Equations Describing Phase Equilibrium Relationships

EQUATIONS DESCRIBING PHASE EQUILIBRIUM RELATIONSHIPS 2.2.1 Liquid Vapor Pressure... 

Furthermore, if the moiecules condense, the total enthalpy given up would include the latent heat. Sherwood et al. describe how the above equations along with atqnopriate mass transfer expressions and phase equilibrium relationships are employed in the analysis of condensers and other processes involving simultaneous heat transfer and mass transfer. 

The remaining boundary conditions are the phase equilibrium relationships at the interface given by the applieable phase diagram. It may be convenient in calculations to describe the eoexistence curve and tie-hnes by empirical equations such as those proposed by Hand (1930). Whatever the details of the method used, the problem is well posed cmee initial compositions, diffusion coefficients, and equihbrimn data are speeified. The solution yields the concentration profiles in both phases. 

The nonlinear algebraic equations that describe a steady-state distiUalion column consist of component balances, energy balances, and vapor-liquid phase equilibrium relationships. These equations are nonlinear, particularly those describing the phase equilibrium of azeotropic systems. Unlike a linear set of algebraic equations that have one unique solution, a nonlinear set can give multiple solutions therefore, the possibility of multiple steady states exists in azeotropic distillation. 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

So far we have made no special assumptions as to the nature of the Phases A and B in deriving Equation (8.10). Evidently the Clapeyron equation is applicable to equilibrium between any two phases of one component at the same temperature and pressure, and it describes the functional relationship between the equilibrium pressure and the equilibrium temperature. 

Equilibrium The physical process (reaction) of adsorption or ion exchange is considered to be so fast relative to diffusion steps that in and near the solid particles, a local equilibrium exists. Then, the so-called adsorption isotherm of the form q = f(Ce) relates the stationary and mobile-phase concentrations at equilibrium. The surface equilibrium relationship between the solute in solution and on the solid surface can be described by simple analytical equations (see Section 4.1.4). The material balance, rate, and equilibrium equations should be solved simultaneously using the appropriate initial and boundary conditions. This system consists of four equations and four unknown parameters (C, q, q, and Ce). 

The concept of equilibrium distribution is another area where names can cause much confusion. The equilibrium distribution of a compound between the gas and liquid phase has been expressed in various forms, i. e. Bunsen coefficientfi, solubility ratio s, Henry s Law constant expressed dimensionless Hc, or with dimensions H. These are summarized in along with equations showing the relationships between them. Another more general term to describe the equilibrium concentrations between two phases is the partition coefficient, denoted by K. It is often used to describe the partitioning of a compound between two liquid phases. 

The composition boundary values entering into Eqs. (All) represent external values for Eqs. (A10). With some further assumptions concerning the diffusion and reaction terms, this allows an analytical solution of the boundary-value problem [Eqs. (A10) and (All)] in a closed matrix form (see Refs. 58 and 135). On the other hand, the boundary values need to be determined from the total system of equations describing the process. The bulk values in both phases are found from the balance relations, Eqs. (Al) and (A2). The interfacial liquid-phase concentrations xj are related to the relevant concentrations of the second fluid phase, y , by the thermodynamic equilibrium relationships and by the continuity condition for the molar fluxes at the interface (57,135). 

The basic equations describing a single stage in a fractionator in which chemical reaction may occur include component material balances, vapor-liquid equilibrium relationships, and energy balance, and restrictions on the liquid vapor phase mol fractions. The model equations for stage j may be expressed as follows ... 

The thermodynamic equilibrium constants shown by Equations 3.15 and 3.16 match the stoichiometric (or concentration based-) constants of stoichiometric models (see Equations 1 and 3 of Reference 1). Since the latter neglect the modulation of the adsorption of a charged species by the surface potential, they are not constant [19] after the addition of the IPR in the mobile phase. Stoichiometric relationships [19] represent only the ratio of equilibrium concentrations and cannot describe equilibrium in the presence of electrostatic interactions. In their stoichiometric approach. 

In gas absorption operations the equilibrium of interest is that between a relatively nonvolatile absorbing liquid (solvent) and a solute gas (usually the pollutant). As described earlier, the solute is ordinarily removed from a relatively large amount of a carrier gas that does not dissolve in the absorbing liquid. Temperature, pressure, and the concentration of solute in one phase are independently variable. The equilibrium relationship of importance is a plot (or data) of x, the mole fraction of solute in the liquid, against y, the mole fraction in the vapor in equilibrium with x. For cases that follow Henry s law, Henry s law constant m, can be defined by the equation... 

As stated above, the analysis of a separation process uses mass balances in conjunction with some specific relation(s) which describe the separation process. For an equilibrium-stage process, this specific relation is the equilibrium relationship that describes the concentration of a component in each phase with respect to each other exiting the stage. Note that the equations can be solved for a single stage once the equilibrium relationship is known. It does not have to be a linear one. 

In the interest of simplicity, the equilibrium relationship given by Eq. (1-10) is used in the following developments. The state of equilibrium for a two-phase (vapor and liquid) system is described by the following equations where any... 

Interfacial Potential. The relationship that describes ionic equilibrium on the ITIES and accounts for all the present ions was first described by Hung (15). The derivation is based on the equality of the electrochemical potentials in either phase for all ions involved and the requirement of electroneutrality in each solvent. The equation is given as a summation over all present ions, i ... 

Equation (2.10), together with the partial equilibrium relationships, determines a fully ignited composition which approximately describes the junction between the ignition phase of the reaction and the purely decelerating association phase which follows it. The sizable concentrations of the intermediates OH, H and O in this composition account for the magnitudes of the essential overshoots of these species which arise early in the main reaction simply because N must decrease in order for the reaction to be completed. To be sure, individual excursions of the concentrations of these intermediates above these fully ignited partial equilibrium values are possible before the individual bimolecular equilibria are approached, but such excursions are short-lived in comparison to the overshoots that depend upon association for their removal. Under many drciunstances, particularly in near-stoichiometric mixtures, no such excursions occur and the fully ignited composition represents upper bounds to the observed concentrations of the intermediates. 

In nonlinear adsorption systems where parallel diffusion mechanisms hold, the mass balance equation given in eq. (9.2-lb) is still valid. The difference is in the functional relationship between the concentrations of the two phases, that is the local adsorption isotherm. In general, this relationship can take any form that can describe well equilibrium data. Adsorption isotherm such as Langmuir, Unilan, Toth, Sips can be used. In this section we present the mathematical model for a general isotherm and then perform simulations with a Langmuir isotherm as it is adequate to show the effect of isotherm nonlinearity on the dynamics behaviour. The adsorption isotherm takes the following functional form ... 

The Young-Laplace equation (equation 5) describes the force balance in terms of capillary pressure for two fluid phases in contact with each other and a surface. If one of the phases is present as a thin film, the equilibrium relationship that accounts for the thin film is the augmented Young-Laplace equation [6-S],... 

The statistical thermodynamic approach to the derivation of an adsorption isotherm goes as follows. First, suitable partition functions describing the bulk and surface phases are devised. The bulk phase is usually assumed to be that of an ideal gas. From the surface phase, the equation of state of the two-dimensional matter may be determined if desired, although this quantity ceases to be essential. The relationships just given are used to evaluate the chemical potential of the adsorbate in both the bulk and the surface. Equating the surface and bulk chemical potentials provides the equilibrium isotherm. 

Above equilibrium is dependent on the mobile-phase pH and the relationship between ionic and nonionic form of the analyte is described by Henderson-Hasselbalch equation... 

For the design of rectification columns, the equilibrium concentrations of the gas and liquid phases at constant pressure are of decisive importance. The McCabe-Thiele diagram (Figure 2.3.2-4), based on Raoult s law, describes this relationship (Equation 2.3.2-9) ... 

Vapor-liquid equilibria can easily be measured experimentally and described mathematically. Equation (3.3-1) is the general relationship for the equilibrium between a liquid phase and an ideal gas phase ... 

Equation (10) is not applicable to the two-phase systems described by regimes II and III. For this case, temperature-vapor pressure equilibrium is the ruling relationship. 

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