Equilibria calculation, 357 relationship

Equation 4.26 defines the relationship between the vapor and liquid mole fractions and provides the basis for vapor-liquid equilibrium calculations on the basis of equations of state. Thermodynamic models are required for (/) and [ from an equation of state. Alternatively, Equations 4.21, 4.22 and 4.25 can be combined to give... 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

Roult s law is known to fail for vapour-liquid equilibrium calculations in polymeric systems. The Flory-Huggins relationship is generally used for this purpose (for details, see mass-transfer models in Section 3.2.1). The polymer-solvent interaction parameter, xo of the Flory-Huggins equation is not known accurately for PET. Cheong and Choi used a value of 1.3 for the system PET/EG for modelling a rotating-disc reactor [113], For other polymer solvent systems, yj was found to be in the range between 0.3 and 0.5 [96],... 

Combining the IAS theory with the Gibbs equation for isothermal adsorption gives the relationship necessary for equilibrium calculations ... 

The ion product, Qsp, is an expression that is identical to the solubility product constant, but its value is calculated using concentrations that are not necessarily those at equilibrium. (The relationship between the expression for solubility product, Kgp, and the expression for the ion product, Qsp, is analogous to the relationship between the equilibrium constant, iQ, and the reaction quotient, Qc.)... 

If a fluid composed of more than one component (e.g., a solution of ethanol and water, or a crude oil) partially or totally changes phase, the required heat is a combination of sensible and latent heat and must be calculated using more complex thermodynamic relationships, including vapor-liquid equilibrium calculations that reflect the changing compositions as well as mass fractions of the two phases. 

Vapor/liquid equilibrium (VLE) relationships (as well as other interphase equilibrium 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 binary 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. 

It is also important to calculate the eqnilibrinm concentration of holes in a doped n-type semicondnctor and of electrons in a doped p-type semicondnctor. These minority carrier concentrations are readily obtained nsing the concepts described above, becanse equations (5) and (6) are actually more general than has been indicated. As mentioned above, the relationship between n and pi in equation (5) is actually an equilibrium constant relationship between the electron and hole concentrations in the sohd. In the discussion above, this expression was derived under the special constraint that = pi, that is under thermal excitation conditions. However, the equilibrium constant relationship on the right-hand side of equations (5) and (6) must hold regardless of the somce of electrons and holes, so it applies to both doped and intrinsic semiconductor samples. We thus obtain... 

Several chemical reactions, including calcium carbonate and hydroxyapatite precipitation, have been studied to determine their relationship to observed water column and sediment phosphorus contents in hard water regions of New York State. Three separate techniques have been used to Identify reactions important in the distribution of phosphorus between the water column and sediments 1) sediment sample analysis employing a variety of selective extraction procedures 2) chemical equilibrium calculations to determine ion activity products for mineral phases involved in phosphorus transport and 3) seeded calcium carbonate crystallization measurements in the presence and absence of phosphate ion. 

The sodium acetate solution is an example of an important general case. For any salt whose cation has neutral properties (such as Na+ or K+) and whose anion is the conjugate base of a weak acid, the aqueous solution will be basic. The Kb value for the anion can be obtained from the relationship Kb = Kw/Ka. Equilibrium calculations of this type are illustrated in Example 7.11. 

To carry out an appropriate flash calculation, the pressure, P, and the temperature, T, must be known. If the values of P and T in the separating vessel are fixed, the value of P must not be so high that the two phases cannot exist at any value of T. Nor must T lie outside the bubble point and dew point range corresponding to P. For a valid two-phase equilibrium calculation, the following relationship must be satisfied ... 

When the equilibrium curve is always concave downward, the minimum reflux ratio can be calculated algebraically. The required relationship can be developed by solving simultaneously equations (6-18), (6-31), and the equilibrium-distribution relationship y =J x). The three unknowns are the coordinates of the point of intersection of the enriching operating line and the q-... 

Write a Mathcad program to calculate the minimum reflux ratio under these conditions and test it with the data of Example 6.4. In this case, the equilibrium-distribution relationship is a table of VLE values therefore, an algebraic relationship of the form y =J x) must be developed by cubic spline interpolation. 

Equilibrium Calculation We now turn our attention to the equilibrium for the ionization of acetic acid, the relationship that determines the buffer pH ... 

Although usually the minimum liquid/gas ratio can be calculated based on the assumption of equilibrium at the bottom of the column where both the gas and liquid have the highest concentration of solute, this is not always the case. Occasionally, the equilibrium curve may be shaped in a manner that causes it to touch the operating line at some point near the middle of the column. Such a case is illustrated in Fig. 6.1-7b with operating line B again representing the minimum UG ratio. This type of operating line-equilibrium curve relationship may be caused by heat effects that alter the equilibrium conditions within the column. 

The last equation is a very important relationship for the thermodynamics of mixtures, and it is the starting point for practically all phase equilibrium calculations. 

For complicated equilibrium calculations, the G-minimization technique is a useful option for the evaluation of phase equilibria, especially if both phase and reaction equilibria are involved. In contrast to the equilibrium conditions, the minimum of G is not only a necessary but a sufficient equilibrium condition. As well, for complicated equilibria it is often the only way to keep the overview. The only knowledge that must be available is the functional relationship for the Gibbs energy g and a clear concept for the minimum evaluation task. The following example shall illustrate the method. 

Table 1 Equilibrium constants relationships, equations to calculate the mole fraction of the monomer (a) and hydrodynamic volume (Fh) for simple and parametric equal K (EK), attenuated K (AK), and incremental K (IK) models. Parameters p, r, and 0 are used to make the first equilibrium constant different from the others. The s)mbol L is given by L = KC, where C is the total molar concentration of the species as a monomer or /-mer.

In order to perform phase-equilibrium calculations, fugacity coefficients can be calculated from the thermodynamic relationship... 

Eor the equilibrium calculation in the decanter, the following relationship is used ... 

Chemical equilibrium calculations are frequently accompanied by determinations of enthalpy changes involved in the overall chemical conversion process. Since such calculations are important for successful industrial application of newly developed processes, we are presenting at least the basic relationships required for judging enthalpy changes. The procedure to be described in the following is in fact a development of the NASA method of determining the coordinates (H, P). 

Before discussing equilibrium calculations, one should understand the nature of the equilibrium relationships in multi-component hydrocarbon systems and the regions in which each calculation is apph-cable. A typical pressure-temperature diagram is shown in Figure 2.1. A specific diagram of this type could be drawn from any system of fixed composition. The actual pressure and temperature coordinates will be different for various compositions. 

In this chapter, we aim to deepen our understanding of chemical equilibrium. We will begin with a brief discussion of the nature of the equilibrium state and then focus on some key relationships involving equilibrium constants. Then we will make qualitative predictions about the condition of equilibrium finally, we will perform various equilibrium calculations. As we will discover throughout the remainder of the text, the equilibrium condition plays a role in numerous natural phenomena and affects the methods used to produce many important industrial chemicals. 

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

In a reservoir at initial conditions, an equilibrium exists between buoyancy forces and capillary forces. These forces determine the initial distribution of fluids, and hence the volumes of fluid in place. An understanding of the relationship between these forces is useful in calculating volumetries, and in explaining the difference between free water level (FWL) and oil-water contact (OWC) introduced in the last section. 

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