Equilibrium, thermodynamic

The equilibrium constant Z of reaction / can be calculated from the change in Gibbs free energy at the temperature of the reaction (Equation 4.38). Also, the change of Gibbs free energy is a function of changes in enthalpy and entropy in the reaction (Equation 4.39)  

In Equations 4.40 and 4.41, H°j, 5 Sj, and ACpj are the standard enthalpy, standard entropy, and heat capacity of reaction y, respectively. By substituting Equations 4.39 through 4.41 in Equation 4.38, the following equation is derived  

Rearranging Equation 4.39, the standard entropy of reaction is obtained, which can be calculated at standard temperature Tq = 298.15 K, as follows  

The equilibrium constant K of reaction / as a function of standard Gibbs energy, standard enthalpy of reaction, and the temperature of the system, is obtained by substitution of Equation 4.43 in Eqnation 4.42  

UG] can be calculated from the standard Gibbs energy of formation (G]j), and 

CEA Chemical Equilibrium with Applications. Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Analysis, Gordon, S. and McBride, B. J. NASA Lewis Research Center, NASA Report 

STANJAN The Element Potential Method for Chemical Equilibrium Analysis Implementation in the Interactive Program STANJAN, W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the element potentials ) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates. 

EQUIL A.E. Lutz and F. Rupley, Sandia National Laboratories, Livermore, CA. A Fortran computer program (eqlib.f) for calculating chemical equilibrium using a modified solution procedure of STANJAN (stanlib.f, W.C. Reynolds, Stanford U.) and CHEMK1N data files for input. For the most recent versions, refer to the Reaction Design website http //www.reactiondesign.com/lobby/ open/index.html. 

GASEQ A Chemical Equilibrium Program for Windows. GASEQ is a PC-based equilibrium program written by C. Morley that can solve several different types of problems including composition at a defined temperature and pressure, adiabatic temperature and composition at constant pressure, composition at a defined temperature and at constant volume, adiabatic temperature and composition at constant volume, adiabatic compression and expansion, equilibrium constant calculations, and shock calculations. More information can found at the website http //www.arcl02.dsl.pipex.com/gseqmain.htm. 

The expression forthe classical distribution function in thennodynamic equilibrium reflects the Boltzmann equilibrium property 

Similarly, for a quantum system in thermal equilibrium, the populations of stationary states are given by the Boltzmann factors Pk and coherences between 

The thermal average of an observable represented by an operator A is, according to Eq. (10.14) 

For future reference we cite here without proof a useful identity that involves the harmonic oscillator Hamiltonian H = p /2m + (1 /2)ma q and an operator of the general formH = explaip + a2q with constant parameters ai and that is, the exponential of a linear combination of the momentum and coordinate operators. The identity, known as the Bloch theorem, states that the thermal average A )t (under the hannonic oscillator Hamiltonian) is related to the thennal average ((aip + Q 2 )2)t according to 

The expression for the classical distribution function inthermod mamic equihbrium reflects the Boltzmann equilibrium property 

Many reactions, such as esterifications, hydrogenations, and dimerizations, proceed to chemical equilibrium. The thermodynamic equilibrium provides information about the maximum possible conversion, which can not be exceeded even with the best catalysts or kinetic tricks. To assess the potential conversion it is essential to know how far the intended reactions are from chemical equilibrium (see Example 3.1.1-2). 

P(Tri), P(FA) = equilibrium partial pressure oftrioxane and formaldehyde, respectively, can be expressed [Busfield 1969] as  

It follows that if the process conditions assumed are T = 353 K and P(FA) = 0.2 bar, a maximum formaldehyde conversion of  

Chemical equilibrium is characterized by the fact that the free enthalpy of reaction is equal to zero (Equation 3.1.2-5)  

V = Stoichiometric coefficients (products -t, starting materials -) the Gibbs standard reaction enthalpy (7 = 298 K, P = 1.013 bar Equation 3.1.2-7)  

Reactive absorption processes occur mostly in aqueous systems, with both molecular and electrolyte species. These systems demonstrate substantially non-ideal behavior. The electrolyte components represent reaction products of absorbed gases or dissociation products of dissolved salts. There are two basic models applied for the description of electrolyte-containing mixtures, namely the Electrolyte NRTL model and the Pitzer model. The Electrolyte NRTL model [37-39] is able to estimate the activity coefficients for both ionic and molecular species in aqueous and mixed solvent electrolyte systems based on the binary pair parameters. The model reduces to the well-known NRTL model when electrolyte concentrations in the liquid phase approach zero [40]. 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

This model also requires different parameters, which are, however, pure interaction parameters, namely binary interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte) and ternary parameters (either two 

The calculation methods for the gas solubility are largely based on the Henry constant, which gives a relationship between the liquid-phase concentration of a physically dissolved gas and its partial pressure. The determination of such coefficients in presence of chemical reactions becomes complicated and, therefore, different estimations based on chemically inert systems are often applied. One of these methods uses the Henry coefficients of similar, but chemically inert, species in order to estimate the solubility of a reactive component An example is represented by the N2O analogy for the determination of CO2 solubility in amine solutions [47]. 

Chemical equilibrium state corresponds to the minimum value of the Gibbs free energy. Hence, the chemical equilibrium composition and the reaction direction can be predicted from the dependence of the Gibbs free energy on the reaction extend. For the reaction 

The fundamental driving force that prompts a molecule to diffuse within a polymer or transfer between a polymer and a surrounding phase is its chemical potential. Like the electrical potential of a battery causes electrons to flow through wires, chemical potential is the driving force in physical chemical phenomena. Substances will naturally tend to move from a higher chemical potential to a lower one. The equation for chemical potential is  

Pi° = chemical potential of substance I at a standard state R = universal gas constant T = temperature, K a = chemical activity 

The drive of a compound to move from a point of high chemical potential to a low one is equivalent, in most cases, to the tendency to equilibrate the compound s chemical potential in a phase, such as within a plastic container or a product, where mass transfer is physically able to occur in other words, the compound tends to reach thermodynamic equilibrium. 

The chemical potential, and thus the chemical activity of a substance is defined with respect to some standard, or reference state. Typically, the reference state chosen for a liquid or solid is the pure substance at the temperature under consideration. Eor a gas, an ideal gas at the temperature under consideration and one atmosphere pressure is the usual reference state. With this choice of reference 

Therefore, in order to understand mass transfer behavior in packaging applications, we have to be able to relate the activity of a substance in one phase to its activity in any contacting phase. As will be seen in the next section, for most packaging applications the activity can be reduced to concentration. 

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If an appreciable current flows between the electrode and the solution, thus disturbing the reversible thermodynamic equilibrium conditions, the electrode is said to be polarized and the system is then operating under irreversible conditions. 

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

The true thermodynamic equilibrium constant is a function of activity rather than concentration. The activity of a species, a, is defined as the product of its molar concentration, [A], and a solution-dependent activity coefficient, Ya. 

The true thermodynamic equilibrium constant, Ksp, for the solubility of AglOa, therefore, is... 

Several features of equation 6.50 deserve mention. First, as the ionic strength approaches zero, the activity coefficient approaches a value of one. Thus, in a solution where the ionic strength is zero, an ion s activity and concentration are identical. We can take advantage of this fact to determine a reaction s thermodynamic equilibrium constant. The equilibrium constant based on concentrations is measured for several increasingly smaller ionic strengths and the results extrapolated... 

A quantitative solution to an equilibrium problem may give an answer that does not agree with the value measured experimentally. This result occurs when the equilibrium constant based on concentrations is matrix-dependent. The true, thermodynamic equilibrium constant is based on the activities, a, of the reactants and products. A species activity is related to its molar concentration by an activity coefficient, where a = Yi[ ] Activity coefficients often can be calculated, making possible a more rigorous treatment of equilibria. 

In this experiment the equilibrium constant for the dissociation of bromocresol green is measured at several ionic strengths. Results are extrapolated to zero ionic strength to find the thermodynamic equilibrium constant. Equilibrium Constants for Calcium lodate Solubility and Iodic Acid Dissociation. In J. A. Bell, ed. Chemical Principles in Practice. Addison-Wesley Reading, MA, 1967. 

Gordus, A. A. Ghemical Equilibrium 1. The Thermodynamic Equilibrium Goncept, /. Chem. Educ. 1991, 68, 138-140. 

The amide formation reaction (highlighted by the circle) leads to the production of a hydrogen-bonded dimer (ZZ) of the reaction product Z with the template Z. The dimer is in thermodynamic equilibrium with free template in the reaction medium. 

Basic Thermodynamics. Equilibrium-phase behavior of mixtures is governed by the free energy of mixing and how this quantity, consisting of enthalpic... 

From a general point of view, the tautomeric studies can be divided into 12 areas (Figure 20) depending on the migrating entity (proton or other groups, alkyl, acyl, metals. ..), the physical state of the study (solid, solution or gas phase) and the thermodynamic (equilibrium constants) or the kinetic (isomerization rates) approach. 

Mass-Transfer Principles Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-hquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equihbrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a hquid are brought into contact. 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

Eroducts of reaction, the membrane reaclor can make conversion eyond thermodynamic equilibrium in the absence of separation. 

A more general, and for the moment, less detailed description of the progress of chemical reactions, was developed in the transition state theory of kinetics. This approach considers tire reacting molecules at the point of collision to form a complex intermediate molecule before the final products are formed. This molecular species is assumed to be in thermodynamic equilibrium with the reactant species. An equilibrium constant can therefore be described for the activation process, and this, in turn, can be related to a Gibbs energy of activation ... 

The anticipated content of impurities in the refined metal may be calculated a priori by assuming thermodynamic equilibrium at both metal/gas interfaces, and using the relevant stabilities of tire gaseous iodides. Adequate thermodynamic data could provide the activities of the impurities widr that of zirconium close to unity, but tire calculation of tire impurity transport obviously requires a knowledge of activity coefficients in the original impure material, which are not sufficiently well known. 

It follows that the position of thermodynamic equilibrium will change along the reactor for those reactions in which a change of tire number of gaseous molecules occurs, and therefore that the degree of completion and heat production or absorption of the reaction will also vaty. This is why the external control of the independent container temperature and the particle size of the catalyst are important factors in reactor design. 

A number of metals, such as copper, cobalt and h on, form a number of oxide layers during oxidation in air. Providing that interfacial thermodynamic equilibrium exists at the boundaries between the various oxide layers, the relative thicknesses of the oxides will depend on die relative diffusion coefficients of the mobile species as well as the oxygen potential gradients across each oxide layer. The flux of ions and electrons is given by Einstein s mobility equation for each diffusing species in each layer... 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

In the original announcement of the workshop the participants were told that everything was to be taken from methanol synthesis except the kinetics. Some may have interpreted this to mean that the known thermodynamic equilibrium information of the methanol synthesis is not valid when taken together with the kinetics. This was not intended, but... 

It is essential, however, to follow a r rous experimental protocol for such applications. To maintain the quantitadve character of NMR spectroscopy, the reped-tion rate of signal averaging experiments has to be at least five times the longest spin-latdce relaxadon dme present in the sample. This waiting period is necessary to ensure that the magnetizadon is probed in a reproducible state, corresponding to thermodynamic equilibrium. 

Local Thermodynamic Equilibrium (LTE). This LTE model is of historical importance only. The idea was that under ion bombardment a near-surface plasma is generated, in which the sputtered atoms are ionized [3.48]. The plasma should be under local equilibrium, so that the Saha-Eggert equation for determination of the ionization probability can be used. The important condition was the plasma temperature, and this could be determined from a knowledge of the concentration of one of the elements present. The theoretical background of the model is not applicable. The reason why it gives semi-quantitative results is that the exponential term of the Saha-Eggert equation also fits quantum-mechanical expressions. 

It is an inference naturally suggested by the general increase of entropy which accompanies the changes occurring in any isolated material system that when the entropy of the system has reached a maximum, the system will be in a state of equilibrium. Although this principle has by no means escaped the attention of physicists, its importance does not seem to have been duly appreciated. Little has been done to develop the principle as a foundation for the general theory of thermodynamic equilibrium (my italics). ... 

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

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