Thermo3-models-final - Copy.docx

CHE 411N Phase and Equilibria of Mixtures TTh 9:00 – 10:30

D e p a r t m e n t o f Chemical Engineering

Term/Academic Year: First Semester of AY 2018-2019

Talamban, Cebu City, Philippines 6000

𝑮𝒆𝒙 expression

Name of Model

Redlich-Kister Expansion

𝑒𝑥

= 𝑥1 𝑥2 [𝐴 + 𝐵 (𝑥1 − 𝑥2 ) + 𝐶 (𝑥1 − 𝑥2 )2 + ⋯ ] 𝑅𝑇𝑙𝑛𝛾1 = 𝐴𝑥22 𝑅𝑇𝑙𝑛𝛾2 = 𝐴𝑥12

Where A≠0 B, C, D, … = 0

Applicability The Redlich-Kister expansion is used for the representation of excess thermodynamic properties. The number of terms retained in this expansion depends on the shape of the Gex curve as a function of composition.

𝐺 𝑒𝑥 = 𝑥1 𝑥2 [𝐴 + 𝐵 (𝑥1 − 𝑥2 ) + 𝐶 (𝑥1 − 𝑥2 )2 + ⋯ ]

𝐺 One-Constant Margules

Activity Coefficient Expression

The one-constant Margules equation provides a satisfactory representation for activity coefficient behavior only for liquid mixtures containing constituents of similar size, shape, and chemical nature

𝐺 𝑒𝑥 = 𝐴𝑥1 𝑥2 .

𝐺 𝑒𝑥 = 𝑥1 𝑥2 [𝐴 + 𝐵 (𝑥1 − 𝑥2 ) + 𝐶 (𝑥1 − 𝑥2 )2 + ⋯ ] Two-Constant Margules

Where A, B ≠ 0 C, D, … = 0 𝐺 𝑒𝑥 = 𝑥1 𝑥2 [𝐴 + 𝐵 (𝑥1 − 𝑥2 )]

Wohl Expansion

𝐺 𝑒𝑥 𝑅𝑇(𝑥1 𝑞1 + 𝑥2 𝑞2 ) = 2𝑎12 𝑧1 𝑧2 + 3𝑎112 𝑧12 𝑧2 + 3𝑎122 𝑧1 𝑧22 + 4𝑎1112 𝑧13 𝑧2 + 4𝑎1222 𝑧1 𝑧23 + 6𝑎1122 𝑧12 𝑧22 + ⋯ Where zi = volume fractions of species i 𝑧𝑖 =

𝑥𝑖 𝑞𝑖 𝑥1 𝑞1 + 𝑥2 𝑞2

𝑅𝑇𝑙𝑛𝛾1 = 𝛼1 𝑥22 + 𝛽1 𝑥23 𝑅𝑇𝑙𝑛𝛾2 = 𝛼2 𝑥12 + 𝛽2 𝑥13

For more complicated systems (mixtures of dissimilar molecults), the excess Gibbs energy of a general mixture will not be a symmetric function of the mole fraction, and the activity coefficients of the two species in a mixture should not be expected to be mirror images.

Where 𝛼𝑖 = 𝐴 + 3(−1)𝑖+1 𝐵 𝛽𝑖 = 4(−1)𝑖 𝐵

Similar to the Redlich-Kister expansion, the Wohl expansion is one that is modeled after the Virial expansion for gaseous mixtures.

Van Laar Equations

𝐺 𝑒𝑥 𝑅𝑇(𝑥1 𝑞1 + 𝑥2 𝑞2 ) = 2𝑎12 𝑧1 𝑧2 + 3𝑎112 𝑧12 𝑧2 + 3𝑎122 𝑧1 𝑧22 + 4𝑎1112 𝑧13 𝑧2 + 4𝑎1222 𝑧1 𝑧23 + 6𝑎1122 𝑧12 𝑧22 + ⋯ Where a12 ≠ 0 a112, a122, … = 0

Wilson Equation

𝐺 𝑒𝑥 = 2𝑎12 𝑧1 𝑧2 (𝑥1 𝑞1 + 𝑥2 𝑞2 ) 𝑅𝑇

𝐺 𝑒𝑥 = −𝑥1 ln(𝑥1 + 𝑥2 Λ12 ) − 𝑥2 ln(𝑥2 𝑅𝑇 + 𝑥1 Λ21 )

𝑙𝑛𝛾1 =

𝛼 𝛼𝑥 [1 + 𝑥1 ]2 𝛽 2

𝑙𝑛𝛾2 =

𝛽 𝛽𝑥 [1 + 𝛼 𝑥2 ]2 1

The liquid-phase activity coefficients for the Wohl expansion can be obtained by taking the appropriate derivatives. They are frequently used to correlate activity coefficient data.

Where α = 2q1a12 β = 2q2a12

𝑙𝑛𝛾1 = − ln(𝑥1 + 𝑥2 Λ12 ) Λ12 + 𝑥2 [ 𝑥1 + 𝑥2 Λ12 Λ 21 − ] 𝑥1 Λ21 + 𝑥2 𝑙𝑛𝛾2 = − ln(𝑥2 + 𝑥1 Λ 21 ) Λ12 − 𝑥1 [ 𝑥1 + 𝑥2 Λ12 Λ 21 − ] 𝑥1 Λ21 + 𝑥2

The ratio of species 1 to species 2 molecules in the vicinity of any molecule is the same as the ratio of their mole fractions. A different class of excess Gibbs energy models can be formulated by assuming that the ratio of species 1 to 2 molecules surrounding any molecule also depends on the differences in size and energies of interaction of the chosen molecule with both species. Thus, around each molecule there is a local composition that differs with the bulk composition

NRTL (NonRandom Two Liquid) Model of Renon and Prausnitz

𝐺 𝑒𝑥 𝜏21 𝐺21 𝜏12 𝐺12 = 𝑥1 𝑥2 ( − ) 𝑅𝑇 𝑥1 + 𝑥2 G21 𝑥2 + 𝑥1 G12

𝑙𝑛𝛾1

2 𝐺21 = ) 𝑥1 + 𝑥2 G21 𝜏12 𝐺12 + ] (𝑥2 + 𝑥1 𝐺12 )2

𝑥22 [𝜏21 (

𝑙𝑛𝛾2

Flory-Huggins Equation

Δ𝑚𝑖𝑥 𝑆 = −𝑅(𝑥1 𝑙𝑛𝜙1 + 𝑥2 𝑙𝑛𝜙2 ) 𝑆 𝑒𝑥 = Δ𝑚𝑖𝑥 𝑆 − Δ𝑚𝑖𝑥 𝑆 𝐼𝑀 Δ𝑚𝑖𝑥 𝐻 = 𝐻 𝑒𝑥 = 𝜒𝑅𝑇(𝑥1 + 𝑚𝑥2 )𝜙1 𝜙2 where χ is an adjustable parameter known as the Flory interaction parameter

𝐺 𝑒𝑥 𝐻 𝑒𝑥 − 𝑇𝑆 𝑒𝑥 = 𝑅𝑇 𝑅𝑇 𝐺 𝑒𝑥 𝜙1 𝜙2 = [𝑥1 𝑙𝑛 + 𝑥2 𝑙𝑛 ] + 𝜒(𝑥1 𝑅𝑇 𝑥1 𝑥2 + 𝑚𝑥2 )𝜙1 𝜙2

2 𝐺12 = 𝑥12 [𝜏12 ( ) 𝑥2 + 𝑥1 G12 𝜏21 𝐺21 + ] (𝑥1 + 𝑥2 𝐺21 )2 𝜙1 1 𝑙𝑛𝛾1 = 𝑙𝑛 + (1 − ) 𝜙2 𝑥1 𝑚 + 𝜒𝜙22

𝑙𝑛𝛾2 = 𝑙𝑛

𝜙2 − (𝑚 − 1)𝜙1 𝑥2 + 𝑚𝜒𝜙12

The Flory and Huggins model is meant to apply to mixtures of molecules of very different sizes (including solutions of polymers).

UNIQUAC (Universal QuasiChemical) Model of Abrams and Prausnitz

𝐺 𝑒𝑥 (𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑜𝑟𝑖𝑎𝑙) 𝑅𝑇 = ∑ 𝑥𝑖 𝑙𝑛 𝑖

𝜙𝑖 𝑥𝑖

𝑧 𝜃𝑖 + ∑ 𝑥𝑖 𝑞𝑖 𝑙𝑛 2 𝜙𝑖 𝑖

𝐺 𝑒𝑥 (𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙) 𝑅𝑇 = − ∑ 𝑥𝑖 𝑞𝑖 ln(∑ 𝜃𝑗 𝜏𝑗𝑖 𝑙𝑛 𝑖

𝑗

𝜃𝑖 ) 𝜙𝑗𝑖

Where ri = volume parameter for species i qi = surface area parameter for species i θi = area fraction of species i 𝑥𝑖 𝑞𝑖 = ∑𝑗 𝑥𝑗 𝑞𝑗 Φi = segment or volume fraction of species i 𝑥𝑖 𝑟𝑖 = ∑𝑗 𝑥𝑗 𝑟𝑗

UNIFAC (UNIQUAC Functional-Group Activity Coefficients) Model

𝐺 𝑒𝑥 𝐺 𝑒𝑥 (𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑜𝑟𝑖𝑎𝑙) = 𝑅𝑇 𝑅𝑇 𝑒𝑥 𝐺 (𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙) + 𝑅𝑇

𝑙𝑛𝛾𝑖 = 𝑙𝑛𝛾𝑖 (𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑜𝑟𝑖𝑎𝑙) + 𝑙𝑛𝛾𝑖 (𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙)

This model is based on the statistical mechanical theory. It allows local compositions to result from both the size and energy differences between the molecules in the mixture.

𝑙𝑛𝛾𝑖 (𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑜𝑟𝑖𝑎𝑙) 𝜙𝑖 = 𝑙𝑛 𝑥𝑖 𝑧 𝜃𝑖 − 𝑞𝑖 𝑙𝑛 2 𝜙𝑖 + 𝑙𝑖 𝜙𝑖 − ∑ 𝑥𝑗 𝑙𝑗 𝑥𝑖 𝑗

𝑙𝑛𝛾𝑖 (𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙) = 𝑞𝑖 [1 = ln(∑ 𝜃𝑗 𝜏𝑗𝑖 ) − ∑ 𝑗

𝑗

𝜃𝑗 𝜏𝑖𝑗 ] ∑𝑘 𝜃𝑘 𝜏𝑘𝑗

Where 𝑙𝑖 =

(𝑟𝑖 − 𝑞𝑖 )𝑧 − (𝑟𝑖 − 1) 2

(𝑖)

𝑙𝑛𝛾𝑖 (𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙) = ∑ 𝑣𝑘 [𝑙𝑛Γ𝑘 𝑘

(𝑖)

− 𝑙𝑛Γ𝑘 ]

The UNIFAC method is used for the prediction of non-electrolyte activity in non-ideal mixtures. The correlation attempts to describe molecular interactions based on the functional groups attached to the molecule. This is done as to reduce the sheer number of binary interactions that would be needed to be measured for the prediction of the system’s state.