



【Prediction Example】 Calculate vaporliquid equilibria for the methanol water CaCl_{2 }(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 CaCl_{2 }to methanol is 15.395 and that to water is 18.7844. 【Solution】 From the given liquid compositions x_{i}'＝ 0.6, x_{salt}＝ 0.04, x_{total,solvent}＝ 0.96 then compositions at salt basis are calculated as x_{１}= 0.6・0.96 = 0.576, x_{2}= 0.4・0.96 = 0.384, respectively. The solvation number for each volatile component is given as S_{10 }= 15.395, S_{20 }= 18.7844 therefore, from relation; S_{i }= S_{io}×x_{i}' they are estimated as S_{1}= 15.395･0.6 = 9.237, S_{2}= 18.7844･0.4 = 7.514, respectively. Next, effective liquid compositions of each volatile component from Eq. (4) is, calculated as x_{1a}' = (0.576  9.237･0.04) / (1 0.04  9.237･0.04  7.514･0.04) = 0.7122 Similarly, x_{2a}' = 0.2878.
Activity coefficients g_{1}^{'} and g_{2}^{'} for effective liquid compositions: x_{1a}', x_{2a}' 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 g_{1}^{' }= ln (x_{1a}' Λ12 x_{2a}') x_{2a}' [Λ12 / ( x_{1a}' Λ12 x_{2a}') Λ2１ / (Λ2１x_{1a}' x_{2a}')] ln g_{2}^{'} = ln (x_{2a}' Λ2１ x_{1a}')  x_{1a}' [Λ12 / ( x_{1a}' Λ12 x_{2a}') Λ2１ / (Λ2１x_{1a}' x_{2a}')] to the compositions x_{1a}' and x_{2a}' , we get ln g_{1}^{'}= 0.0418, ln g_{2}^{'} = 0.3142 then g_{1}^{'} = 1.0427, g 2' = 1.3692.
Second, determine the activity coefficient for vapor pressure lowering g_{mix,solvent} . From the solvation number for each pure solvent S_{10}, S_{20}, Eq. (2) estimates activity coefficients for respective components: g_{ 1,solvent }and g_{2, solvent}. g_{ 1,solvent} = (0.96  0.04・15.395) / (1  0.04・15.395) / 0.96 = 0.9332 g_{2, 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 g_{1 }= 1.0427・0.9096・0.7122・0.96 / 0.576 = 1.1257 g_{2 }= 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 A_{1 }= 8.07919、B_{1 }= 1581.341、C_{1 }= 239.65 A_{2 }= 8.02754、B_{2 }= 1705.616、C_{2 }= 231.405. The Equilibrium temperature can be determined by bubble point calculation as 72.58 ℃. Then the methanol vapor pressure is P_{1 }= 10 ( 8.07919  1581.341 / (72.58 239.65) ) = 1034.01 mmHg and that of water is P_{2 }= 10 ( 8.02754  1705.616 / (72.58 231.405) ) = 261.03 mmHg. Therefore, methanol and water partial pressures are p_{1 }= 1034.01・1.1257・0.576 = 670.46 mmHg p_{2 }= 261.03・0.8961・0.384 = 89.82 mmHg. The total pressure is then π = p_{1} p_{2} = 760.28 mmHg. Vapor compositions for the components y_{1}, y_{2} are y_{1 }= p_{1} /π = 670.46 / 760.28 = 0.882, y_{2} = 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. 
©, Shuzo Ohe, 2010, All rights reserved.