applied to a single-phase fluid in a closed system wherein no chemical reactions occur.

Transcription

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2 The basc relaton connectng the Gbbs energy to the temperature and pressure n any closed system: (ng) (ng) d(ng) d dt (nv)d (ns)dt T T,n appled to a sngle-phase flud n a closed system wheren no chemcal reactons occur. Consder a sngle-phase, open system:,n Chemcal potental (ng) (ng) (ng) d(ng) d dt dn T n T,n,n,T,n j

3 Defne the chemcal potental: (ng) n,t,n j The fundamental property relaton for sngle-phase flud systems of constant or varable composton: d(ng) (nv)d (ns)dt dn When n =, dg Vd SdT dx G V T,x G S TT,x

4 Consder a closed system consstng of two phases n equlbrum: d(ng) (nv) d (ns) dt dn d(ng) (nv) d (ns) dt dn nm (nm) (nm) d(ng) (nv)d (ns)dt dn dn

5 d(ng) (nv)d (ns)dt dn dn The two phase system s closed thus: d(ng) (nv)d (ns)dt dn dn 0 dn dn ( )dn 0 Multple l phases at the same T and are n equlbrum when chemcal potental of each speces s the same n all phases.

6 Defne the partal molar property of speces : M (nm) n The chemcal potental and the partcle molar Gbbs energy are dentcal: G For thermodynamc property M:,T,n nm M(,T, n, n,..., n,...) j M M d(nm) n d n dt Mdn T T,n,n

7 M M d(nm) n d n dt M dn T T,n,n d nm ndmmdn dn x dn ndx M M ndm Mdn n d n dt M (xdn ndx ) T T,n,n M M dm d dt Mdx n M xm dn 0 Tn T,n T,n n M M dm d dt M dx 0 T T, n, n Summablty relatons nm n M 0 M x M 0 Calculaton of mxture propertes from partal propertes

8 M M dm d dt Mdx 0 T T,n,n dm x dm M dx M M d dt xdm 0 T T,n,n The Gbbs/Duhem equaton Ths equaton must be satsfed for all changes n, T and the caused by changes of state n a homogeneous phase. For the mportant specal case of changes at constant T and : x dm 0 M

9 M x M x M For bnary systems Const. and T, usng Gbbs/Duhem equaton dm x dm Mdx xdm M dx dm x x M dx M dx dm dx M M M M x M dm dm M M x dx dx M M x d (B) x (A)

10 The three knds of propertes used n soluton thermodynamcs p p p G,S,, H, U Soluton propertes M, for example M artal propertes,, for example G,S, H, U ure-speces propertes M, for example G,S, H, U

11 EXAMLE Descrbe a graphcal nterpretaton of equatons (A) and (B) dm M I dx x dm I I I I dx 0 dm I M x dx I M x dm dx I M I M

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13 The need arses n a laboratory for 000 cm 3 of an antfreeze soluton consstng of 30 mol-% methanol n water. What volumes of pure methanol and of pure water at 5 C must be mxed to form the 000 cm 3 of antfreeze at 5 C? The partal and pure molar volumes are gven ml Methanol : V cm mol V cm o cm m Water : V cm mol V 8.06 ol V x V xv V (0.3)(38.63) (0.7)(7.765) 4.05 cm 3 / mol n V V t mol n (0.3)(83.46) mol n (0.7)(83.46) mol V t nv 4.974)(40.77) 07 ( cm V t n V ( 58.7)(8.068) 053 cm 3 3

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15 The enthalpy of a bnary lqud system of speces and at fxed T and s: H 400x 600x xx (40x 0x) Determne expressons for H and H as functons of x, numercal values for the pure-speces enthalpes H and H, and numercal values for the partal enthalpes at nfnte dluton and H H H H 400x 600x xx (40x 0x) x x H x 0x 3 H H x dh dx H 60x x 3 H x 3 x 0 x x H 40 J mol H 640 J mol

16 d ( ng) ( nv ) d ( ns) dt G dn G (ns) G T n,n,t,n j (nv) n T,n,T,n j G T, x S G T,x V dg V d S dt

17 H UV nh nu(nv) (nh) (n U) (n V) n n n,t,n,,t,n j, j,t,n, j H U V For every equaton provdng a lnear relaton among the thermodynamc propertes of a constant-compostoncomposton soluton there exsts a correspondng equaton connectng the correspondng partal propertes p of each speces n the soluton. هر معادلهكه رابطه اي خطي بين خواص ترموديناميكي يك محلول با تركيب ثابت برقرار نمايد ا داراي يك معادله المثني است كه خواص جزيي متناظر هر يك از اجزاء مخلوط را بههم مربوط مي نمايد.

18 n V t p n y n p n V t p y,,...,n

19 Gbbs s theorem A partal molar property (other than volume) of a consttuent هر خاصيت جزيي مولي ) غير از حجم ( يك جزء تشكيلدهنده مخلوط گاز ايدهآل the correspondng موليto جزء equal خالصs بهصورت mxture گاز ايدهآل در an deal-gas دمايn مخلوط speces ولي برابر همان خاصيت ميباشد. molar property در the speces مخلوط مof gas at فشاري deal معادل فشار a pure as جزيي آن theدر mxture temperature but at a pressure equal to ts partal pressure n the mxture. M g g ( T, ) M ( T, p ) (nv g ) g (n ) n V n n n T,,n T,,n n j j j g V g V M g V g

20 Snce the enthalpy of an deal gas s ndependent of pressure: g g g H (T, ) H (T,p ) H (T, ) M y M M M y M roperty change of mxng H g g g g H H y H 0 y H g Enthalpy change of mxng Smlarly U g y U g Internal energy change of mxng g g g U U y U 0

21 The entropy of an deal gas does depend on pressure g g dt ds C R T d ds Rdln (T const.) g S (T,) S (T,p ) Rln Rln Rlny g g p y M (T,) M (T,p ) g g S (T,) S (T,) R ln y g g g g S ys R yln y S S y S R y ln y g g g

22 G H TS g g g H (T,) g H (T,) g G H TS lny g g g g g S (T, ) S (T, ) Rlny G G lny g g g g g dg V d d d ln (const. T) G (T) ln g (T) ln y g g G y (T) yln y

23 Chemcal potental: provdes fundamental crteron for phase equlbra however the Gbbs energy hence μ s defned n relaton to however, the Gbbs energy, hence μ, s defned n relaton to the nternal energy and entropy - (absolute values are unknown).

24 g ( ) G (T) ln Fugacty: a quantty that takes the place of μ G (T) ln f Wth unts of pressure

25 G (T) lnf G (T) ln g For deal gases G G R ln R 0 g G G ln g f f Fugacty coeffcent ln R G d ln (Z ) (const. T) 0

26 When the compressblty factor s gven by two terms vral equaton Z B d ln (Z 0 ) (const. T) ln B 0 d B ln

27 Saturated vapor: Saturated lqud: v v G (T) lnf l l G (T) lnf G v G l v v l f l G G ln f f ln 0 f v l f v f l f sat = = v l sat For a pure speces coexstng lqud and vapor phases are n equlbrum p p g q p p q when they have the same temperature, pressure, fugacty and fugacty coeffcent.

28 The fugacty of pure speces as a compressed lqud: G (T) lnf sat sat G (T) lnf sat f G G ln f sat dg = V d S dt sat G G V d (const. T) sat Snce V s a weak functon of ln f f sat sat V d l sat f V (- ) sat ln = f f = sat sat sat l sat sat sat f= exp V( V(- )

29 For H O at a temperature of 300 C and for pressures up to 0,000 ka (00 bar) calculate values of f and from data n the steam tables and plot them vs.. For a state at : For a low pressure reference state: G G * ( T ) ln * ( T ) ln f f f f * H H * ( S S ) R T ln * G H TS * * f ka f ln * ( G G f G * ) The low pressure (say ka) at 300 C: H * J S * g J gk f 8.05 H ln (S 0.345)

30 For dfferent values of up to the saturated pressure at 300 C, one obtans the values of f,and hence. Note, values of f and at ka are obtaned = 4000 ka T = 300 ºC H = 96 J/g S = J/gK f ln ( ) f 36 ka 36/

31 T = 300 ºC sat = ka sat f ln ( ) H = 75 J/g S = J/gK V =.403 cm 3 /g f sat ka sat /

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33 Values of f and at hgher pressure l sat sat sat V( ) f exp l 3 V (.403)(8.05) 5.8 cm / mol f exp 5.8( 859.7) = 0000 ka 5.8( ) f exp ka /

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35 Fugacty and fugacty coeffcent: speces n soluton For speces n an deal gas mxture: g (T) ln y For speces n a mxture of real gases or n a soluton of lquds: (T) ln f ˆ Fugacty of speces n soluton (replacng the partal pressure) Multple phases at the same T and are n equlbrum when the fugacty of each consttuent speces s the same n all phases: fˆ f ˆ... fˆ

36 The resdual property: R M M M g The partal resdual property: R g M M M R g G G G G (T) lnfˆ g g ( ) y G (T) ln y The fugacty coeffcent of speces n soluton g G G ln y ˆ ˆf ˆf y R G lnˆ

37 For deal gas R R G 0 G lnˆ ˆf g ˆ g y ˆf g y

38 Fundamental resdual-property ng ng d d(ng) dt G HTS relaton d(ng) (nv)d (ns)dt dn ng nv nh G d d dt dn ng f (, T, n ) G/ as a functon of ts canoncal varables allows evaluaton of all other thermodynamc propertes, and mplctly contans complete property nformaton. ng nv nh G d d dt dn g g g g G ng nv nh d d dt dn or R R R ng nv nh R G d d dt dn R R R ng nv nh d d dt ln ˆ dn

39 R R R ng nv nh d d dt ln ˆ dn R R V (G /) Tx T,x R H (G /) T T R,x ln ˆ R (ng / ) n,t,n j

40 Develop a general equaton for calculaton of ln ˆ values form compressbltyfactor data. R (ng ) ln ˆ n,t,n j ˆ (nz n) d ln R n ng 0 d 0,T,n j ( nz n) (nz) Z n n n lnˆ 0 ( Z ) d Integraton at constant temperature and composton

41 Generalzed correlatons for the fugacty coeffcent d r r ln ( Z ) ( const. Tr ) 0 Z r Z Z 0 ln d r 0 r r r ( Z ) Z ( const. Tr 0 0 r ln d 0 r r d r 0 r ( Z ) 0 r ) ln ln 0 ln ( ) 0 ln r 0 Z For pure gas d r Table E3-E6 r

42 Estmate a value for the fugacty of -butene vapor at 00 C and 70 bar. T.77 0 r Table E5 and E r ( ) f (0.638)(70) 44.7 bar

43 Generalzed correlatons for the fugacty coeffcent r B Z T B c B r c c B 0 r c r 0 Z (B B ) dr ln (Z ) (const. T 0 r) r T r r 0 ln = (B +ωb ) T r For pure gas 0 B B T.6 r 0.7 T 4. r

44 Fugacty coeffcent from the vral E.O.S The vral equaton Z B The mxture second vral coeffcent B B j y y j B j For a bnary mxture B y yb y yb y yb y yb

45 Fugacty coeffcent from the vral E.O.S nb Z n nz,,, ) ( ) ( n T n T n nb n nz Z,,, n T n T ) ( ) ( ˆ nb d nb, 0, ) ( ) ( ln n T n T n nb d n nb

46 B yyb yyb yyb yyb B y (y )B yyb yyb y (y )B B y B y B y y (B B B ) B B B B ybyb yy y n / n / n n n n B B B n n n n nb n B n B nn n (nb) nnnn B n n T,n

47 (nb) nn nn B ( ) n n n T,n T,n (nb) n n n B ( ) n n n n n (nb) (nb) B (y yy ) nn n T,n n T,n B y (y ) (nb) n T,n B y (nb) ln ˆ n T,n ln ˆ = B +y δ ˆ ln B y Smlarly:

48 For mult-component systems ˆ ln k Bkk yy j( k j) j Where: B B B Where: k k kk

49 ω +ω j B j= (B +ωω B) ωj= cj 0 j cj T cj= (TcT cj)(-k j) = cj Z cj V cj cj Emprcal lnteracton t parameter Z = cj Z +Z c cj V = cj V +V /3 /3 c cj 3

50 Determne the fugacty coeffcents for ntrogen and methane n N ()/CH 4 () mxture at 00K and 30 bar f the mxture contans 40 mol-% N. B B B ( 59.8) ln ˆ cm 3 mol 30 B 35. (0.6) (0.6) 0. (83.4)(00) 050 y ˆ 0.95 ln ˆ 30 B y (0.4) (0.6) (83.4)(00) ˆ

51 ˆ ˆ Estmate and for an equmolar mxture of methyl ethyl ketone () / toluene () at 50 C and 5 ka. Set all k j = 0. B j j cj cj j ( B 0 Z cj Z B j c ) Z T cj cj cj Z cj V cj cj ( TcTcj )( kj ) V cj V / 3 c V / 3 cj B B B 3 ˆ B y ln ˆ B y ln ˆ ˆ 0.983

52 The deal soluton Serves as a standard to be compared: G d cf. G G g G g ln x ln y M d x M d x M G d x G x ln x S d G T d G T, x R ln x S d S R ln x S d x S R x ln x V d d G T, x G T d V V V xv d H d G d TS d G ln x TS ln x H d H d H x H

53 The Lews/Randall Rule For a specal case of speces n an deal soluton: ( T ) ln fˆ d G d ( T ) ln fˆ d G G d G ( T ) ln ln x f The Lews/Randall rule fˆ d x f ˆ d The fugacty coeffcent of speces n an deal soluton s equal to the fugacty coeffcent of pure speces n the same physcal state as the soluton and at the same T and.

54 Excess propertes The mathematcal formalsm of excess propertes s analogous to that of the resdual propertes: M E M M where M represents the molar (or unt-mass) value of any extensve thermodynamc property (e.g., V, U, H, S, G, etc.) Smlarly, we have: d ng nv nh E E E d d dt G E dn The fundamental excess-property relaton

55 The excess Gbbs energy and the actvty coeffcent The excess Gbbs energy s of partcular nterest: c.f. G E G G G E R E G G G G ln d d ( T ) ( T ) fˆ x f ln fˆ x f lnˆ ln ln fˆ x f The actvty coeffcent of speces n soluton. A factor ntroduced nto Raoult s law to account for lqud-phase non-dealtes. For deal soluton, 0, E G

56 E E E ng nv nh d d G dt E dn ng d E nv E d nh E dt ln dn T, E E V ( G / ) T H E E ( G / ) T E ( ng / ) ln n x,x, T, n j Expermental accessble values: actvty coeffcents from VLE data, V E and H E values come from mxng experments. E G x ln Important applcaton n phase-equlbrum x d ln 0 ( const. T, ) thermodynamcs.