a ( 2-2 )( 5 -)( ) 2 6 mol L bar mol 0 a bar 0 m L a m a J mol 6 - ( a m mol ) - 0x0-3 m 3 mol x0-3 m 3 mol //8 95

96 2005//8 96 Determe partal molar volume Example 4.2 Calculato of the molar etropy of the sothermal expaso of a va er Waals gas s: Derve the molar etropy of sothermal expaso of a va er Waals gas: - - results a Maxwell relato: b R 2 a b R 2 2 b R b b R 2 l

97 2005//8 97 Chap 4 Fuametal Equatos of hermoyamcs o eal wth ervatves of the volume of flu: α α cubc expaso coeffcet for κ κ compressblty sothermal for κ α κ κ α π - as

98 2005//8 98

99 2005//8 99 Chap 4 Fuametal Equatos of hermoyamcs C π as ( ) C C α π π C C C ( ) [ ] ( ) ( )( ) ( ) κ α α κ α α π α α π 2 C C

100 2005//8 00 Chap 4 Fuametal Equatos of hermoyamcs ( ) α as p - as κ α as For eal gas / R/ ( / ) R/ so ( / ) - 0 ( / )

101 2005//8 0 Chap 4 Fuametal Equatos of hermoyamcs ( ) ( ) ( ) ( ) C C C C C C C C p J J J J κ α κ α

102 2005//8 02 Chap 4 Fuametal Equatos of hermoyamcs [ ] ( ) ( ) C C C C C α α α - J lso J s a state fucto:

103 2005//8 03 Example p p p C p κ α Chap 4 Fuametal Equatos of hermoyamcs