Equilibrium constant simplified equation

A low effectiveness factor — see Figure 3.23 — implies that the effectiveness factor and thus the effective rate for the steam reforming of methane is inversely proportional to the Thiele modulus [199] and hence the equivalent particle diameter assuming that the particle is isotherm. For a first-order equilibrium rate expression, a general effectiveness factor can be evaluated as shown in [199] [389]. For a large equilibrium constant, this equation can be simplified to ... 

To construct such a diagram, a set of defect reaction equations is formulated and expressions for the equilibrium constants of each are obtained. The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. To simplify matters, only one defect reaction is considered to be dominant in any particular composition region, this being chosen from knowledge of the chemical attributes of the system under consideration. The simplified equilibrium expressions are then used to construct plots of the logarithm of defect concentration against an experimental variable such as the log (partial pressure) of the components. The procedure is best illustrated by an example. 

As with the expression in Equation (6.6), this equilibrium constant can be simplified by incorporating the water term into K, thereby yielding a new constant which we will call Kh, the basicity constant ... 

The esterification of TPA with EG is a reaction between two bifunctional molecules which leads to a number of reactions occurring simultaneously. To simplify the evaluation of experimental data, model compounds have been used for kinetic and thermodynamic investigations [18-21], Reimschuessel and coworkers studied esterification by using EG with benzoic acid and TPA with 2-(2-methoxyethoxy) ethanol as model systems [19-21], The data for the temperature dependency of the equilibrium constants, AT, = K,(T), given in the original publications are affected by printing errors. The corrected equations are summarized in Table 2.3. 

The equilibrium constant for the carbamate reaction (eq.VIII) was simplified by assuming a o = an< re placing all other activities by molalities. Numbers for Kg(T) at 20, 40 and 60 oc were determined from experimental results. (Van Krevelen et al. only report discrete numbers or diagrams for some constants. For inter- and extrapolation these numbers were replaced by equations, wherein the dimensions of m and T are moles/ kg H2O and Kelvin, respectively.) ... 

In this study the authors develop simplified equations relating equilibrium fractionations to mass-scaling factors and molecular force constants. Equilibrium isotopic fractionations of heavy elements (Si and Sn) are predicted to be small, based on highly simplified, one-parameter empirical force-field models (bond-stretching only) of Sip4, [SiFJ, SnCl4, and [SnCl,] -. 

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%. 

The Brpnsted equations relate a rate constant k to an equilibrium constant K.,. In Chapter 6 we saw that the Marcus equation also relates a rate term (in that case AG ) to an equilibrium term AG°. When the Marcus treatment is applied to proton transfers99 between a carbon and an oxygen (or a nitrogen), the simplified 011 equation (p. 216)... 

Hence, equilibrium constants of homogeneous electron-transfer reactions between (A) and B are evidently connected to the differences in reduction potentials of A and B. This connection reflects a definite physical phenomenon. Namely, if two redox systems are in the same solution, they react with each other until a unitary electric potential is reached. For the transfer of only one electron at room temperature, the following simplified equation can be employed ... 

This leads to some complicated differential equations which are usually solved numerically. To simplify things, let us assume that the surface reaction (Eq. (2.24)) is the rate-determining step, while the adsorption and the desorption steps are at equilibrium (i.e., the net change in Eqs. (2.4) and (2.6) is zero). In this case, Eq. (2.26) apply, where KA and /adsorption equilibrium constants for A and B, respectively. 

Strictly speaking, equations 1.2-7 and 1.2-9 should have c° in the denominator, where c° = 1M is the standard concentration, to make the equilibrium constant dimensionless (Mills et al., 1993). However, the c° is omitted in this book in order to simplify expressions for equilibrium constants. Nevertheless, equilibrium constants are still considered to be dimensionless, so their logarithm can be taken. 

Considering the literature review above, it is necessary to develop a comprehensive view that may apply to any system, whatever the luminescent probe is, each system being considered as one specific case of a more general theory. In this section, we first discuss the general scheme of (photo)chemical equations needed to describe a TRES experiment as well as a simplified scheme without photophysical reactions. The theoretical conditions to derive equilibrium constants from TRES data are discussed for the R(III), U(VI), and Cm(III) cases and are related to the experimental data presented above. 

Equation (5) is greatly simplified the true thermodynamic partition coefficient would be the quotient of analyte activities, not concentrations. Furthermore, the equation assumes that analyte A is present in only one form (one molecular structure or ion). When this is not realized in practice, a more complex equilibrium constant must be used. 

A simple equation between Cbz and Cbs may be obtained by means of various simplifying assumptions. If we take the case of an ideal exchange (both the exchanger and the solution are ideal), and which has only a single category of site, the equilibrium constant Aa can be written as... 

The concentration of any of these species depends on the total concentration of dissolved aluminum and on the pH, and this makes the system complex from the mathematical point of view and consequently, difficult to solve. To simplify the calculations, mass balances were applied only to a unique aluminum species (the total dissolved aluminum, TDA, instead of the several species considered) and to hydroxyl and protons. For each time step (of the differential equations-solving method), the different aluminum species and the resulting proton and hydroxyl concentration in each zone were recalculated using a pseudoequilibrium approach. To do this, the equilibrium equations (4.64)-(4.71), and the charge (4.72), the aluminum (4.73), and inorganic carbon (IC) balances (4.74) were considered in each zone (anodic, cathodic, and chemical), and a nonlinear iterative procedure (based on an optimization method) was applied to satisfy simultaneously all the equilibrium constants. In these equations (4.64)-(4.74), subindex z stands for the three zones in which the electrochemical reactor is divided (anodic, cathodic, and chemical). 

This equation simplifies to kex = kQ if kc << 1c, and to kex = kQk,/kc if 1c, <conformational openings are the rate-limiting step of the exchange and the latter condition is referred to as EX2, where the equilibrium constant for local opening together with the intrinsic exchange rate determine the H-D exchange process. 

The stability of the metal complex can be expressed as the equilibrium constant (see also Chapter 4). For example, if Equation (5.43) is simplified to... 

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. 

The thermodynamic equilibrium constant can be evaluated by using the simplified form of the Gaines-Thomas equation [201], assuming that the change in water content in the exchanger is negligible. The equation, with defined as the corrected selectivity coefficient by Eq. (4), follows-. 

Values of and can deviate significantly from unity typical values for range from 0.7 to 0.9 at a pressure of 7.5 MPa and typical MSR temperatures from 200°C to 320 C, while values for range from 0.8 to 0.95. Estimates of may be obtained from an equation of state, but such calculations are beyond the scope of the present discussion. Furthermore, computational alternatives often require a choice between simplified and more realistic, but complex, models. An instance that illustrates this situation is estimation of and for use in determining reaction equilibrium constants. 

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. 

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