Equilibrium Relationships

The vapor-liquid equilibrium relationship can be deflned in terms of K values by... 

The vapor pressure of a crude oil at the wellhead can reach 20 bar. If it were necessary to store and transport it under these conditions, heavy walled equipment would be required. For that, the pressure is reduced (< 1 bar) by separating the high vapor pressure components using a series of pressure reductions (from one to four flash stages) in equipment called separators , which are in fact simple vessels that allow the separation of the two liquid and vapor phases formed downstream of the pressure reduction point. The different components distribute themselves in the two phases in accordance with equilibrium relationships. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Thiourea (II) may be obtained from ammonium thiocyanate (I) by an isomeric change analogous to ammonium cyanate, but the equilibrium relationship is very different (compare Section 111,133) ... 

The fundamental equilibrium relationships we have discussed in the last sections are undoubtedly satisfied to the extent possible in polymer crystallization, but this possibility is limited by kinetic considerations. To make sense of the latter, both the mechanisms for crystallization and experimental rates of crystallization need to be examined. 

H. E. Bamer, H. Beisswenger, and K. E. Bamer, "Chemical Equilibrium Relationships AppHcable in Fluid Bed Combustion," Proceedings of the Ninth International Conference on Fluidi d-Bed Combustion, Boston, Mass., May 4—7,1987. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

In the rectifying section, the equilibrium relationship for component at any stage n can be expressed in terms of component flow rate in the distillate d = Dxd and component absorption factor A = hjKiS... 

When chemical equilibrium is achieved qiiickly throughout the liquid phase (or can be assumed to exist), the problem becomes one of properly defining the physical and chemical equilibria for the system. It sometimes is possible to design a plate-type absorber by assuming chemical-equilibrium relationships in conjunction with a stage efficiency factor as is done in distillation calculations. Rivas and Prausnitz [Am. Tn.st. Chem. Eng. J., 25, 975 (1979)] have presented an excellent discussion and example of the correct procedures to be followed for systems involving chemical equihbria. 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

Equilibrium concentrations of HOU and OCl depend on the pH of the wastewater. Increasing the pH shifts the preceding equilibrium relationships to the right, causing the formation of higher concentrations of HOCl. 

Solution characteristics eomposition, equilibrium relationships (solubility), metastable zone width, purity, partition eoeffieient, liquid density, viseosity, and their temperature dependenee (Chapter 3). 

Equation (5-69) describes rate-equilibrium relationships in terms of a single parameter, the intrinsic barrier AGo, which therefore assumes great importance in interpretations of such data. It is usually assumed that AGo is essentially constant within the reaction series then it can be estimated from a plot of AG vs. AG° as the value of AG when AG = 0. Another method is to fit the data to a quadratic in AG and to find AGq from the coefficient of the quadratic term. ... 

Figure 5-17. Rate-equilibrium relationship according to Eq. (5-69) for a hypothetical system with AG = 20.

There are several equations other than the Marcus equation that describe rate-equilibrium relationships. Murdoch writes all of these equations in the general form... 

Although it is not the intent of this chapter to evaluate the methods and techniques for establishing the equilibrium relationships, selected references will be given for the benefit of the designer s pursuit of more detail. This subject is so detailed as to require specialized books for adequate reference such as Prausnitz [54]. 

In reaction (7), all of the molecular species involved in the equilibrium are in the solution as dissolved species. Though the equilibrium relationship that exists among the concentrations is a little more complicated than in the solubility product expressions, the guiding principles are the same. 

In pure water, where the only source of ions is reaction (6), the concentrations of H+(aq) and OH (aq) must be equal. But what if we add some HC1 to the solution We have already noted that HQ is a strong electrolyte, dissolving to give the ions H+(aqJ and G (aq). Thus, hydrogen chloride adds H+(aq) but not OH (aq) to the solution. The concentrations [H+] and [OH-] are no longer equal. However, they are still found to be tied together by the equilibrium relationship... 

Adsorption, like extraction, depends on equilibrium relationships. Isothermal adsorption is projected by Langmuir isotherms. The model is shown in Figure 7.14, which is based on the linear model of the following equation ... 

Changes in free energy and the equilibrium constants for Reactions 1, 2, 3, and 4 are quite sensitive to temperature (Figures 2 and 3). These equilibrium constants were used to calculate the composition of the exit gas from the methanator by solving the coupled equilibrium relationships of Reactions 1 and 2 and mass conservation relationships by a Newton-Raphson technique it was assumed that carbon was not formed. Features of the computer program used were as follows (a) any pressure and temperature may be specified (b) an inert gas may be present (c) after... 

In Scheme 7-1 kx and kn refer to the rate constants for a benzene derivative, in our case the benzenediazonium ion, bearing a substituent X in the 3- or 4-position, and the corresponding unsubstituted benzene derivative respectively. The term p is Hammett s reaction constant for the reaction, and o is Hammett s substituent constant which is, at least in principle, independent of the nature of the reaction but different for the 3- and 4-positions. A plot of log kx (or log kx - log kH) versus o should give a straight line. Its slope (positive or negative) corresponds to the reaction constant p. Equilibrium relationships are treated analogously. 

In this section, consideration will be given to the equilibrium relationships between shear stress and shear rate for fluids exhibiting non-Newtonian behaviour. Whenever the shear stress or the shear rate is altered, the fluid will gradually move towards its new equilibrium state and, for the present, the period of adjustment between the two equilibrium states will be ignored. 

In processing, it is frequently necessary to separate a mixture into its components and, in a physical process, differences in a particular property are exploited as the basis for the separation process. Thus, fractional distillation depends on differences in volatility. gas absorption on differences in solubility of the gases in a selective absorbent and, similarly, liquid-liquid extraction is based on on the selectivity of an immiscible liquid solvent for one of the constituents. The rate at which the process takes place is dependent both on the driving force (concentration difference) and on the mass transfer resistance. In most of these applications, mass transfer takes place across a phase boundary where the concentrations on either side of the interface are related by the phase equilibrium relationship. Where a chemical reaction takes place during the course of the mass transfer process, the overall transfer rate depends on both the chemical kinetics of the reaction and on the mass transfer resistance, and it is important to understand the relative significance of these two factors in any practical application. 

Each of these processes is characterised by a transference of material across an interface. Because no material accumulates there, the rate of transfer on each side of the interface must be the same, and therefore the concentration gradients automatically adjust themselves so that they are proportional to the resistance to transfer in the particular phase. In addition, if there is no resistance to transfer at the interface, the concentrations on each side will be related to each other by the phase equilibrium relationship. Whilst the existence or otherwise of a resistance to transfer at the phase boundary is the subject of conflicting views"8 , it appears likely that any resistance is not high, except in the case of crystallisation, and in the following discussion equilibrium between the phases will be assumed to exist at the interface. Interfacial resistance may occur, however, if a surfactant is present as it may accumulate at the interface (Section 10.5.5). 

The relation between CAi[ and CAi2 is determined by the phase equilibrium relationship since the molecular layers on each side of the interface are assumed to be in equilibrium with one another. It may be noted that the ratio of the differences in concentrations is inversely proportional to the ratio of the mass transfer coefficients. If the bulk concentrations, CAt> and CA02 are fixed, the interface concentrations will adjust to values which satisfy equation 10.98. This means that, if the relative value of the coefficients changes, the interface concentrations will change too. In general, if the degree of turbulence of the fluid is increased, the effective film thicknesses will be reduced and the mass transfer coefficients will be correspondingly increased. 

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

Besmann, T. M., SOLGASMIX-PV, A Computer Program to Calculate Equilibrium Relationship in Complex Chemical Systems, ORNL/TM-5775, Oak Ridge National Laboratory, Oak Ridge, TN( 1977)... 

Independent Reactions. In this section, we consider the number of independent reactions that are necessary to develop equilibrium relationships between N chemical species. A systematic approach is the following ... 

The conversion is carried out using the equilibrium relationship between the gas- and liquid-phase concentrations. Usual practice is to assume Henry s law. Thus, the gas-phase concentration that is equivalent to u is Kh ai, where Kh is... 

The mass transfer coefficients, Kg and Ky, are overall coefficients analogous to an overall heat transfer coefficient, but the analogy between heat and mass transfer breaks down for mass transfer across a phase boundary. Temperature has a common measure, so that thermal equilibrium is reached when the two phases have the same temperature. Compositional equilibrium is achieved at different values for the phase compositions. The equilibrium concentrations are related, not by equality, as for temperature, but by proportionality through an equilibrium relationship. This proportionality constant can be the Henry s law constant Kh, but there is no guarantee that Henry s law will apply over the necessary concentration range. More generally, Kyy is a function of composition and temperature that serves as a (local) proportionality constant between the gas- and liquid-phase concentrations. 

Membrane Reactors. Consider the two-phase stirred tank shown in Figure 11.1 but suppose there is a membrane separating the phases. The equilibrium relationship of Equation (11.4) no longer holds. Instead, the mass transfer rate across the interface is given by... 

These component balances are conceptually identical to a component balance written for a homogeneous system. Equation (1.6), but there is now a source term due to mass transfer across the interface. There are two equations (ODEs) and two primary unknowns, Og and a . The concentrations at the interface, a and a, are also unknown but can be found using the equilibrium relationship, Equation (11.4), and the equality of transfer rates. Equation (11.5). For membrane reactors. Equation (11.9) replaces Equation (11.4). Solution is possible whether or not Kjj is constant, but the case where it is constant allows a and a to be eliminated directly... 

To study the equilibrium relationship between the dye uptake and the final pH, the pH edge experiments were carried out according to previously reported method [5]. 

The equilibrium relationships found by Sorrell (1977) were valid only for room temperature (22+2 °C) and, because samples were allowed to cure in sealed containers, for equilibrium water vapour pressures determined by the assembly of phases present. The phases which exist under such conditions were quite unequivocally found to be 4 1 5 and 1 1 2. However Sorrell pointed out that it is entirely possible that lower hydration states of either phase could be stable at higher temperatures or lower humidities. In particular the 4 1 4 phase (Feitknecht, 1933) may well be such a phase, particularly as one of the five waters of hydration is known to be held only loosely in the structure. Indeed, Sorrell reported that he observed a slight shoulder on the larger dehydration peak of the DTG curve of the 4 1 5 phase that might be assigned to the loss of this first water molecule. He did not, however, succeed in isolating or characterizing a 4 1 4 phase. 

Since temperature of formation of carbonates can be estimated from homogenization temperature of fluid inclusions in carbonates, we can place a limit of CO2 from the above equilibrium relationships. The estimated CO2 range is 1-0.01 mol/kgH20. 

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