Outline o Bubble/Dew Calculations using MRL Making the function for - PowerPoint PPT Presentation

Outline o Bubble/Dew Calculations using MRL Making the function for G E realistic: o 11.4 Van der Waals perspective Van Laar Scatchard/Hildebrand Flory, Flory-Huggins o 11.5 Theory - Skip o 11.6 Local Composition Theory Wilson UNIQUAC UNIFAC 1

11.4 Van der Waals perspective Regular Solutions (V E = 0, S E = 0) energetics of mixing are described by the same energy equation for mixtures that we previously developed in discussing the simple basis for mixing rules. ∑ ∑ − = − ρ ig U U x x a i j ij Noting that 1/ ρ = V = Σ x i V i according to “regular solution theory,” ∑ ∑ – x i x j a ij U ig ( ) U – = - - - - - - - - - - - - - - - - - - - - - - - - - - - - ∑ x i V i For the pure fluid, taking the limit as x i → 1, 2

a ii x i a ii U ig U is U ig ( ) i ⇒ ( ) ∑ U – = – - - - - - – = – - - - - - - - - - - V i V i For a binary mixture, subtracting the ideal solution result to get the excess energy gives, 2 a 11 2 a 22   a 11 a 22 x 1 + 2 x 1 x 2 a 12 + x 2 U E   = x 1 - - - - - - - + x 2 - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -   V 1 V 2 x 1 V 1 + x 2 V 2 3

van Laar 2 F a a I x x V V ≡ − ⇒ = 11 22 E 1 2 1 2 Q U x V Q G J + V V x V H K 1 2 1 1 2 2 A A x x = = E E 12 21 1 2 G U + ( x A x A ) 1 12 2 21 A 12 γ 1 ln = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2 A 12 x 1 1 + - - - - - - - - - - - - - A 21 x 2 4

Fitting Van Laar to a single experiment: γ 2 2 x 2 ln γ 1 1 A 12 = ln + - - - - - - - - - - - - - - - γ 1 x 1 ln γ 1 x 1 ln 2 γ 2 1 A 21 = ln + - - - - - - - - - - - - - - - γ 2 x 2 ln Azeotrope point can be used. See example 11.6 5

Infinite dilution can be used. See example 11.7 6

Scatchard Hildebrand a 12 = a a 11 22   x 1 x 2 V 1 V 2 - a 11 a 22 2 a 11 - a 22 U E   = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - - – - - - - - - - - - - - - -   2 2 2 2 x 1 V 1 + x 2 V 2 V 1 V 2 V 1 V 2 2   x 1 x 2 V 1 V 2 a 11 a 22 U E   = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - -   x 1 V 1 + x 2 V 2 V 1 V 2 = = Φ Φ δ − δ + 2 E E G U f ( x V x V ) a 1 2 1 2 1 1 2 2 = = ΦΦ δ − δ + 2 E E f ( ) G U xV x V a 1 2 1 2 1 1 2 2 Â F i ∫ / is known as the "volume fraction" x V x V i i i i 7

d i ∫ a / V is known as the "solubility parameter" ii i ∆ ∆ − vap vap U H RT δ i ≡ = V V i i This is a predictive technique valid for nonpolar substances. See table 11.1 for parameters. Also can make adjustable. 8

Van Laar and Scatchard-Hildbrand 355 350 345 T(K) Scatchard-Hildebrand 340 k ij =-0.038 335 van Laar 330 0 0.2 0.4 0.6 0.8 1 MeOH x,y methanol + benzene 9

Flory’s Equation Recall Ideal Solution result (pg 95) V T V T ∆ S 1 = n 1 R ln - - - - - - = n tot x 1 R ln - - - - - - i i V 1 V 1 V T V T ∆ = ln - - - - - - = ln - - - - - - S 2 n 2 R n tot x 2 R i i V 2 V 2 10

  V T V T ∆ ∆ ∆   = + = ln - - - - - - + ln - - - - - - S S 1 S 2 n tot R x 1 x 2  i i  V 1 V 2 11

Φ Φ = − = + E E E 1 2 G H TS RT x ( ln x ln ) 1 2 x x 1 2 12

Flory-Huggins Model Φ 1 Φ 2   G E Φ 1 Φ 2 x 1 ( )χ RT = RT x 1 ln - - - - - - + x 2 ln - - - - - - + + x 2 R   x 1 x 2 13

11.6 Local composition models (nonrandom) Common Features o Lattice Model - Fixed number of neighbors = 10 o Uses G E = A E approximation (good) o Local Composition 14

x 21 x 2 x 12 x 1 - Ω 21 - Ω 12 - - - - - - - = - - - - - - - - - - - = - - - - x 11 x 1 x 22 x 2 o Two-fluid Theory ( ) ( ) 1 2 U ig U ig U ig ( ) ( ) ( ) U – = x 1 U – + x 2 U – - x 1 x 2 Ω 21 ε 21 ( ε 11 ) x 2 x 1 Ω 12 ε 12 ( ε 22 ) N A z – – U E = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - x 2 Ω 21 x 1 Ω 12 2 x 1 + + x 2 A E U E – - T d ∫ - - - - - - - = - - - - - - - - - - - - - - - + C RT RT T 15

Wilson’s equation V 1 – A   Ω 12 Λ 21 21 = = - - - - - - exp - - - - - - - - - - -   V 2 RT Λ 12 Λ 21   γ 1 ( x 2 Λ 12 ) ln = – ln x 1 + + x 2 - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - -   x 2 Λ 12 x 1 Λ 21 x 1 + + x 2 16

UNIQUAC A z ε ij ( ε jj ) – N – q i q i – a q i     Ω ij - τ ij ij = - - - - exp - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = - - - - exp - - - - - - - - - = - - -     q j 2 RT q j T q j r i q i θ i τ ij 17

E G = Φ Φ ln / + x ln / x x x a f a f 1 1 1 2 2 2 RT − Φ θ + Φ θ 5 ln / ln / q x q x a f a f 1 1 1 1 2 2 2 2 − θ + θ τ − θ τ + θ ln( ) ln( ) q x q x 1 1 1 2 21 2 2 1 12 2 18

Φ 1 Φ 1 Φ 1 Φ 1     γ 1 ln = ln - - - - - - + 1 – - - - - - - – 5 q 1 ln - - - - - - + 1 – - - - - - -     θ 1 θ 1 x 1 x 1 θ 2 τ 12 θ 1 ( θ 1 θ 2 τ 21 ) + q 1 1 – ln + – - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - θ 1 θ 2 τ 21 θ 1 τ 12 θ 2 + + Φ 2 Φ 2 Φ 2 Φ 2     γ 2 ln = ln - - - - - - + 1 – - - - - - - – 5 q 2 ln - - - - - - + 1 – - - - - - -     θ 2 θ 2 x 2 x 2 θ 1 τ 21 θ 2 ( θ 1 τ 12 θ 2 ) + q 2 1 – ln + – - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - θ 1 θ 2 τ 21 θ 1 τ 12 θ 2 + + 19

Group parameters for the UNIFAC and UNIQUAC equations. AC in the table means aromatic carbon. (DIFFERS SLIGHTLY FROM TEXT) Main Sub- R(rel.vol.) Q(rel.area) Example Group group CH2 CH3 0.9011 0.8480 CH2 0.6744 0.5400 n-hexane: 4 CH2+2 CH3 CH 0.4469 0.2280 isobutane: 1CH+3 CH3 C 0.2195 0 neopentane: 1C+ 4 CH3 C=C CH2=CH 1.2454 1.1760 1-hexene: 1 CH2=CH+3 CH2+1 CH3 CH=CH 1.1167 0.8670 2-hexene: 1 CH=CH+2 CH2+2 CH3 CH2=C 1.1173 0.9880 CH=C 0.8886 0.6760 C=C 0.6605 0.4850 ACH ACH 0.5313 0.4000 benzene: 6ACH AC 0.3652 0.1200 benzoic acid: 5ACH+1AC+1COOH ACCH2 ACCH3 1.2663 0.9680 toluene: 5ACH+1ACCH3 ACCH2 1.0396 0.6600 ethylbenzene: 5ACH+1ACCH2+1CH2 ACCH 0.8121 0.3480 OH OH 1.0000 1.2000 n-propanol: 1OH+1 CH3+2 CH2 20

Group parameters for the UNIFAC and UNIQUAC equations. AC in the table means aromatic carbon. (DIFFERS SLIGHTLY FROM TEXT) CH3O CH3OH 1.4311 1.4320 methanol is an independent group H water H2O 0.9200 1.4000 water is an independent group furfural furfural 3.1680 2.4810 furfural is an independent group DOH (CH2OH) 2.4088 2.2480 ethylene glycol is an independent 2 group ACOH ACOH 0.8952 0.6800 phenol: 1ACOH+5ACH CH2CO CH3CO 1.6724 1.4880 dimethylketone: 1 CH3CO+CH3 CH2CO 1.4457 1.1800 diethylketone=1 CH2CO+2 CH3+1 CH2 CHO CHO 0.9980 0.9480 acetaldehyde: 1CHO+1 CH3 CCOO CH2COO 1.9031 1.7280 methyl acetate: 1 CH3COO+1 CH3 CH2COO 1.6764 1.4200 methyl propanate: 1 CH2COO+2 CH3 benzoic acid: COOH COOH 1.31013 1.2240 5ACH+1AC+1COOH 21

UNIFAC Pure 2-propanol CHOHCH 3 CH 3 CH 3 CHOHCH 3 ( ) 1 o ( ) µ CH 3 µ CH 3 SOG 1 µ CH 3 µ CH 3 – – CH 3 CHOHCH 3 O H 2 CH 3 CH 3 CHOHCH 3 CH 3 H 2 O CH CH 3 CHOHCH 3 3 CH 3 Hypothetical H O 2 solution of SOG o µ CH 3 µ CH 3 “pure” CH 3 – Real mixture (SOG) 22

g = g + g COMB RES ln ln ln k k k ( ) ( ) SOG 1 SOG o 1 o µ CH 3 µ CH 3 µ CH 3 µ CH 3 µ CH 3 µ CH 3 – – – ( ) 1 Γ CH 3 Γ CH 3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = ln – ln RT RT RT o µ 1 µ 1 – ( ) ( ) resid 1 1 γ 1 ν m [ Γ m Γ m ] ∑ ln = - - - - - - - - - - - - - - - - - = ln – ln RT m 23