Equilibrium constant simplifying approximations

Hydrogen electrodes are approximately non-polarizable, which implies that the solution and the interface are in equilibrium. This simplifies the task of maintaining a constant reference potential. In an ideally non-polarizable electrode, the electrode... 

Understanding the meaning of a small equilibrium constant can sometimes help to simplify a calculation that would otherwise involve a quadratic equation. When Kc is small compared with the initial concentration, the value of the initial concentration minus x is approximately equal to the initial concentration. Thus, you can ignore x. Of course, if the initial concentration of a substance is zero, any equilibrium concentration of the substance, no matter how small, is significant. In general, values of Kc are not measured with accuracy better than 5%. Therefore, making the approximation is justified if the calculation error you introduce is less than 5%. 

With symmetric boundary conditions at the chosen time t = 0, the microscopic formulation conforms to time reversible laws as expected. The same conclusion follows from an analogous examination of the Liouville equation. In this setting, the initial data at time, t = 0, is a statistical density distribution or density matrix. Although there are celebrated discussions on the problem of the approach to equilibrium, we nevertheless observe that without course graining or any other simplifying approximations the exact subdynamics would submit to the same physical laws as above, i.e., time reversibility and therefore constant entropy. 

Although the equilibrium constant can be evaluated in terms of kinetic data, it is usually found independently so as to simplify finding the other constants of the rate equation. With Ke known, the correct exponents of Eq. (7-64) can be found by choosing trial sets until ki comes out approximately constant. When the exponents are small integers or simple fractions, this process is not overly laborious. 

A frequent complication is that several simultaneous equilibria must be considered (Section 3-1). Our objective is to simplify mathematical operations by suitable approximations, without loss of chemical precision. An experienced chemist with sound chemical instinct usually can handle several solution equilibria correctly. Frequently, the greatest uncertainty in equilibrium calculations is imposed not so much by the necessity to approximate as by the existence of equilibria that are unsuspected or for which quantitative data for equilibrium constants are not available. Many calculations can be based on concentrations rather than activities, a procedure justifiable on the practical grounds that values of equilibrium constants are obtained by determining equilibrium concentrations at finite ionic strengths and that extrapolated values at zero ionic strength are unavailable. Often the thermodynamic values based on activities may be less useful than the practical values determined under conditions comparable to those under which the values are used. Similarly, thermodynamically significant standard electrode potentials may be of less immediate value than formal potentials measured under actual conditions. 

Because the activities of species in the exchanger phase are not well defined in equation 2, a simplified model—that of an ideal mixture—is usually employed to calculate these activities according to the approach introduced bv Vanselow (20). Because of the approximate nature of this assumption and the fact that the mechanisms involved in ion exchange are influenced by factors (such as specific sorption) not represented by an ideal mixture, ion-exchange constants are strongly dependent on solution- and solid-phase characteristics. Thus, they are actually conditional equilibrium constants, more commonly referred to as selectivity coefficients. Both mole and equivalent fractions of cations have been used to represent the activities of species in the exchanger phase. Townsend (21) demonstrated that both the mole and equivalent fraction conventions are thermodynamically valid and that their use leads to solid-phase activity coefficients that differ but are entirely symmetrical and complementary. 

Dunken and Fritzsche (6) have summarized all the equilibrium constant calculations from infrared data used by earlier workers. They have shown that within the limits of accuracy in measurement, agreement can be reached between the results of two methods of calculation which contradict each other in their assumptions. By making infrared measurements at different temperatures, they also show that these chance agreements can come about at one temperature and not at another. Dunken and Fritzsche consider that simplified treatments of the type enumerated by Lussan are only approximations to the truth, and one should always employ a general model of association in which all the associated species (up to a certain maximum size) are present. The authors discuss the evidence for the cyclic dimer form but do not refer specifically to it in their calculations nor do they suggest that any of the higher multimers are cyclic. 

We use the hydrolysis of A into P and Q as an illustration. Examples are the hydrolysis of benzylpenicillin (pen G) or the enantioselective hydrolysis of L-acetyl amino acids in a DL-mixture, which yields an enantiomerically pure L-amino acid as well as the unhydrolysed D-acetyl amino acid. In concentrated solutions these hydrolysis reactions are incomplete due to the reaction equilibrium. It is evident that for an accurate analysis of weak electrolyte systems, the association-dissociation reactions and the related phase behaviour of the reacting species must be accounted for precisely in the model [42,43]. We have simplified this example to neutral species A, P and Q. The distribution coefficients are Kq = 0.5 and Kp = K = 2. The equilibrium constant for the reaction K =XpXQ/Xj = 0.01, where X is a measure for concentration (mass or mole fractions) compatible with the partition coefficients. The mole fraction of A in the feed (z ) was 0.1, which corresponds to a very high aqueous feed concentration of approximately 5 M. We have simulated the hydrolysis conversion in the fractionating reactor with 50-100 equilibrium stages. A further increase in the number of stages did not improve the conversion or selectivity to a significant extent. Depending on the initial estimate, the calculation requires typically less than five iterations. 

By writing the law of mass action for each of the independent transformations and the conservation of species, we obtain analytical expressions (sometimes making certain approximations to simplify the calculations) for the molar factions (or partial pressures) of each of the components as a function of the equilibrium variables, which may include molar fi-actions (or partial pressures of certain components) and equilibrium constants. 

The steady-state approximation helps us relate the rate law determined from the RDS with the rate law as determined from experiment. Recall that the rate law of a mechanism is dictated by the stoichiometry of the RDS, but how can we know if this rate law is consistent with the experimental rate law We can use the fact that the preceding step(s) is/are at equilibrium to derive a rate law in terms of the original reactants (whose amounts or concentrations we can measure). There are two ways to do this. First, we will adopt a simplified approach. If the first step in the above two-step process is in fact in equilibrium, then we can write an equilibrium constant expression for it... 

To simplify equilibrium calculations, we sometimes assume that the concentrations of one or more species are negligible and can be approximated as 0.00 M. In a sum or difference assuming a concentration is 0.00 M leads to an appropriate result. In contrast, if we were to simplify and equilibrium constant expression by assuming on or more concentrations are zero, we would be multiplying or dividing by 0.00, which would render the expression meaningless. 

In fact, the system thus obtained is complex to solve, particularly when we do not have the numerical values of the equilibrium constants. We can simplify the calculation by using Brouwer s majority defect approximation, which, remember, involves considering only the two largest terms (one with each sign) in the expression of electrical neutrality [3.36]. Thus, in our example, we would have four Brouwer cases, defined by ... 

For some equilibrium problans of the type shown in Example 14.9,14.10, and 14.11, we can make an approximation that simplifies the calculations without any significant loss of accuracy. For example, if the equihbrium constant is relatively small, the reaction will not proceed very far to the right Therefore, if the initial reactant concaitration is relatively large, we can make the assumption that x is small relative to the initial concentration of reactant. To see how this approximation works, consider again the simple reaction A 2 B. Suppose that, as before, we have a reaction mixture in which the initial concentration of A is 1.0 M and the initial concentration of B is 0.0 M and that we want to find the equilibrium concentrations. However, suppose that in this case the equilibrium constant is much smaller, sty K = 3.3 X 10 The ICE table is identical to the one we set up previously ... 

However, there are other features of the kinetic system of differential equations that may simplify the situation. The application of kinetic simplification principles (see Sect. 2.3) may result in the situation where it is not that the individual parameters have an influence on the solution, but only some combinations of these parameters. A simple example occurs when species B is a QSS-species within the A B C reaction system, and its concentration depends only on ratio kilk2-Also, when the production rate of species C is calculated using the pre-equilibrium approximation (see Sect. 2.3.2) within reaction system A B C, it depends only on equilibrium constant K = kjk2 and does not depend on the individual values of ki and 2-... 

We will thoroughly study the equilibrium corresponding to quasi-chemical reactions. From section 3.2.4, we can define the associated properties of these reactions and therefore equilibrium constants. However, frequently, the number of superimposed equilibriums is important in that it leads to very complex calculations. We can simplify these with the help of certain justified approximations. These approximations are of two orders ... 

When the equilibrium constant for a reaction is very small (K very large (K >t> 1), we can often use approximations to simplify our calculations because ... 

The equilibrium constant for this reaction is quite small, so we anticipate that very little of the H2S will react and the equilibrium concentration of H2S will be close to the initial concentration [H2S]gq [H2S]o- This approximation may prove helpful for simplifying the equilibrium constant expression. 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

Thus, the simplified Two-Network experiment shows by a direct comparison of forces at constant length that the trapped entangled structure of a well cross-linked elastomer contributes to the equilibrium modulus by an amount that is approximately equal to the rubber plateau modulus. The modulus contribution from the trapped entangled structure will be less for lower molecular weights and especially at low degrees of cross-linking (14). 

A quantitative description of interdependent fluxes and forces is given by irreversible thermodynamics, a subject that treats nonequilibrium situations such as those actually occurring under biological conditions. (The concepts of nonequilibrium and irreversibility are related, because a system in a nonequilibrium situation left isolated from external influences will spontaneously and irreversibly move toward equilibrium.) In this brief introduction to irreversible thermodynamics, we will emphasize certain underlying principles and then introduce the reflection coefficient. To simplify the analysis, we will restrict our attention to constant temperature (isothermal) conditions, which approximate many biological situations in which fluxes of water and solutes are considered. 

Lehr and Mewes [67] included a model for a var3dng local bubble size in their 3D dynamic two-fluid calculations of bubble column flows performed by use of a commercial CFD code. A transport equation for the interfacial area density in bubbly flow was adopted from Millies and Mewes [82]. In deriving the simplified population balance equation it was assumed that a dynamic equilibrium between coalescence and breakage was reached, so that the relative volume fraction of large and small bubbles remain constant. The population balance was then integrated analytically in an approximate manner. 

Bigelow, Glass and Zisman advanced explanations for both the concentration-influenced and the concentration-constant portions of the curves shown in Fig. 10-5. Their derivation of a model for the concentration-governed portion of the wetting curve can be simplified at the outset by treating the oriented monolayer as a solid phase in equilibrium with the solution. We can then write the following approximation ... 

Instead of using the quadratic equation, we may use the method of successive approximations. In this procedure, we will first neglect c compared to the initial concentrations to simplify calculations, and calculate an initial value of jc. Then we can use this first estimate of x to subtract from Ca and Cb to give an initial estimate of the equilibrium concentration of A and B, and calculate a new x. The process is repeated until x is essentially constant. 

The initial rate equation derived by steady-state analysis is of the second degree in A and B (SO). It simplifies to the form of Eq. (1) if the rates of dissociation of substrates and products from the complexes are assumed to be fast compared with the rates of interconversion of the ternary complexes k, k )] thus, the steady-state concentrations of the complexes approximate to their equilibrium concentrations, as was first shown by Haldane (14)- The kinetic coefficients for this rapid equilibrium random mechanism (Table I), together with the thermodynamic relations KeaKeab — KebKeba and KepKepq — KeqKeqp, suffice for the calculation of k, k and all the dissociation constants Kea = k-i/ki, Keab = k-i/ki, etc. 

While the design of distillation columns can be quite complicated, we will consider only the simplest case here. The simplificadons we will use are that vapor-liquid equilibrium will be assumed to exist on each tray (or equilibrium stage) and in the reboiler, that the column operates at constant pressure, that the feed is liquid and will enter the distillation column on a tray that has liquid of approximately the same composition as the feed, that the molar flow rate of vapor V is the same throughout the column, and that the liquid flow rate L is constant on all trays above the feed tray, and is constant and equal to L -b F below the feed tray, where F is the molar flow rate of the feed to the column, here assumed to be a liquid. The analysis of this simplified distillation column involves only the equilibrium relations and mass balances. This is demonstrated in the illustration below. 

---