Phase Modeling Tools

Name: Phase Modeling Tools | Applications to Gases
Brand: Wiley-ISTE
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Applications to Gases

Michel Soustelle(Autor*in)

Wiley-ISTE (Verlag)

Erschienen am 5. August 2015

300 Seiten

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978-1-119-17847-7 (ISBN)

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Inhalt

Cover
Title Page
Copyright
Contents
Preface
Notations
Symbols
1: Thermodynamic Functions and Variables
1.1. State variables and characteristic functions of a phase
1.1.1. Intensive and extensive conjugate variables
1.1.2. Variations in internal energy during a transformation
1.1.3. Characteristic function associated with a canonical set of variables
1.2. Partial molar parameters
1.2.1. Definition
1.2.2. Properties of partial molar variables
1.3. Chemical potential and generalized chemical potentials
1.3.1. Chemical potential and partial molar free enthalpy
1.3.2. Definition of generalized chemical potential
1.3.3. Variations in the chemical potential and generalized chemical potential with variables
1.3.3.1. Variations with intensive variables
1.3.3.2. Variations with composition
1.3.3.3. Total differential
1.3.4. Gibbs-Duhem relation
1.3.5. Generalized Helmholtz relations
1.3.6. Chemical system associated with the general system
1.4. The two modeling scales
2: Macroscopic Modeling of a Phase
2.1. Thermodynamic coefficients and characteristic matrices
2.1.1. Thermodynamic coefficients and characteristic matrix associated with the internal energy
2.1.2. Symmetry of the characteristic matrix
2.1.3. The thermodynamic coefficients needed and required to thermodynamically define the phase
2.1.4. Choosing other variables: thermodynamic coefficients and characteristic matrix associated with a characteristic function
2.1.5. Change in variable from one characteristic matrix to another
2.1.6. Relations between thermodynamic coefficients and secondary derivatives of the characteristic function
2.1.7. Examples of thermodynamic coefficients: calorimetric coefficients
2.2. Partial molar variables and thermodynamic coefficients
2.3. Common variables and thermodynamic coefficients
2.3.1. State equation
2.3.2. Expansion coefficients
2.3.2.1. Cubic expansion coefficient (or coefficient of relative increase in volume)
2.3.2.2. Coefficient of pressure increase at a constant volume
2.3.2.3. Isothermal compressibility coefficient at constant temperature T
2.3.2.4. Linear expansion coefficient
2.3.3. Molar heat capacities
2.3.4. Young's Modulus
2.3.5. Electric permittivity
2.3.6. Volumic and area densities of electric charge
2.4. Thermodynamic charts: justification of different types
2.4.1. Representation of a variable as a function of its conjugate
2.4.2. Representation of a characteristic function as a function of one of its natural variables
2.5. Stability of phases
2.5.1. Case of ensemble E0 of extensive variables
2.5.2. Coefficients associated with ensemble En
2.5.3. Case of other ensembles of variables
2.5.4. Conclusion: stability conditions of a phase in terms of thermodynamic coefficients
2.5.5. Example - applying stability conditions
2.6. Consistency of thermodynamic data
2.7. Conclusion on the macroscopic modeling of phases
3: Multi-Compound Phases - Solutions
3.1. Variables attached to solutions
3.1.1. Characterizing a solution
3.1.2. Composition of a solution
3.1.2.1. Molar fractions
3.1.2.2. Mass fractions
3.1.2.3. Concentrations or molarities
3.1.2.4. Molalities
3.1.3. Peculiar variables and mixing variables
3.1.3.1. Definitions
3.1.3.2. Molar variables of mixing
3.1.3.3. Molar Gibbs energy of mixing
3.1.3.4. Other molar variables of mixing
3.2. Recap of ideal solutions
3.2.1. Thermodynamic definition
3.2.2. Molar Gibbs energy of mixing of an ideal solution
3.2.3. Molar enthalpy of mixing of the ideal solution
3.2.4. Molar entropy of mixing of the ideal solution
3.2.5. Molar volume of mixing
3.2.6. Molar heat capacity of ideal solution: Kopp's law
3.3. Characterization imperfection of a real solution
3.3.1. Lewis activity coefficients
3.3.1.1. Coefficients of activity and reference states
3.3.1.2. Relation between the coefficients of activity of the components of a solution
3.3.1.3. Influence of temperature on the coefficients of activity
3.3.1.4. Influence of other intensive variables on the coefficients of activity
3.3.1.5. Another expression of the chemical potential in a condensed solution
3.3.1.6. Influence of composition on the coefficients of activity
3.3.1.7. Ideal solutions and coefficients of activity
3.3.2. Characterizing the imperfection of a real solution by the excess Gibbs energy
3.3.2.1. Definition of excess variables
3.3.2.2. Excess Gibbs energy
3.3.2.3. Excess entropy
3.3.2.4. Excess enthalpy
3.3.2.5. Excess heat capacity
3.3.3. Other ways to measure the imperfection of a solution
3.3.3.1. The osmotic coefficient method
3.3.3.2. The coefficient method F
3.4. Activity of a component in any solution: Raoult's and Henry's laws
3.5. Ionic solutions
3.5.1. Chemical potential of an ion
3.5.2. Relation between the activities of ions and the overall activity of solutes
3.5.3. Mean concentration and mean ionic activity coefficient
3.5.4. Obtaining the activity coefficient of an individual ion
3.5.5. Ionic strength
3.6. Curves of molar variables as a function of the composition in binary systems of a solution with two components
4: Statistics of Object Collections
4.1. The need to statistically process a system
4.1.1. Collections, system description - Stirling's approximation
4.1.2. Statistical description hypothesis
4.1.3. The Boltzmann principle
4.2. Statistical effects of distinguishable non-quantum elements
4.2.1. Distribution law
4.2.2. Calculation of a
4.2.3. Determining coefficient ß
4.2.4. Energy input to a system
4.2.5. The Boltzmann principle for entropy
4.3. The quantum description and space of phases
4.3.1. Wave functions and energy levels
4.3.2. Space of phases: discernibility of objects and states
4.3.3. Localization and non-localization of objects
4.4. Statistical effect of localized quantum objects
4.5. Collections of non-localized quantum objects
4.5.1. Eigen symmetrical and antisymmetric functions of non-localized objects
4.5.2. Statistics of non-localized elements with symmetrical wave functions
4.5.3. Statistics of non-localized elements with an asymmetric function
4.5.4. Classical limiting case
4.6. Systems composed of different particles without interactions
4.7. Unicity of coefficient ß
4.8. Determining coefficient a in quantum statistics
5: Canonical Ensembles and Thermodynamic Functions
5.1. An ensemble
5.2. Canonical ensemble
5.2.1. Description of a canonical ensemble
5.2.2. Law of distribution in a canonical ensemble
5.2.3. Canonical partition function
5.3. Molecular partition functions and canonical partition functions
5.3.1. Canonical partition functions for ensembles of discernable molecules
5.3.2. Canonical partition functions of indiscernible molecules
5.4. Thermodynamic functions and the canonical partition function
5.4.1. Expression of internal energy
5.4.2. Entropy and canonical partition functions
5.4.3. Expressing other thermodynamic functions and thermodynamic coefficients in the canonical ensemble
5.4.3.1. Helmholtz free energy
5.4.3.2. Pressure
5.4.3.3. Gibbs free energy
5.4.3.4. Chemical potential
5.4.3.5. Heat capacity at constant volume
5.5. Absolute activity of a constituent
5.6. Other ensembles of systems and associated characteristic functions
6: Molecular Partition Functions
6.1. Definition of the molecular partition function
6.2. Decomposition of the molecular partition function into partial partition functions
6.3. Energy level and thermal agitation
6.4. Translational partition functions
6.4.1. Translational partition function with the only constraint being the recipient
6.4.2. Translational partition function with the constraint being a potential centered and the container walls
6.5. Maxwell distribution laws
6.5.1. Distribution of ideal gas molecules in volume
6.5.2. Distribution of ideal gas molecules in velocity
6.6. Internal partition functions
6.6.1. Vibrational partition function
6.6.1.1. Diatomic molecules
6.6.1.2. Complex molecules
6.6.2. Rotational partition function
6.6.2.1. Heteronuclear diatomic molecules
6.6.2.2. Homonuclear diatomic molecules
6.6.2.3. Complex molecules
6.6.3. Nuclear partition function and correction of symmetry due to nuclear spin
6.6.4. Electronic partition function
6.7. Partition function of an ideal gas
6.8. Average energy and equipartition of energy
6.8.1. Mean translational energy
6.8.2. Mean rotational energy
6.8.3. Mean vibrational energy
6.9. Translational partition function and quantum mechanics
6.10. Interactions between species
6.10.1. Interactions between charged particles
6.10.1.1. Pairing interaction model
6.10.1.2. Ionic atmosphere
6.10.2. Interaction energy between two neutral molecules
6.10.2.1. The hard sphere model without force of interaction
6.10.2.2. The hard sphere model without Keesom repulsion force
6.10.2.3. The van der Waals force model
6.11. Equilibrium constants and molecular partition functions
6.11.1. Gaseous phase homogeneous equilibria
6.11.2. Liquid phase homogeneous equilibria
6.11.3. Solid phase homogenous equilibria
6.12. Conclusion on the macroscopic modeling of phases
7: Pure Real Gases
7.1. The three states of the pure compound: critical point
7.2. Standard state of a molecular substance
7.3. Real gas - macroscopic description
7.3.1. Pure gas diagram (P-V)
7.3.2. "Cubic" state equations
7.3.2.1. Van der Waals state equation
7.3.2.2. The RK state equation of Redlich and Kwong
7.3.2.3. SRK state equation of Soave, Redlich and Kwong
7.3.2.4. Peng and Robinson state equation
7.3.3. Other state equations
7.3.3.1. Dieterici equation
7.3.3.2. Development of the virial
7.3.3.3. Beattie-Bridgeman state equation
7.3.4. The theorem of corresponding states and the generalized compressibility chart
7.3.5. Molar Gibbs energy or chemical potential of a real gas
7.3.6. Fugacity of a real gas
7.3.6.1. Definition
7.3.6.2. Calculating the fugacity coefficient
7.3.6.3. Approximate value of the fugacity coefficient
7.3.6.4. Generalized chart for fugacity coefficients
7.3.7. Heat capacities of gases
7.3.7.1. Heat capacities at constant pressure
7.3.7.2. Heat capacities at constant volume
7.3.7.3. Variations in heat capacities with pressure or volume
7.4. Microscopic description of a real gas
7.4.1. Canonical partition function of a fluid
7.4.2. Helmholtz energy and development of the virial
7.4.3. Forms of the second coefficient of the virial
7.4.3.1. Hard sphere model without interaction force
7.4.3.2. Keesom hard-sphere model without repulsion force
7.4.3.3. The Lennard-Jones model
7.4.4. Macroscopic state equations and microscopic description
7.4.5. Chemical potential and fugacity of a real gas
7.4.6. Conclusion on microscopic modeling of a real gas
7.5. Microscopic approach of the heat capacity of gases
7.5.1. Classical theorem from the equipartition of energy
7.5.2. Quantum theorem of heat capacity at constant volume
8: Gas Mixtures
8.1. Macroscopic modeling of gas mixtures
8.1.1. Perfect solutions of perfect gases
8.1.1.1. Thermodynamic definition
8.1.1.2. Property
8.1.2. Mixture of real gases
8.2. Characterizing gas mixtures
8.2.1. Method of the state equations of gas mixtures
8.2.1.1. Van der Waals state equation
8.2.2. The Beattie-Bridgeman state equation
8.2.2.1. Development of the virial
8.2.2.1.1. Definitions of the virial coefficients for the mixture
8.2.2.1.2. Calculating the second coefficient of the virial of a mixture
8.2.3. Calculating the compressibility coefficient of a mixture
8.2.3.1. From coefficients of pure gases
8.2.3.2. Using reduced charts
8.2.4. Method using activity coefficients of solutions
8.3. Determining activity coefficients of a solution from an equation of state
8.3.1. Methodology
8.3.2. Studying solutions using the PSRK method
8.3.2.1. The modified state equation of Soave-Redlich-Kwong and the mixing laws
8.3.2.2. Parameters of the model and overview of results
8.3.3. VTPR Model
8.3.4. VGTPR Model
Appendices
Appendix 1: The Method of Lagrange Multipliers
A1.1. An overview of the problem
A1.2. Solving the problem using the method of Lagrange multipliers
A1.3. Determining the multiplier values
Appendix 2: Moments of Inertia of Molecules
Appendix 3: Mathematical Complements
A3.1. Homogeneous functions and Euler functions
A3.2. Exact total differentials
A3.2.1. Definition and identity of Maxwell
A3.2.2. Integrating factor
A3.3. Cross partial derivatives
A3.4. Elements of combinatorial analysis
A3.4.1. Arrangements
A3.4.1.1. Definition
A3.4.1.2. Arrangements with repetition
A3.4.1.3. Simple arrangements or without repetition
A3.4.2. Permutations
A3.4.2.1. Permutations without repetition
A3.4.2.2. Permutations with repetitions
A3.4.3. Combinations
A3.4.3.1. Definitions
A3.4.3.2. Simple combinations and combinations without repositioning
A3.4.3.3. Combinations with repetition
A3.4.3.4. Properties of combinations
Appendix 4: Constants and Units
A4.1. The international system of units (SI)
A4.2. Electrostatic formulae
Appendix 5: Function G
A5.1. Definition
A5.2. Recurrence relation and table of values
A5.3. Extension to negative values
Bibliography
Index

Symbols

A : Area of a surface or interface

: Hamaker constant between two media 1 and 2

A : affinity

: electrochemical affinity

AM : molar area

Am : molecular area

a : cohesion pressure of a gas; radius of the unit cell of a liquid

A, B, . : components of a mixture

a mix and b mix : terms for mixing of constants of a state equation

: coefficient of the virial in the increase in pressure

Bi : ith coefficient of the virial

b : co-volume of a gas, co-surface of an adsorbed gas

C : concentration, trace concentration of a voltage-pH diagram

: excess molar heat capacity at constant pressure

Ci : molar concentration (or molarity) of a component i

: mean concentration of ions in an ionic solution

: contribution of free electrons of a metal to the molar heat capacity

Cv(r) : contribution of rotational movements to the heat capacity at constant volume

Cv(t) : contribution of translational movements to the heat capacity at constant volume

Cv(v) : contribution of vibrational movements to the heat capacity at constant volume

CV, CP : heat capacity at constant volume and pressure

c : capacity of a capacitor; number of independent components

D : dielectric constant of the medium; protection or contact diameter of a molecule

: Debye function

d : distance between two liquid molecules

deS : exchange of entropy with the external environment

di : degree of oxidation i of an element A

diS : internal production of entropy

d : element of volume

E : energy of the system

E : Young's modulus

: Einstein function

E0 : internal energy associated with a reaction at temperature 0 K

E0 : standard electric potential or standard electromotive force of an electrochemical cell

Eabs : reversible electrical voltage of an electrochemical cell

Eb : balanced equation

: total mean energy of an element of a canonical ensemble

EC : total energy of canonical ensemble

EI : potential energy due to interactions

Ej : energy of an element j of canonical ensemble

Ekin : molar kinetic energy of electrons in a metal

Ep : set of variables at p intensive variables chosen to define a system

e : relative potential of an electrode

e0 : standard electrical potential (or normal potential) of an electrode

e0 : potential of equi-activity or equi-concentration of an electrode

eabs : absolute potential of an electrode

F : Helmholtz free energy

: excess molar free energy

: excess partial molar Helmholtz free energy of a component i

: partial molar Helmholtz free energy of mixing of a component i

: Helmholtz free energy, partial molar Helmholtz free energy of a component i

Fel : contribution of free electrons to molar Helmholtz free energy

: electrochemical Helmholtz free energy

: heterogeneous spreading parameter

fi : fugacity of component i in a gaseous mixture

Fm : molar Helmholtz free energy

: faraday

: molar Helmholtz free energy of a pure component i

: fugacity of a pure gas i

: excess Gibbs free energy

: electro-capillary Gibbs free energy

: electrochemical Gibbs free energy

: excess partial molar Gibbs free energy of a component

G : Gibbs free energy

: Gibbs free energy, partial molar Gibbs free energy of i, generalized Gibbs free energy

Gm : molar Gibbs free energy

: molar Gibbs free energy of mixing

: excess partial molar Gibbs free energy of a component

g : osmotic coefficient; acceleration of gravity; coefficients of degeneracy or multiplicity or statistical weight

molar Gibbs free energy of a component i

ga : statistical weight of fundamental electron level of the nucleus a

gi : coefficient of multiplicity of state i

g(e) : statistic weight of electron levels

g(r) : radial distribution function

: distribution of velocity onents according to Ox

g* : molar Gibbs free energy of gas i under a pressure of 1 atmosphere in a mixture

: standard molar enthalpy of formation at temperature T

: enthalpy, partial molar enthalpy of i

H : Hamiltonian

Hi, : resonance integral between two identical neighboring atoms

Hi,i : Coulombian integral between two identical neighboring atoms

H : magnetic field

: electrochemical enthalpy

: excess molar enthalpy

: molar enthalpy of mixing

: excess partial molar enthalpy of a component i

: partial molar enthalpy of mixing of component i

ht : spreading coefficient

h : stoichiometric coefficient of protons in an electrochemical reaction

h : Planck's constant

: molar enthalpy of a pure component i

hét : spreading coefficient of a liquid on another Harkins liquid

I : ionic strength of an ionic solution

Im : ionic strength reduced to molarities

I, I1, I2 I3 : moments of inertia.

II : configuration integral of the canonical translational partition function

i : Van 't Hoff factor

: partial molar value of the variable J relative to component i

: value of mixing of size relative to component

: partial molar value of mixing of size J relative to component i

: size of J relative to component i in a perfect solution

: partial molar value of size relative to component i in a perfect solution

: value of size J for the pure component i in the same segregation state

j : quantum rotation number

: Thermodynamic coefficient associated with the set of variables Ep, Xj is its variable of definition and Yi its function of definition

: equilibrium constant of the phase transformation Tr for component i

Kij : local composition weighting factor

Kads : adsorption equilibrium constant

KAX : solubility product of solid AX

: partition coefficient of compound i between both phases a and ß

Kd : dissociation constant

Kfe : adsorption equilibrium function

: equilibrium constant relative to concentrations

: equilibrium constant relative gacities

: equilibrium constant relative to partial pressures

Kr : equilibrium constant

Ks : solubility product

k : wave number

kB : Boltzmann constant

Lt : latent heat accompanying the transformation t

lc : capillary length

M : molar mass

M : magnetic moment; Madelung constant

m, ms : mass of solute s in grams per kg of solvent

m : total mass

mi : mass of component i

N : number of molecules in an ensemble

Na : Avogadro's number

NA : number of molecules of the component A

NC : number of elements in the canonic ensemble

Nc : Total number of cells in a liquid.

n : total number of moles in a solution or a mixture; quantum translation number

ni : number of objects i of system with energy ei.

: total number of moles in phase:a

: mean number of neighboring holes in a molecule in a liquid

ni : number of moles of component i

NL : total number of holes in a liquid

: critical pressure of a mixture

P : pressure of a gas

: sublimation vapor pressure of component i

: saturation vapor pressure of component i

: relative pressure of the mixture

Pc : critical pressure

pF : Fermi impulsion

Pi : partial pressure of component i

Pj : fraction of number of elements in a state j

p : number of external physical variables; spreading parameter

Q : heat balance

Qa : reaction quotient in terms of activities

QP : heat of transformation at constant pressure ; reaction quotient in terms of partial pressure

Qr : reaction quotient of transformation r

QV : heat of transformation at constant volume

q? : equilibrium heat of adsorption

qd : differential heat of adsorption.

qi : volumic fraction parameter

qisost : isosteric heat of adsorption

: reaction rate

R : ideal gas constant

R : radius of mean curvature of a surface; distillation reflux ratio

rA : radius of ionic atmosphere

r0 : distance of minimal energy between two molecules

rc : radius of a cylindrical tube

ri : parameter of volumic fraction

rK : Kelvin's radius

s : parameter of order of an alloy

: molar entropy of mixing

: excess partial molar...

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