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
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
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
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.
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
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
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
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)
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
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
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. 3-3 - ml Methanol : V 38.63 cm mol V 40.77 cm o 3-3 - 8cm m Water : V 7.765 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 000 4.05 83.46 mol n (0.3)(83.46) 4. 974 mol n (0.7)(83.46) 58. 7 mol V t nv 4.974)(40.77) 07 ( cm V t n V ( 58.7)(8.068) 053 cm 3 3
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 600 80x 0x 3 H H x dh dx H 60x 40 40 x 3 H 600 40x 3 x 0 x x H 40 J mol H 640 J mol
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
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.
n V t p n y n p n V t p y,,...,n
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
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
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
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
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).
g ( ) G (T) ln Fugacty: a quantty that takes the place of μ G (T) ln f Wth unts of pressure
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
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
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.
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(- )
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 * 3076. 8 J S * 0. 3450 g J gk f 8.05 H 3076.8 ln (S 0.345) 8.34 573.5
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 859.7 ka are obtaned = 4000 ka T = 300 ºC H = 96 J/g S = 6.364 J/gK f 8.05 96 3076.8 ln (6.364 0.345) 8.34 573.5 f 36 ka 36/ 4000 0.908
T = 300 ºC sat = 859.7 ka sat f 8.05 75 3076.8 ln (5.708 0.345) 8.34 573.5 H = 75 J/g S = 5.708 J/gK V =.403 cm 3 /g f sat 6738.9 ka sat 6738.9 / 859.7 0.7843
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 6738.9exp 5.8( 859.7) 834 573.5 = 0000 ka 5.8(0000 859.7) f 6738.9exp 6789.4 ka 834 573.5 6789.4 /0000 0.6789
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
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
For deal gas R R G 0 G lnˆ ˆf g ˆ g y ˆf g y
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
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
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
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
Estmate a value for the fugacty of -butene vapor at 00 C and 70 bar. T.77 0 r Table E5 and E6 0.67 r.73.096 0.9 ( ) 0 0.638 f (0.638)(70) 44.7 bar
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 0.083 B 0.39 0.4 T.6 r 0.7 T 4. r
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
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
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
(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:
For mult-component systems ˆ ln k Bkk yy j( k j) j Where: B B B Where: k k kk
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
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) 35. 05.0 0. 6 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 05.00 (0.4) (0.6) 0. 835 (83.4)(00) ˆ 0. 834
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 0. 08 ln ˆ B y 0. 07 ln ˆ 0.987 ˆ 0.983
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
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.
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
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
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.