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 (007. 5. 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
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
34 w Table. Slope of < X/ > vs x for 5 different lattice structures Linear Square Cubic BCC FCC 0000.00008.0000 3.0008 4.0004 6.00057 0000.00004.0000 3.0005 4.0008 6.0003 40000.0000.00005 3.00008 4.0000 6.0004 80000.00000.00003 3.00004 4.00005 6.00008 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 0000.00003.446.7330.99870.44947 0000.0008.4365.73.0000.44845 40000 0.99954.446.7339.99907.44864 80000.000.437.7370.99986.450 average 0.99997.4406.7385.9994.4490 deviation 0.0009 0.00046 0.0009 0.00063 0.0008 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
, 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
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.34 0.48.5 0.3.30 0.44 EtOH+chloroform at 35,45,55 C.03..5.7.66.97 MeOH+benzene at 35,55 C 3.7.03.46.89.78.68 benzene+methylcyclohexane at atm 0.5 0.5 0.7 0.6 0.8 0.8 cyclohexane+toluene at atm 0.9 0.8 0.9 0.8 0.9 0.9 cyclopentane+benzene at atm 0.38 0.7 0.39 0.8 0.38 0.38 propanol+benzene at atm.30 0.86.5 0.96.6.08 EtOH+benzene at atm.66.9.4 0.79.55 0.87 methylcyclopentane+benzene at atm 0. 0.09 0.3 0.09 0. 0. methylcyclopentane+toluene at atm 0.40 0.40 0.40 0.40 0.40 0.40 average [%].08 0.69 0.94 0.64.09 0.74 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 0.048 0.0 0.09 0.09 0.045 0.0 EtOH+chloroform at 35,45,55 C 0.0064 0.008 0.03 0.0094 0.03 0.045 MeOH+benzene at 35,55 C 0.055 0.039 0.083 0.06 0.08 0.0 benzene+methylcyclohexane at atm 0.0040 0.0039 0.004 0.0039 0.004 0.004 cyclohexane+toluene at atm 0.0044 0.0043 0.0044 0.0043 0.0044 0.0044 cyclopentane+benzene at atm 0.005 0.008 0.005 0.008 0.005 0.005 propanol+benzene at atm 0.000 0.007 0.0088 0.0076 0.0097 0.008 EtOH+benzene at atm 0.058 0.008 0.0 0.0077 0.03 0.0049 methylcyclopentane+benzene at atm 0.007 0.006 0.008 0.006 0.007 0.007 methylcyclopentane+toluene at atm 0.004 0.004 0.004 0.004 0.004 0.004 average of y 0.0090 0.0067 0.0080 0.0067 0.0090 0.0069 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
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, 954.. Guggenheim, E.GA. Mixtures; Clarendon Press, Oxford, 95. 3. Prausnitz, J. M.; Lichtenthaler, R..; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, nd Ed.; Prentice-Hall, 986. 4. Wilson, J. M. J. Am. Chem. Soc. 964, 86, 7. 5. Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications, 987. 6. Mendenhall, W.; Sincich, T. Statistics for Engineering and the Sciences, 4th Ed.; Prentice-Hall, 995. 7. Flory, P. J. Principles of Polymer Chemistry; Cornell Univ. Press, 953. 8. Brown, I; Smith, F. Aust. J. Chem. 957, 0, 43. 9. Scatchard, G.; Raymond, C. J. Am. Chem. Soc. 938, 60, 79. 0. Scatchard, G.; Wood, S.; Mochel, J. J. Am. Chem. Soc. 946, 68, 957.. Myers, H. S. Ind. Eng. Chem. 956, 48, 04.. Wehe, A.; Coates, J. AICHE J. 955,, 4. 3. Barker, J. A. Aust. J. Chem. 953, 6, 07. 4. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 4th Ed.; McGraw- Hill, 988. 5. Hayden, J. G.; O Connell, J. P. Ind. Eng. Chem., Process Des. Dev. 975, 4, 09. 6. Prausnitz, J. M. et al. Computer calculations for multicomponent vapor-liquid and liquid-liquid equilibria; Prentice-Hall, 980. 007, Vol. 5, o. 4