Chap 6 hyscal Transformatons hase ( ): A form of matter that s unform n both chemcal composton and physcal state throughout one phase. hase Transton (

Transcription

1 Chap 6 hase Equlbrum Table of Contents 6. hase Dagrams of One-Component Systems 6. The Clapeyron Equaton 6.3 The Clauss-Clapeyron Equaton Vapor-Lqud Equlbrum of Bnary Lqud Mxtures 6.5 Vapor ressure of Nondeal Mxtures and Henry s Law 6.6 Actvty Coeffcents 6.7 Collgatve ropertes 6.8 Two-Component Systems Consstng of Sold and Lqud hases 6.9 Specal Topc: Effect of Surface Tenson on the Vapor ressure 005//3

2 Chap 6 hyscal Transformatons hase ( ): A form of matter that s unform n both chemcal composton and physcal state throughout one phase. hase Transton ( ): A spontaneous converson of one phase nto another phase. Transton Temperature (,T trs ): Temperature at whch the chemcal potentals of two phases are equal. hase Dagram ( ): A map showng the regons of ntensve varables, ( vs. T ) or (x J vs. T ) where each phase of a system s thermodynamcally the most stable form. 005//3

3 Chap 6 hase Transton The general regons of and T where sold, lqud, or gas s stable (at mnmum molar Gbbs energy) are shown on ths phase dagram. For example, the sold phase s the most stable phase at low temperatures and hgh pressures. 005//3 3

4 Chap 6 hase Equlbrum The hase Boundares ( ): The lnes on phase dagram show the condtons under whch the two phases separated by the boundary are n equlbrum. A metastable phase ( ): s thermodynamcally unstable but prevented by knetc reason from a transton to a stable phase. The Vapor ressure ( ): The pressure above a condensed phase s exerted by ts vapor n dynamc equlbrum wth the condensed phase at a specfc temperature. 005//3 4

5 Chap 6 hase Equlbrum The vapor pressure of a lqud or sold s the pressure exerted by the vapor n equlbrum wth the condensed phase. 005//3 5

6 Chap 6 hase Equlbrum The crtcal temperature (, T C ): the temperature at whch the densty of a lqud and the vapor above t become equal and the surface between the two phases dsappears. The crtcal pressure (, C ): the vapor pressure at the crtcal temperature. The supercrtcal flud ( ): a flud at a temperature hgher than ts T C but wth a densty that s comparable to that of a normal lqud. A supercrtcal flud has propertes that are ntermedate between those of a lqud and a gas. The Supercrtcal Flud Chromatography (SFC) s a form of chromatography where the moble phase s a supercrtcal flud. The superflud flows wthout any vscosty. 005//3 6

7 Chap 6 hase Equlbrum (a) A lqud n equlbrum wth ts vapor. (b) When a lqud s heated n a sealed contaner, the densty of the vapor phase ncreases and that of the lqud decreases slghtly. There comes a stage, (c), at whch the two denstes are equal and the nterface between the fluds dsappears. Ths dsappearance occurs at the crtcal temperature. The contaner needs to be strong: the crtcal temperature of water s 374 and the vapor pressure s then 8 atm. 005//3 7

8 Chap 6 hase Equlbrum Sublmaton Vapor ressure ( ): The pressure of a vapor n equlbrum wth the sold. The varaton of the sublmaton vapor pressure wth temperature s the soldvapor boundary on the phase dagram. Bolng temperature (,T b ): (vapor pressure ambent pressure) Normal bolng pont ( ): the temperature when vapor pressure atm Standard bolng pont ( ): the temperature at whch the vapor pressure reach bar(0.987 atm), whch s slghtly lower than ts normal bolng pont (99.6 vs. 00 for water). 005//3 8

9 Chap 6 hase Equlbrum The meltng temperature (,T m ): the temperature at whch the sold and lqud phases of a substance are n equlbrum at a gven pressure. The normal meltng pont ( ): the meltng temperature when the pressure s atm. The freezng temperature (,T f ): T f s the same temperature as the meltng temperature n all but certan materals. The normal freezng temperature ( ): s the freezng temperature at atm. The standard freezng temperature ( ): s the freezng temperature at bar. 005//3 9

10 Chap 6 hase Equlbrum The trple pont (, T 3 ) and ts sgnfcance: Trple pont of a substance s the pont on a phase dagram at whch three phases are n mutual equlbrum. For a sngle component system, the phase rule mples that the trple pont s an nvarant pont of substance. The temperature of the trple pont of water s used to defne the sze of the Kelvn for t s defned as lyng at T K, exactly. 005//3 0

11 Chap 6 hase Dagram The expermentally determned phase dagram for water showng the dfferent sold phases. Note the change of vertcal scale at atm. For water: T C 647. K C 0.5 bar T K bar 005//3

12 Fg 6.(a) Fg 6.a hase Dagram for Water The vapor pressure of water s gven as a functon of T by the lne between lqud and vapor areas. The dashed extenson of ths lne gves the vapor pressure of supercooled water. 005//3

13 Fg 6.(b) Fg 6.b hase Dagram for Water At hgher pressure, four other crystal forms of ce are formed. The freezng pont of ce I s lowered to - when the pressure s rased to 000 bar, but hgher pressures lead to other crystal forms of ce (III, V, VI) for whch d/dt s postve. 005//3 3

14 Fg 6.(c) Fg 6.c The expermental phase dagram for carbon doxde The equlbrum between sold and gas at one bar s at -78. Note that, as the trple pont les at pressures well above atmospherc, lqud carbon doxde does not exst under normal condtons (only above a pressure of at least 5. bar). 005//3 4

15 Chap 6 hase Dagram The expermental phase dagram for carbon doxde A pressure of at least 67 atm must be appled at room temperature to produce the lqud carbon doxde. An easly accessble crtcal flud s obtaned at 3 and 74 bar. 005//3 5

16 Fg 6.(d) Fg 6.d hase Dagram for Carbon The graphte and damond are n equlbrum at room temperature only at pressure above 0000 bar. 005//3 6

17 Chap 6 hase Equlbrum If only pressure-volume work are nvolved, the number of degrees of freedom for the descrpton of the ntensve state of the system s F C- +. For a one-component system, F 3 -, no more than three phases s n equlbrum. 005//3 7

18 Chemcal otental Varaton wth T Snce dg -S dt + V d, The chemcal potental of a pure substance decreases as ts T s rased. Snce S (g) > S (l) > S (s), the slope s more negatve for gas than for the lqud and more negatve for lqud than for the sold. µ T p S 005//3 8

19 Chap 6 hase Equlbrum The temperature dependence of the chemcal potentals of the sold, lqud, and gas phases of a substance. The phase wth the lowest chemcal potental at a specfed temperature s the most stable one at that temperature. The transton temperatures, the meltng and bolng temperatures, are the temperatures at whch the chemcal potentals of two phases are equal. (In practce, the lnes should be slghtly concave downward, snce entropy ncreases wth ncreasng temperature) 005//3 9

20 Chemcal otental Varaton wth In general, V (l ) > V (s ), so an ncrease n pressure ncrease the chemcal potental of the lqud more than the sold, so t rasng the meltng temperature. However, f V (l ) < V (s ), as s the case for water, the opposte s true, so the meltng temperature s lowered when pressure ncreases. µ T V For gas phase, f t follows the deal gas law, V (g) RT/, then - ºRT ln / º 005//3 0

21 Fg 6. Chemcal otentals Dependence on T Fg 6. Dependence of the chemcal potentals of sold, lqud, and gas phases on temperature at constant pressure. The dash lnes n (b) are for lower pressure. 005//3

22 Chap 6 hase Equlbrum The pressure dependence of the chemcal potental of a substance depends on the molar volume of the phase. The lnes show schematcally the effect of ncreasng pressure on the chemcal potentals of the sold and lqud phases and the correspondng effects on the freezng temperatures. (a) In ths case the molar volume of the sold s less than that of the lqud and (s) ncreases less than (l). As a result, the freezng temperature rses. 005//3

23 Chap 6 hase Equlbrum (b) Here the molar volume s greater for the sold than the lqud (as for water), (s) ncreases more strongly than (), and the freezng temperature s lowered. 005//3 3

24 Chap 6 hase Equlbrum ressure may be appled to a condensed phase ether by (a) compressng the condensed phase or (b) subjectng t to an nert pressurzng gas. When pressure s appled, the vapor pressure of the condensed phase ncreases. 005//3 4

25 Chap 6 hase Equlbrum When two or more phases are n equlbrum, the chemcal potental of a substance (and, n a mxture, a component) s the same n each phase and s the same at all ponts n each phase. At equlbrum the unform chemcal potental throughout a system s followed from the general condtons for equlbrum at constant temperature and pressure: (dg ),T 0 Gbbs 005//3 5

26 Example Assessng the effect of pressure on the chemcal potental Calculate the effect on the chemcal potentals of ce and water of ncreasng the pressure from.00 bar to.00 bar at 0. The densty of ce s 0.97 g cm -3 and that of lqud water s g cm -3 under these condtons. Answer: T The change n chemcal potental of an ncompressble substance when the pressure s changed by p s V m p The molar volume of two phases of water come from the mass densty,, and the molar mass, M, by usng V m M/. ( ) A µ V ( - - )( 5.80x0 kg mol.00x0 a ) -3 ( 97 kg m ) M p µ ce +.97 ρ ( ) A ( - - )( 5.80x0 kg mol.00x0 a ) -3 ( 999 kg m ) M p µ water +.80 ρ mol 005//3 6 J J - mol -

27 Chap 6 hase Equlbrum When pressure s appled to a system n whch two phases are n equlbrum (at a), the equlbrum s dsturbed. It can be restored by changng the temperature, so movng the state of the system to b. It follows that there s a relaton between d and dt that ensures that the system remans n equlbrum, G0, as ether varable s changed. 005//3 7

28 Fg. 6.3 Fg 6.3 Coexstence curve for a one-component system. Along ths lne ( ) ( ). If the temperature s changed dt, the pressure has to be changed d as ndcated to mantan equlbrum. At pont : ( ) ( ), At pont : ( ) ( ) + d ( ) ( ) ( ) + d ( ) so that d ( ) d ( ) 005//3 8

29 Chap 6 hase Equlbrum For one component system wth two phases, ( and ), f the T and are both changed n such a way to keep the chemcal potentals of two phases equal to each other, the two phases wll contnue to coexst: d d Along phase boundary for each phase, d dg V d - S dt, Thus V d - S dt V d - S dt, Along the coexstence curve n the -T plot, the necessary relaton for d/ dt was derved by Clapeyron: d dt S V β β -S -V α α S H V T V 005//3 9

30 Chap 6 The locaton of phase boundares The Clapeyron Equaton s a relaton between the change n and T for two phases that reman n equlbrum when ether varables s modfed. It can therefore be regarded as an expresson for the SLOE of a phase boundary n a phase dagram: d dt T trs S V trs trsh T V trs 005//3 30

31 Chap 6 hase Equlbrum A typcal sold-lqud phase boundary slopes steeply upwards. Ths slope mples that, as the pressure s rased, the meltng temperature rses. Most substances behave n ths way. d trsh dt T V trs 005//3 3

32 Clapeyron equaton Clapeyron equaton: d d T T snce the unt of molar quanttes cancel n the equaton, the S, H and V per unt mass may also be used. The transton may apply to vaporzaton, sublmaton, fuson, or between two sold phases. The molar enthalpes of sublmaton, fuson and vaporzaton at trple pont are related by sub H fus H + vap H For water at fuson transton, t expands on freezng, so that fus V and thus d/dt for meltng s negatve. trs H trs V 005//7 3

33 Chap 6 hase Equlbrum Example 6. Change n the bolng pont of water wth pressure Ans: What s the change n the bolng pont of water at a 00 per a change n atmosphere pressure. The molar enthalpy of vaporzaton s kj mol -, the molar volume of lqud water s 0.09x0-3 m 3 mol -, and the molar volume of steam s 30.99x0-3 m 3 mol -, all at 00 and.035 bar: d dt T V vap H ( - ) g V l ( J mol ) ( )( K 30.80x0 m mol ) 363a K - Thus dt/d.768x0-4 K a - 005//7 33

34 Chap 6 hase Equlbrum Example 6. Change n the freezng pont of water wth pressure Calculate the change n pressure requred to change the freezng pont of water. At 0 the heat of fuson of ce s J g -, the densty of water s g cm -3, and the densty of ce s g cm -3. Ans: The recprocals of the denstes,.000 and.0908, are the volumes n cubc centmeters of g. The volume change on freezng ( V ) s therefore -9.06x0-8 m 3 g - l -V s. For small changes fus H, T, and V l -V s are vrtually constant, so that ( ) d dt T V fus H ( -V ) l s ( J g ) ( )( K -9.06x0 m g ) -.348x0 7 a K - The change n the freezng pont of water per bar pressure s dt/d K bar - 005//7 34

35 Chap 6 hase Equlbrum Example 6.3 Calculate the vapor pressure from thermodynamc tables Usng the data n Table C. determne the vapor pressure of H O(l) and Br (l) at 98.5 K. These vaporzaton processes may be handled as f they were chemcal reactons. Ans: H O(l) H O(g) vap Gº (-37.9) kj mol - Now vap Gº 98 -RT ln / º bar Br (l) Br (g) vap Gº (0) 3.0 kj mol - Now vap Gº 98 -RT ln / º 0.85 bar 005//7 35

36 Chap 6 hase Equlbrum Example 6.4 Calculate the pressure to make damond from graphte Calculate the equlbrum pressure for the converson of graphte to damond at 5. The denstes of graphte and damond may be taken to be.5 and 3.5 g cm-3, respectvely, ndependent of pressure, n calculatng the change of G wth pressure. Ans: C(graphte) C(damond) Gº J mol V - x0 m mol -.9x0-6 m 3 mol G V d G V d G ( ) - G V - T - G - G J mol a V -.9x0 m mol.5x0 9 a or.5x0 4 bar 005//7 36

37 6.3 Clausus-Clapeyron equaton The formula for the sold-lqud boundary s obtaned by ntegratng the Clapeyron equaton for fuson wth approxmaton: H T trs dt H T trs d ln * V T* T V T* trs trs * + trs trs H T ln V T* 005//7 37

38 6.3 Clausus-Clapeyron Equaton An approxmaton form of the Clapeyron equaton appled when one phase s condensed and the other s a gas that may be treated as beng perfect. V vap V ( g) V RT g The Clausus-Clapeyron equaton for a lqud-vapor or a sold-vapor phase boundares s based on approxmatons: d d T T V vap H ( ) g-v l vap T V H g vap RT H d dln vap RT H d T ln H - vap + C RT 005//7 38

39 Vapor pressure of water Fg 6.4 A plot of ln (/º) versus /T should be lnear, and ths s borne out by data on both vaporzaton and sublmaton. (Table 6.) ln (/º) T - /0 3 K - Std B: standard Bolng ont. F: Freezng ont ln - vap H + RT C 005//7 39

40 Table 6. Vapor ressure of Ice and Water t/ / ka t/ / ka //7 40

41 005//7 4 Integratng between two lmts n, T: The Clausus-Clapeyron equaton can be regarded as an equaton for the temperature varaton of the vapor pressure of lqud. vap exp T T R H Chap 6 hase Equlbrum T T T T R H d d vap / / ln ( ) vap vap - T R T T T H T T R H ln vap T T R H ln ln

42 Chap 6 hase Equlbrum The slope of the lqud-vapor boundary s less steep than that of the sold-lqud boundary. If the enthalpy of vaporzaton s assumed constant over the T and ranges of nterest, the Clausus-Clapeyron equaton can be ntegrated to * exp H R T * vap T For the slope of the sold-vapor boundary: the enthalpy of sublmaton s used n the Clapeyron equaton gvng a steeper boundary than for vapor. 005//7 4

43 Chap 6 hase Equlbrum Near the pont where they concde (at the trple pont), the sold-gas boundary has a steeper slope than the lqudgas boundary because the enthalpy of sublmaton s greater than the enthalpy of vaporzaton and the temperatures that occur n the Clausus-Clapeyron equaton for the slope have smlar values. 005//7 43

44 Chap 6 hase Equlbrum A typcal lqud-vapor phase boundary. The boundary can be regarded as a plot of the vapor pressure aganst the temperature. ln vap R H T T T T 005//7 44

45 Chap 6 hase Equlbrum The dependence of the enthalpy of vaporzaton on temperature s based on the of approxmaton: vap H A + BT + CT Snce vap H RT d ln dt vap vap ( /T ) dvap vaph A B dlnvap dt + + C vap RT R T T A ln vap - + B ln T + CT + D R T d ln -R d dt At trple pont, ln sub ln vap ln sub H R sub T sub T ln vap vap R H T vap T 005//7 45

46 Fg 6.5 vap H RT dln dt vap dln R d vap ( T ) Fg 6.5 The enthalpy of vaporzaton approaches zero as the temperature approaches the crtcal temperature, T C. 005//7 46

47 Vapor-Lqud Equlbrum of Bnary Lqud Mxtures Raoult 884 :, A A *, A * A. A, B x +x, ( x ) A B A* x B A*- *, Raoult s Law. A-A, A-B, B-B, Raoult s Law., y : y x *,. a x. A A : µ () () l µ l + RT lnx x * 005//7 47

48 Vapor-Lqud Equlbrum of Bnary Lqud Mxtures Roault : µ (g) µ ( l ) : : : ( ) ( ) g µ g + RT ln µ () l µ () l + RT a µ ln ( ) µ () g + RT ln µ l + RT lna 005//7 48

49 Vapor-Lqud Equlbrum of Bnary Lqud Mxtures µ g ln ( ) + RT ln µ () l + RT a :µ *(g) µ *(l ) a * *. : ( ) * µ () g + RT ln µ l : RT ln RT lna * a * 005//7 49

50 005//7 50 Vapor-Lqud Equlbrum of Bnary Lqud Mxtures Raoult,, : + x *+ x * * + x ( * - *) Bubble pont lne. Raoult : ( ) * * * * * * * x x x x x y ( ) y * *- * * y x x + ( ) y *- * * * y x x + ( ) x x x x x y * * * * * * * + + +

51 Fg 6.7(a) (a) lot of total pressure for the benzene()-toluene() versus x, as gven by equaton: * + ( *- *) x 005//7 5

52 Vapor-Lqud Equlbrum of Bnary Lqud Mxtures dew pont lne. y x * * x * * y + y * * * + y - * y * + x * * * - * * ( *- *) y ( *- ) * + * y 005//7 5

53 Fg 6.7(b) (b) lot of the total pressure for the benzene()- toluene() versus y, as gven by equaton: (x /y ) * * */ [ *+ ( *- *) y ]. 005//7 53

54 Chap 6 hase Equlbrum From y * + x * ( ) * * x ( ) *- x * + x y * y * So x * + y * ( ) *- * y Also * + x - * *- * ( *- * ) x 005//7 54

55 Chap 6 hase Equlbrum Te lnes ( ): onts on the dew pont lne and bubble pont lne at the same pressure represent the compostons of vapor and lqud phases that are n equlbrum. These pont are connected by a horzontal lne referred to as a te lne. x n L x B n V y B (n L +n V ) x TB The lever rule: (n L +n V ) x TB n L x B + n V y B nl ( yb -xtb ) (x-v) n x -x ( l- V ( ) x) TB B 005//7 55

56 Fg 6.8 Benzene () - toluene () system at 60. (a) artal and total pressures. (b) Lqud and vapor compostons. 005//7 56

57 Example 6.5 Calculaton of partal pressure for benzene and toluene. At 60 the vapor pressures of pure benzene and toluene are 0.53 and 0.85 bar, respectvely. What are the equatons of the bubble pont lne and dew pont lne? For a soluton wth 0.60 mole fracton toluene, what are the partal pressure of toluene and benzene, and what s the mole fracton of toluene n the vapor? Ans: Let toluene be component, * 0.85 bar; * 0.53 bar Bubble pont lne: + x *+ x * * + ( * - *) x 0.53 bar + (-0.38 bar) x Dew pont lne: (x /y ) * * */ [ *+ ( *- *) y ] bar /[0.85 bar + (0.38 bar ) y ] x 0.60 x * (0.60)(0.85 bar) 0. bar x * (0.40)(0.53 bar) 0.05 bar * + ( * - *) x 0.53 bar + (-0.38 bar) (0.60) 0.36 y x * / [ *+ ( *- *) x ] 0. bar/0.36 bar //7 57

58 Example 6.6 Calculaton of actvtes of benzene and toluene n a soluton. At 60 the vapor pressures of pure benzene and toluene are 0.53 and 0.85 bar, respectvely. What are the actvtes of toluene (component ) and benzene (component ) n a soluton contanng mole fracton toluene? Ans: * 0.85 bar; * 0.53 bar x 0.60 x * (0.60)(0.85 bar) 0. bar x * (0.40)(0.53 bar) 0.05 bar a / * (0. bar)/(0.85 bar) 0.60 a / * (0.05 bar)/(0.53 bar) 0.40 Snce these are deal soluton, the actvtes are equal to the mole fractons. 005//7 58

59 Chap 6 hase Equlbrum : Gbbs : G x + µ xµ RT x lnx G/T H -T x H T G S - x S -R x lnx G V T xv T, : mxg mxs mxh 0 ( ) mxv 0 ( ) µ (l) µ (l) + RT lnx G- x µ RT x lnx S- x S -R x lnx ( ) 005//7 59

60 Table 6. Vapor ressure of Toluene() and Benzene() t/ */bar */bar //7 60

61 Example 6.7 Calculaton of mole fractons of benzene and toluene n a soluton that bols at 00 and atm (.03 bar). What s the mole fracton x of toluene (component ) n the toluene-benzene (component ) soluton that bols at 00, and what s the mole fracton y of toluene n the vapor? Ans: at 00, * 0.74 bar; *.800 bar x - * *- * The mole fracton n the vapor phase: y x */ (0.744)(0.74 bar) /.03 bar These ponts are labeled a and b n Fg For nondeal solutons the ponts have to be obtaned expermentally. 005//7 6

62 Fg 6.8 Bolng pont dagram at.03 bar ( atm) Fg 6.8 Benzene()- toluene() bolng ponts: lqud and vapor compostons. The lqud bols at the temperature gven by the lower curve. 005//7 6

63 Fg 6.9 Fg 6.9 Threedmensonal plot for a two-component lqud and vapor system. The plot conssts of two surfaces (bubble pont surface and dew pont surface). 005//7 63

64 Fg 6.0 The number of theoretcal plates n a column s equal to the number of successve nfntesmal vaporzaton at equlbrum requred to gve the separaton that s actually acheved. Fg 6.0 Bubble-cap Fractonatng Column. 005//7 64

65 Chap 6 hase Equlbrum K Henry s Law 803 Henry x y K Example 6.8 The solublty of carbon doxde n water x Usng the Henry s law constant, calculate the solublty of carbon doxde n water at 5 n mole per lter at a partal pressure of CO over the soluton of bar. Assume that lter of soluton contans practcally 000 g of [ ] a 000 [ ] 0 a x0 a CO + [ CO ] 8.0 [ CO ] x ( mol L ) a CO 0.67x0 a 5 3.3x0 moll //8 65

66 Ideal-dlute soluton: Solutons of dssmlar lquds can show strong devatons from Raoult s law unless a substance has x B < 0.0. However, the vapor pressure usually starts off as a straght lne. Ths s emboded n Henry s law, p B K B x B and the slope of the lne s the Henry s-law constant K B. A soluton whch obeys Henry s law s called an deal-dlute soluton. 005//8 66

67 Ideal-dlute soluton: Idealdlute soluton s one n whch the solvent obeys Raoult law but the solute obey Henry s law, p J x J K J (Henry s law) K J s an emprcal constant obtaned by extrapolatng the low concentraton data to x J. If the solute obeys Henry s law then the solvent obeys Raoult s law. 005//8 67

68 Chap 6 hase Equlbrum p : ( ) ( ) ( ) µ l µ g µ g + RT ln K x + RT ln ( g) µ K ( ) µ g + RT ln + RT lnx µ l + RT lnx where () () ( ) l µ g K + RT ln µ 005//8 68

69 Fg 6.(a) (a) Lqud mxture wth a negatve azeotrope: Chloroform()- Acetone() at //8 69

70 Fg 6.(b) (b) Lqud mxture wth a postve azeotrope: Carbon dsulfde()- Acetone() at //8 70

71 Fg 6. Vapor Lqud Vapor (a) Bolng pont curve at constant pressure for a maxmum bolng pont azeotrope. (b)bolng pont curve at constant pressure for a mnmum bolng azeotrope. Lqud 005//8 7

72 Fg 6.3 Vapor pressure curve for one component of a bnary lqud mxture at constant temperature. 005//8 7

73 Example 6.9 roof that f Henry s law holds for the solute, Raoult s law holds for the solvent. Show that f Henry s law holds for the solute (component ), Raoult s law holds for the solvent (component ). Ans: The Gbbs-Duhem equaton provdes a relatonshp between the dfferentals of the chemcal potentals of components and at constant temperature and pressure. If µ + dµ µ RT ln a x - x dµ Snce x + x, dx -dx then If µ µ x RT lnx + µ * + RT lnx Kx RT µ +RT ln * RT - dx x dx d µ ln,then µ µ *. dµ dx x constant Therefore µ + * RT ln * RT RT d x x 005//8 73

74 Chap 6 hase Equlbrum Actvty coeffcent apply to nondeal soluton wth () l µ () + RT ( γ x ) µ l ln γ as x If there are postve dervatons from Raoult s law, s greater than unty; and f there are negatve dervatons from Raoult s law, s less than unty. As actvty a / *, here To calculate the actvty coeffcent of from expermental data: a a γ x γ * x γ x * y x * 005//8 74

75 Table //8 75

76 Example 6.0 Calculate actvty coeffcents n ether-acetone soluton Calculate the actvty coeffcents for ether() and acetone() n ether-acetone soluton at 30. The expermental data are gven n Table 6.5 and are plotted n Fgure 6.4. Ans: At 0.5 mole fracton acetone, the actvty coeffcents of the two components are gven by γ γ x * x * 5.ka ( 0.5)( 86.ka).4 ka ( 0.5)( 37.7 ka) The actvty coeffcents of both components, calculated n ths way at other mole fractons, are summarzed n Table 6.5. Note: as the mole fracton of ether component approaches unty, ts actvty coeffcent approaches unty, snce the vapor pressure asymptotcally approaches that gven by Raoult s law. 005//8 76

77 Fg 6.4 Fg 6.4 artal pressure of ether-acetone soluton at //8 77

78 γ 'K x γ ' / K x y K x γ ' as x 0 Apparent Henry's constant : K ' A plot of K ' versus x and extraplotng to x x x 0 005//8 78

79 Fg 6.5 Evaluaton of Henry s law constant K as K at x 0 for acetone n ether-acetone solutons at //8 79

80 For 0.5 mole fracton of acetone, the actvty coeffcent of acetone based on the dervaton from Henry s law s γ ' y K x.4 ka 0.5 ( )( 78.4 ka) 0.57 a * γ ' K * x The two types of actvty coeffcents are related by: γ ' K γ * γ or γ ' * K 005//8 80