Equilibrium calculations simplifying assumptions

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

Dynamic Performance. Most models do not attempt to separate the equilibrium behavior trom the mass-transfer behavior. Rather they treat adsorption as one dynamic process with an overall dynamic response of the adsorbent bed to the feed stream. Although numerical solutions can be attempted for the rigorous partial differential equations, simplifying assumptions are often made to yield more manageable calculating techniques. 

One method of obtaining equilibrium concentrations of the various species involves no simplifying assumptions. In this method as many independent equations relating the unknown quantities are obtained as there are unknowns, in which a mathematical statement for each concentration and a mathematical equation for each equilibrium reaction can be written. These are then solved. For complex systems the calculations may be cumbersome. Sillen developed a computer program HALTAFALL (available in FORTRAN) by which equilibrium concentrations of species can be calculated thout simplifying assumptions. Precise values of equilibrium constants and activity coefficients are required for best results. 

These calculations yield, subject to some simplifying assumptions, relative T-site alumimun substitution energies computed (1) for the thermodynamic equilibrium state, (2) at zero K and (3) for models devoid of non-firamework species. Framework zeolites, metastable structures, are produced under luetic control and if, as indicated by the most recent calculations, the relative T-site substitution energies for the (Cerent sites are not grossly disparate, the actual distributions in reed materitds will be determined by the particular conditions of synthesis. As the molecular-level mechanisms of zeolite sjmthesis remain obscure, we especially need some experimental indicator of which sites are actually adopted by aluminum in real MFI-framework materials. 

Many students find Step 7a to be troublesome because they fear that making invalid approximations will lead to serious errors in their computed results. Such fears are groundless. Experienced scientists are often as puzzled as beginners when making an approximation that simplifies an equilibrium calculation. Nonetheless, they make such approximations without fear because they know that the effects of an invalid assumption will become obvious by the time a computation is completed (see Example 11-6). It is a good idea to try questionable assumptions early during... 

The IAS theory was later extended to account for the adsorption of gas mixtures on heterogenous surfaces [52,53]. It was also extended to calculate the competitive adsorption isotherms of components from hquid solutions [54]. At large solute loadings, the simplifying assumptions of the LAS theory must be relaxed in order to account for solute-solute interactions in the adsorbed phase. The IAS model is then replaced by the real adsorbed solution (RAS) model, in which the deviations of the adsorption equilibrium from ideal behavior are lumped into an activity coefficient [54,55]. Note that this deviation from ideal beha dor can also be due to the heterogeneity of the adsorbent surface rather than to adsorbate-adsorbate interactions, in which case the heterogeneous IAS model [55] should be used. 

In order to calculate the spectrum, the chemical-shielding tensors of all spins, as well as their relative orientation and distances, have to be known. The assumption of a quasi-equilibrium state simplifies Equation (4.28) considerably. For long mixing times t , all elements in exp(Wr ,) have equal intensity. This leads to the following signal function for a single-crystallite orientation ... 

We neglected in the above calculation any consumption of ammonia in forming the complexes. We see that 20% of it was consumed, as a first approximation. If we were to recalculate the P s at 0.08 M NH3, P2 would still be equal to 1.0, and most of the silver would still exist as AgCNHs)". The relative values of Po and Pi would change, however. This is an iterative procedure or method of successive approximations. It can be used in any equilibrium calculation in which assumptions are made to simplify the calculations, including simple acid-base equilibria where the amount of acid dissociated is assumed negligible compared to the... 

If the initial concentration of reactant is large relative to the change in concentration to reach equilibrium, we can make the simplifying assumption that the change can be neglected in performing calculations. 

Jaques and Furter [95] derived an equation for the salt effect on the water-vapour equilibrium in binary mixtures which correlates the temperature and the liquid concentration of the three components ethanol, water and salt. The equation has 6 constants. The theory of the salt effect has been discussed by Furter and Meranda [96]. On the basis of simplifying assumptions Sada et al. [97] have established a relation for the calculation of vapour-liquid equilibria for non-aqueous binary systems in which the salt is dissolved only in one component e.g., benzene-ethanol with lithium or calcium chloride). 

Weak acids and weak bases do not ionize (or protolyze) completely in aqueous solution. The approach used to solve for the concentrations of solution components for weak acid or base solutions is similar to that used for strong acids and strong bases, but we are not able to make the simplifying assumption in theKa orKt, equilibrium equations that complete dissociation takes place. Typical calculations for weak acid and weak base systems are illustrated in the following example. 

A closed form solution of this equation is not possible. An additional simplifying assumption made in the analysis is the so-called depletion approximation . In this approximation, the free carrier concentrations are assumed to fall abruptly from their equilibrium values Hq and po in the bulk neutral region to a negligibly small value in the barrier space charge region. In reality, this transition occurs smoothly over a distance in which the bands bend by about 3 kT, but the calculations made using the depletion approximation are sufficiently accurate for most purposes. TIius, using the depletion approximation, Equation [3.26] can be written as ... 

Comment In Sample Exercise 17.3, the calculated pH is the same whether we solve exactly using the quadratic equation or make the simplifying assumption that the equilibrium concentrations of acid and base are equal to their initial concentrations. The simplifying assumption works because the concentrations of the acid-base conjugate pair are both a thousand times larger than K. In this Sample Exercise, the acid-base conjugate pair concentrations are only 10-100 as large as Therefore, we cannot... 

Mathematical equations that model these processes enable us to calculate the concentrations of a chemical released to the environment in air, water, soil, and sediment. Application of these equations requires numerous simplifying assumptions about the nature of the process (whether equilibrium/ steady state or not, ideal or not), the physicochemical properties of the substance and its reactivity, and the characteristics of the environment itself. 

If the initial concentration of a reactant, [AJ, is much larger than the change in its concentration to reach equilibrium, x, we make the simplifying assumption that xcan be neglected in calculations. (Section 17.5)... 

Making a Simplifying Assumption to Calculate Equilibrium Concentrations... 

Do the problem in two parts. First, you assume that the H30 ion from the strong acid and the conjugate base from the buffer react completely. This is a stoichiometric calculation. Actually, the HgO ion and the base from the buffer reach equilibrium just before complete reaction. So you now solve the equilibrium problem using concentrations from the stoichiometric calculation. Because these concentrations are not far from equilibrium, you can use the usual simplifying assumption about x. 

Can you make any assumptions that allow you to simplify the equilibrium calculations ... 

Apply the general method for solution equilibrium calculations outlined on page 761-762 to determine the pH values of the following solutions. In applying the method, look for valid assumptions that may simplify the numerical calculations. 

For all but the few smallest clusters, the number of possible structures is virtually unlimited. In order to be able to treat the larger systems, quite restrictive assumptions about their geometry has to be made. For those clusters where well-defined equilibrium structures do exist, these are likely to possess a non-trivial point-group symmetry (in many cases the highest possible symmetry). It therefore seemed justified to focus the study on high-symmetric systems. Symmetry can also be used to simplify the calculation of electronic structure, and reduces the number of geometrical degrees of freedom to be optimized. 

Consider a few examples where the equilibrium concentrations of components of one of the phases [e.g., (y)] are calculated when their concentrations in the other phase (a) are given. Assumptions will be made that simplify the notation we assume that the activities of the ions appearing in the thermodynamic equations are equal to their concentrations, and we consider systems where the ions have like valencies z. 

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