Study of Excess Gibbs Energy for a Lattice Solution by Random Number Simulation
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1 Journal of the Korean Chemical Society 007, Vol. 5, o. 4 Printed in the Republic of Korea ù x mw ½ w š w * w yw ( Study of Excess Gibbs Energy for a Lattice Solution by Random umber Simulation Hae-Young Jung* Department of Chemistry, DukSung Women's University, Seoul 3-74, Korea (Received May, 007. ù x ww w w s y w ³ s w. s l yw ½ G E w w. w -» sx w š» w. : yw, ½, ABSTRACT. Performing random number simulations, we approximated that the distribution of the number of ways for arranging molecules randomly on a lattice is a normal distributon for, the number of interactions between the nearest neighbors of different molecules. From this distribution, an approximate equation of the excess Gibbs energy G E for a lattice solution of nonrandom mixing was derived. Using the equation, liquid-vapor equilibria of several binary solutions were calculated and compared with the calculated result of other equations. Keywords: onrandom Mixing, Excess Gibbs Energy, Lattice w w w» w cell, hole, free volume (lattice. p wƒ w ƒ w x s š. yw w, w yw w w š. ù y w ywz w t wùƒ quasi-chemical. ƒ ywsx ƒ ¾ quasi-chemical š. k w 3, Wilson 4 w -» sx x w š. z Wilson UIQUAC 3 œw w ù ywz yw wš w. ù» w 5ƒ w w x ww. w w 3
2 s y w ³ s w š, y w ½ G E w w. w 0ƒ -» w sx w š, Redlich- Kister, Van Laar, Wilson 3 w. ù x mw s x ù» mw ù w w. x, s,,, 5ƒ w x. ƒ z, 4, 6, 8,. x w yw wƒ ù w x sw g. x j w yw y w - - y w. 5 ù x mw ½ w š 33 Fig.. Probability distribution of X/ from 0 6 trials when x =0.3, =0648 for a cubic lattice. HX!! ( ! ; X = = ( ( X! ( X!X! ( +! ( y s»w s w w X w s³ t r. 6 s³ X = ; = + ( Fig.. σ X/, X/ vs x from 0 6 trials when =0648 for a cubic lattice. t r (3 σ X = X x (4 x 0000~80000, - x = / 0., 0., 0.3, 0.4, 0.5 w ww. w wz 0 6 w. Fig. ƒ j, X w y s ³ s w, t ƒ j s w 6 w ³ s w w. š Fig. X s³ r, x ƒ w. σ X x (5 x mw (4, (5 Table,. Table, Linear, Square, Cubic, BCC, FCC ƒƒ x, s,,, ù kü x wš. Table, x w yw (, (3 mw (4, (5 ù 007, Vol. 5, o. 4
3 34 w Table. Slope of < X/ > vs x for 5 different lattice structures Linear Square Cubic BCC FCC in the st column are approximate values except for a linear lattice Table. Slope of σx/ vs x for 5 different lattice structures Linear Square Cubic BCC FCC average deviation in the st column are approximate values except for a linear lattice w ƒƒ z/, z., f w X y s s³ r ³ s w. z X = x x σ X = z -----x x (6 (7 (6, (7 j w y s ù w z w š. ³ s w ½ G E X w y s ³ s w» w., w Ω. ( Ω(,, X +! exp!! πσ -- X X σ X X (8 w m w w Q. 5 Q Ω,, X βe ( e ( X = ; β = kt { X} (9 (9 E(X -i,j y εij w t. (0 ( Q (9 w w. 5 ( X * w X. (8, (0, (3 l (8, (, (4 l EX ( = β εx + βz ( ε + ε ε + ε ε = ε ; ε ij < 0 ( Q Ω(,, X * e βex* [ ln Ω(,, X βex ( ] = 0 X X * = X β εσ X ( (3 (4 ( lnq +! ln ln( πσ X β ε X β ε = + σ X!! βz ( ε + ε (5 Journal of the Korean Chemical Society
4 , ƒ j (5 w. lnq lnx lnx β ε X β ε + σ X (6 (6, (7 w w 007, Vol. 5, o. 4 ù x mw ½ w š 35 (6 (7 yw v y Vmix wš m w œ 3 w (7 l yw½ Gmix G E t. (8, (9 βz ( ε + ε lnq x lnx x lnx zβ εx x ( β εx x zβ ( ε + ε G mix = x lnx + x lnx + x x ( A Bx x G E = x x ( A Bx x A = zβ ε, B = zβ ε = A z (8 (9 (0 (0 z š j (9 -. B yw ùkù z ùkü w. š (9 3- Redlich-Kister B'=0 w. G E = x x [ A + B ( x x + C ( x x ] ( (9 x =0.5 w e ùkü j», š». x, j» š w ( w» w. r φ = = ; r = --- r + r x + rx r ( ( r, r š w Flory- x r Huggins 7 š - ¼ ùkü j», š w x ƒ š. (9, ( l G E = φ φ ( A Bφ φ, B = A z (3 (3 z A, r. A, B yw w z ùkü, r j», w z ùkü w. (3 (8 l A, B ƒƒ x ƒw A, B, r 3- w ww š. CH 3O +CCl 48, EtOH +chloroform 9, MeOH+benzene 0, benzene+methylcyclohexane, cyclohexane +toluene, cyclopentane+benzene, propanol+benzene, EtOH+benzene, methylcyclopentane+benzene, methylcyclopentane+toluene 0ƒ w sx w. (3 sw w Redlich-Kister, Van Laar, Wilson w. Redlich-Kister, Van Laar, Wilson w š x 3. š ƒ ùkù yw» w» w x s³s (4 y j Barker 3 w. P = data i = ( P i P calc data (4 (4» Pcalc w w. P calc = γ x P + γ x P (5 γ, γ y ½ G l E. 3 (5
5 36 w Table 3. relative error % in vapor pressure calculated for 0 binary mixtures RK- RK-3 Eqn.(3 Eqn.(3-B Van Laar Wilson CH 3O +CCl 4 at 45 C EtOH+chloroform at 35,45,55 C MeOH+benzene at 35,55 C benzene+methylcyclohexane at atm cyclohexane+toluene at atm cyclopentane+benzene at atm propanol+benzene at atm EtOH+benzene at atm methylcyclopentane+benzene at atm methylcyclopentane+toluene at atm average [%] Table 4. y calculated for 0 binary mixtures RK- RK-3 Eqn.(3 Eqn.(3-B Van Laar Wilson CH 3O +CCl 4 at 45 C EtOH+chloroform at 35,45,55 C MeOH+benzene at 35,55 C benzene+methylcyclohexane at atm cyclohexane+toluene at atm cyclopentane+benzene at atm propanol+benzene at atm EtOH+benzene at atm methylcyclopentane+benzene at atm methylcyclopentane+toluene at atm average of y P P P s exp v s ( B ( P calc P P calc δ y = P s exp v s ( B ( P calc P P calc δ y = δ = B B B (6 (7 (6 v, v v modified Rackett 4 w, P s, P s ƒ s x» š x 4. B, B, B ƒ Hayden O Connell 5 w y v w š x 6. Table 3, 4 ùkü. Table 3» w %, Table 4» w s³s yƒ. Table 3, 4 RK-, RK-3 ƒƒ ƒ, 3 Redlich-Kister ùkü Eqn. (3-B (3 B w 3 w ùkü. Table 3, 4 3 w (3 ƒ š, (3 Redlich-Kister š. w Wilson ƒ š. ù x ww w s w ³ s w. l yw ½ G E w x œ w, ùkù Journal of the Korean Chemical Society
6 3 ƒƒ, yw, j» z ùkü ƒ š.» Redlich- Kister. w p y w z š w š w ƒ. 006 w ü w. x ù x mw ½ w š 37. Hirschfelder, J.G O.; Curtiss, C.G F.;G Bird, R.G B. The Molecular Theory of Gases and Liquids; John Wiley & Sons, Guggenheim, E.GA. Mixtures; Clarendon Press, Oxford, Prausnitz, J. M.; Lichtenthaler, R..; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, nd Ed.; Prentice-Hall, Wilson, J. M. J. Am. Chem. Soc. 964, 86, Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications, Mendenhall, W.; Sincich, T. Statistics for Engineering and the Sciences, 4th Ed.; Prentice-Hall, Flory, P. J. Principles of Polymer Chemistry; Cornell Univ. Press, Brown, I; Smith, F. Aust. J. Chem. 957, 0, Scatchard, G.; Raymond, C. J. Am. Chem. Soc. 938, 60, Scatchard, G.; Wood, S.; Mochel, J. J. Am. Chem. Soc. 946, 68, Myers, H. S. Ind. Eng. Chem. 956, 48, 04.. Wehe, A.; Coates, J. AICHE J. 955,, Barker, J. A. Aust. J. Chem. 953, 6, Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 4th Ed.; McGraw- Hill, Hayden, J. G.; O Connell, J. P. Ind. Eng. Chem., Process Des. Dev. 975, 4, Prausnitz, J. M. et al. Computer calculations for multicomponent vapor-liquid and liquid-liquid equilibria; Prentice-Hall, , Vol. 5, o. 4
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