# Vapor-Liquid Equilibria by the Solvation Method

## Shuzo Ohe, Ph.D. Prof. Sci. Univ. Tokyo

 【Prediction Example】 Calculate vapor-liquid equilibria for the methanol water CaCl2 (4 mole %) system at 1 atm by solvation method when the liquid composition of methanol is 60 mole % at salt free basis and the solvation number of CaCl2 to methanol is 15.395 and that to water is 18.7844. 【Solution】 From the given liquid compositions xi'＝ 0.6, xsalt＝ 0.04, xtotal,solvent＝ 0.96 then compositions at salt basis are calculated as x１= 0.6・0.96 = 0.576, x2= 0.4・0.96 = 0.384, respectively. The solvation number for each volatile component is given as 　　　　　S10 = 15.395,　　S20 = 18.7844 therefore, from relation; 　Si = Sio×xi'　they are estimated as S1= 15.395･0.6 = 9.237, S2= 18.7844･0.4 = 7.514, respectively. Next, effective liquid compositions of each volatile component from Eq. (4) is, calculated as x1a'　= (0.576 - 9.237･0.04) / (1- 0.04 - 9.237･0.04 - 7.514･0.04) 　　　= 0.7122　 Similarly,　x2a' = 0.2878. 　 Activity coefficients g1' and g2'　for effective liquid compositions: x1a', x2a' are calculated by the Wilson equation, using the following parameters for the methanol water system: 　　　Λ12 = 0.5515 and Λ2１ = 0.8978. By applying the Wilson equation: 　 ln g1'　= -ln (x1a' Λ12 x2a') x2a' [Λ12 / ( x1a' Λ12 x2a') -Λ2１ / (Λ2１x1a' x2a')] 　 ln g2'　= -ln (x2a' Λ2１ x1a') - x1a' [Λ12 / ( x1a' Λ12 x2a') -Λ2１ / (Λ2１x1a' x2a')] to the compositions x1a' and x2a' , we get ln g1'= 0.0418, ln g2' = 0.3142 then g1' = 1.0427, g 2' = 1.3692. 　 Second, determine the activity coefficient for vapor pressure lowering　gmix,solvent . From the solvation number for each pure solvent S10, S20, Eq. (2) estimates activity coefficients for respective components: g 1,solvent　and g2, solvent. g 1,solvent　= (0.96 - 0.04・15.395) / (1 - 0.04・15.395) / 0.96　= 0.9332 g2, solvent = (0.96 - 0.04・15.395) / (1- 0.04・18.7844) / 0.96　= 0.8741 Eq. (3) calculates activity coefficients of solvent mixture g mix, solvent. 　 g mix, solvent = 0.9332･0.6 0.8741･0.4 = 0.9096 Total activity coefficients for each volatile component are calculated from Eq. (7) as　 　 g1　= 1.0427・0.9096・0.7122・0.96 / 0.576 = 1.1257 　 g2　= 1.3692・0.9096・0.2878・0.96 / 0.384 = 0.8961. 　 Calculate the vapor pressures for each volatile component from the Antoine equation. Antoine constants for methanol and water are A1　= 8.07919、B1　= 1581.341、C1　= 239.65 A2　= 8.02754、B2　= 1705.616、C2　= 231.405. The Equilibrium temperature can be determined by bubble point calculation as 72.58 ℃. Then the methanol vapor pressure is P1 = 10 ( 8.07919 - 1581.341 / (72.58 239.65) ) = 1034.01 mmHg and that of water is P2 = 10 ( 8.02754 - 1705.616 / (72.58 231.405) ) = 261.03 mmHg. Therefore, methanol and water partial pressures are p1 = 1034.01・1.1257・0.576 = 670.46 mmHg p2 = 261.03・0.8961・0.384 = 89.82 mmHg. The total pressure is then 　π = p1 p2 = 760.28 mmHg. Vapor compositions for the components y1, y2 are y1 = p1 /π = 670.46 / 760.28 = 0.882, y2 = 0.118. The observed bubbling point is 72.6 ℃、and the vapor phase composition of methanol is 0.884 mole fraction. Absolute errors are 0.02 ℃ and 0.002, which indicate a high degree of accuracy.