ChemE 123 LE2 Conceptual

Partial molar volume in an ideal-gas mixture

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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

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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)

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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

1. All excess properties become **zero** as either species approaches **purity**.
2. Although G_E vs. x1 is approximately parabolic in shape, both H_E and TS_E exhibit individualistic composition dependencies.
3. 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

1. r : volume parameter → Relative number of segments per molecule
2. 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).