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Partial molar volume in an ideal-gas mixture
Gibbs’ Theorem (Ideal-gas properties)
A partial molar property (other than volume) of a constituent species in an ideal-gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.
Gibbs’ Theorem (Ideal-gas properties) Formula
Partial molar enthalpy in ideal-gas mixture
Pressure has no effect on enthalpy
Partial molar entropy in ideal-gas mixture
Partial molar Gibbs’ energy in ideal-gas mixture (1)
Simplest possible composition dependence, based on entropy increase due to random intermixing of molecules
Partial molar Gibbs’ energy in ideal-gas mixture (2)
Solution property (summability relation)
Property change of mixing (at the same conditions, constant T and P)
Volume change of mixing
ΔV_mix = V - x1V1 - x2V2
Enthalpy change of mixing
Entropy change of mixing
Always positive (mixing = more molecular disorder)
Gibbs Energy Change of Mixing
Always negative (mixing = spontaneous)
Gibbs Energy Change of Mixing (Summability relation)
Overall change in property
ΔM = ΔM_i + ΔM_j + … + ΔM,ig_mix
Property change of demixing
ΔM,ig_demix = -ΔM,ig_mix
Phase Equilibrium (Gibbs energy)
For α- and β-phases of a pure species in equilibrium, Gibbs free energies ARE equal (G𝜶 = G𝜷)
Phase equilibrium (chemical potential)
For multiple π phases with N number of species at same T and P in equilibrium, µ𝜶_i = µ𝜷_i = … = µπ_i
Vapor-liquid equilibrium (chemical potential)
µV_i = µL_i
Chemical potential as equilibrium criterion
𝜇𝑖 = 𝐺𝑖_bar; Gibbs energy only defined as (𝑈) and entropy (𝑆). U = no absolute values, 𝜇𝑖 = no absolute values
Chemical potential in ideal gas mixtures
As either P or yi approaches zero, μi_ig becomes negative infinity which is not only applicable to ideal gas but to all kind of gas; not applicable to dilute systems (yi → 0)
Fugacity as equilibrium criterion
New criterion for equilibrium
Fugacity
For real, pure species (f_i); analogous to pressure, also in pressure units
Fugacity of pure, ideal gas
fi_ig = P
Fugacity coefficient of pure species
Φi = fi/P
Fugacity coefficient of pure species and residual Gibbs energy
ln (Φi) = Gi_R/RT
Fugacity coefficient for pure ideal gas
Φi = fi/P = 1
Residual gas volume
Difference between the real gas volume, Vi and the ideal gas volume, Vi_ig.
Residual Gibbs energy
In terms of compressibility factor Z
Fugacity coefficient of Pure Gas (VEOS) [Pitzer]
Fugacity coefficient of Pure Gas (VEOS) [Lee-Kesler]
Fugacity coefficient of Pure Gas (CEOS)
Z_int = 1
Gibbs energy at equilibrium
Delta_G = 0
Fugacity of Vapor and Liquid Phases at Equilibrium
fi_v = fi_l = fi_sat; for a pure species, the liquid and vapor phases are in equilibrium when they have the same temperature, pressure, and fugacity (criterion).
Fugacity Coefficient of Vapor and Liquid Phases at Equilibrium
Φi_v = Φi_l = Φi_sat
Fugacity of a Pure Subcooled Liquid
For Φi_sat: use EOS and evaluate at Pi_sat
Assumption for fugacity of pure subcooled liquid
Vi_l is considered as a property having weak dependence on pressure at a temperature less than the species’ critical temperature; assumed constant and equal to the saturated liquid-phase molar volume, V_sat
Chemical potential of species in solution
Fugacity of species in solution
Fugacity of ideal gas species in solution
fi_hat_ig = y_i*P = p_i
Phase equilibrium (fugacity)
Multiple 𝜋 phases with 𝑁 number of species at same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases. fi_hat_𝜶 = fi_hat_ 𝜷 = … = fi_hat_ π
Vapor-liquid equilibrium (fugacity)
fi_hat_v = fi_hat_l
Partial residual property
Partial residual Gibbs energy
Fugacity coefficient of species in solution
Φi_hat = fi_hat/yi*P
Fugacity coefficient of pure species in solution and partial residual Gibbs energy
Fundamental residual property relation (derivation)
Fundamental residual property relation (Gibbs energy)
At constant temperature, pressure, and, 𝑛_𝑗≠𝑖
Fugacity coefficient of species in solution (Z-terms)
Second virial coefficient (B)
For pure gas, only a function of temperature; for gas mixtures, function of both temperature and composition
Cross coefficient (B_ij)
Characterizes the bimolecular interaction between species i and j
Fugacity coefficient of species in solution (VEOS)
Vapor side expression of VLE criterion
For the vapor and liquid phase with N number of species at same T and P in equilibrium:
fi_hat_v = fi_hat_l
fi_hat_v = yi*Φi_hat*P
fi_hat_v = f(T, P, yi)
Ideal solution
Ideal mixture of gases, liquids, or solids
Ideal solution model partial molar Gibbs energy
mole fraction is denoted as 𝑥𝑖 since this model is often used for mixture of liquids; ideal gas mixture: xi → yi
Ideal solution volume
Ideal solution entropy
Ideal solution enthalpy
Lewis/Randall Rule (fugacity)
Applies to each species (solid, liquid, or gas) present in an ideal solution; fugacity of each species in an ideal solution (fid_hat) is directly proportional to its composition (xi), with proportionality constant equal to pure species fugacity (fi)
Lewis/Randall Rule (fugacity coefficient)
Fugacity coefficient of a species in an ideal solution, 𝜙i_hat_id, is equal to the pure species fugacity coefficient, 𝜙i.
Activity coefficient of species in ideal solution
γi = fi_hat/xi*fi
Excess property
Difference between real solution property and ideal solution property
Activity coefficient and excess Gibbs energy
Chemical potential in different solution models
Molecular interactions in different solution models
Fugacity and fugacity coefficients in different solution models
Gamma/Phi formulation of VLE
Assuming that Vi_l is weakly dependent on pressure at a temperature lower than the species’ critical temperature
Excess Properties and Residual Properties
Fundamental excess property relation
Equations for excess properties
Nature of Excess Properties
No meaning for pure species
Excess properties are often strong functions of temperature.
At normal temperatures, they are not strongly influenced by pressure.
Plot of G_E , H_E , and S_E vs. composition
All excess properties become zero as either species approaches purity.
Although G_E vs. x1 is approximately parabolic in shape, both H_E and TS_E exhibit individualistic composition dependencies.
When an excess property M_E has a single sign, the extreme value of M_E (maximum or minimum) often occurs near the equimolar composition.
VLE Data
At xi = yi = 0, fi_hat = 0
At xi = yi = 1, fi_hat = fi_ig = Pi_sat
For ideal gas, at xi = 1, ɣi = 1, at xi = 0, ɣi = indeterminate
Rational functions
No theoretical foundation
Cannot be extended to mixtures with more than two components
No temperature dependence
Local composition
Composition in a local volume around a molecule
May be different from overall mixture composition
Presumed to account for the short-range order and nonrandom molecular orientations arising from the molecule’s size and IMFs.
Local composition models
Founded on statistical mechanics instead of being arbitrary
Can be extended to multi-component systems while requiring only binary interaction parameters
Incorporates temperature dependence
Wilson Equation
Works well for mixtures of polar and nonpolar species, e.g. alcohols and alkanes.
Works well for hydrocarbon mixtures and is readily extended to multicomponent mixtures.
Wilson parameters, Λij and Λji however, must be positive to be valid for infinite dilution cases.
Use when components in the liquid phase are completely miscible over the entire composition range
Unable to describe systems exhibiting partial miscibility
NRTL Equation
Widely used for liquid-liquid extraction
The parameter α characterizes the tendency of species j and i to be distributed nonrandomly.
Not very appealing from a theoretical perspective, but its flexibility has led to a broad range of applications including combinations with electrolyte models.
Can be used for highly nonideal systems as well as for partially miscible systems
Preferred when limited solubility is a feature of the system
UNIQUAC Model
Uses local area fraction θij as the primary concentration variable
Comprised of
combinatorial effects due to differences in size and shape
residual effects due to differences in intermolecular forces
Often gives good representation of VLE and LLE for binary and multicomponent mixtures containing nonelectrolytes
For hydrocarbons, ketones, esters, water, amines, alcohols, nitriles, etc.
Local area fraction
Determined by representing a molecule by a set of bonded segments.
Each molecule is characterized by two structural parameters determined relative to a standard segment.
Local area fraction structural parameters
r : volume parameter → Relative number of segments per molecule
q : surface parameter → Relative surface area
UNIFAC Model
Theoretically-based way to estimate activity coefficients
Predictions can be made over a temperature range of 275-425 K and for pressures to a few atmospheres
All components must be condensable at near-ambient conditions
A molecule is viewed as an aggregate of functional groups, each comprised of subgroups.
The relative volume (Rk) and relative surface area (Qk) are properties of the subgroups.
A fluid mixture property is viewed as the sum of contributions from the molecules’ subgroups instead of the entire molecules.
Simple mixtures, with components similar in chemical nature and molecular size
One-parameter Margules
Moderately non-ideal binary mixtures
Margules, van Laar, Wilson, NRTL, UNIQUAC
Strongly non-ideal mixtures (e.g., alcohols + hydrocarbons)
Wilson
Solutions with miscibility gap (LLE)
NRTL, UNIQUAC
Thermodynamic consistency
High consistency if avg. |δln(ɣ1/ɣ2)| < 0.03
Accept if avg. |δln(ɣ1/ɣ2)| < 0.10
Barker’s method
An alternative method requiring regression that provides better thermodynamic consistency.
Find values of the parameters that minimize the error: P* - P
Data interpretation
Even if data satisfy the tests, the reliability of experimental data is still in question.
Stability criterion (G)
The equilibrium state of a closed system is that state for which the total Gibbs energy is a minimum with respect to all possible changes at the given T and P.
all irreversible processes occurring at constant T and P proceed in such a direction as to cause a decrease in the Gibbs energy of the system (negative, spontaneous).
mixed state must be the one of lower Gibbs energy with respect to the unmixed state.
Complete miscibility
Graph of ΔG vs x1 is all concave up (positive second derivative)
Immiscibility
ΔG vs x1 has concave down (negative second derivative)
when mixing occurs, a system can achieve a lower value of the Gibbs energy by forming two phases than by forming a single phase
Stability criterion for single-phase binary mixture
At fixed temperature and pressure, a single-phase binary mixture is stable if and only if ΔG and its first and second derivatives are continuous functions of x1, and the second derivative is positive.
Stability criterion (G_E/RT)
use activity coefficient models to test
If true, composition has single phase
Stability criterion (activity coefficient)
Stability criterion (fugacity)
Stability criterion (chemical potential)
Liquid-liquid equilibrium
When stability criterion breaks, liquid-liquid equilibrium exists. (unstable, 2 liquid phases)
Solubility diagram
T vs x1
Curve UAL - α phase (rich in species 2)
Curve UBL - β phase (rich in species 1)
x1_α and x1_β phase
At each temperature, these compositions are those for which the curvature of the ΔG vs. x1 curve changes sign.
Between these compositions, it is concave down (negative second derivative) and outside them it is concave up.
At these points, the curvature is zero (inflection points on ΔG vs. x1 curve).