Equilibrium Constant-Based Calculations

The reactions introduced in Equations (4.6)-(4.10) are equilibrium reactions, as indicated by the double arrow. This means that reactants and products approach a stable mixture that is defined by the thermodynamic equilibrium. This occurs theoretically at infinite residence time. Because of high temperatures, equilibrium may be also approached in technical gasification systems at finite residence time. However, some deviations remain. 

The easiest way to describe the thermodynamic equilibriiun is the formulation of the equilibrium constants for each reaction from tabulated values for enthalpy of formation AfH, and standard entropy S,. For a given reaction, the reaction enthalpy Ar// and reaction entropy ArS° - both are fimctions of temperature - are calculated as follows  

the index i represents the participating species and i/ the respective stoichiometric coefficients, which are negative for reactants and positive for products. The standard equilibrium constant is computed by 

In Equations Equations (5.4) and (5.5), Pq represents the reference pressure for determination of Af//° and which is usually 1 bar or 10 Pa. The molar fraction equilibrium constant kx is also a function of the system pressure p. 

In the next step, the law of mass action can be used to calculate the according amounts of gas species, that is, in terms of partial pressure. 

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In Equation (5.13), the temperature T must be provided in K, whereas the coefficients a to e regarding reaction j are given in Table 5.2. The range of validity is 500 to 1800°C. The equilibrium constant-based calculations have the main advantages in terms of simplicity of the calculation. Deviations from equilibrium can be easily implemented by calculating equilibrium constants at other than the physical process temperature (see Section 5.6.1). However, the main drawback is that trace components may be not accurately predicted, and the implementation of non-ideal gas behavior is difficult. In more complex calculation frameworks, the equilibrium constant-based approach may also show disadvantages regarding mathematical behavior. 

A quantitative solution to an equilibrium problem may give an answer that does not agree with the value measured experimentally. This result occurs when the equilibrium constant based on concentrations is matrix-dependent. The true, thermodynamic equilibrium constant is based on the activities, a, of the reactants and products. A species activity is related to its molar concentration by an activity coefficient, where a = Yi[ ] Activity coefficients often can be calculated, making possible a more rigorous treatment of equilibria. 

Whatever the exact form of carbon deposition may be, it must be recognized and taken into account in future calculations. The deposited material is called Dent carbon, and equilibrium constants based on its free energy are included in Figure 1. The deposition of carbon as Dent carbon was confirmed qualitatively by Pursley et al. (4). In other recently reported equilibrium calculations (5, 6, 7), it was assumed that... 

The rest of this chapter is a variation on a theme introduced in Chapter 9 the use of equilibrium constants to calculate the equilibrium composition of solutions of acids, bases, and salts. We shall see how to predict the pH of solutions of weak acids and bases and how to calculate the extent of deprotonation of a weak acid and the extent of protonation of a weak base. We shall also see how to calculate the pH of a solution of a salt in which the cation or anion of the salt may itself be a weak acid or base. 

Table 2.3 Corrected equations for the calculation of temperature-dependent equilibrium constants based on publications of Reimschuessel and co-workers [21]...

Calculations of isotope effects and isotopic exchange equilibrium constants based on the Born-Oppenheimer (BO) and rigid-rotor-harmonic-oscillator (RRHO) approximations are generally considered adequate for most purposes. Even so, it may be necessary to consider corrections to these approximations when comparing the detailed theory with high precision high accuracy experimental data. 

According to C Hj yield and concentrations of the four components, the equilibrium constant of the reaction in Scheme 2.6 was determined as 4 X 10 . The value calculated from the potential difference was 4.4 X 10. Consequently, there is a coincidence between the calculated equilibrium constant based on the electrode potentials and the equilibrium constant determined from the liquid-phase experiment. 

The very low value of Ashmore and Burnett is difficult to explain. It is easy to demonstrate that the discrepancy is not resolved by assuming the N03 intermediate in nitrogen dioxide decomposition is the pernitrite radical, in contradistinction to the symmetric nitrate radical. Their calculation of k5 depended on an experimentally obtained value for k 5 and an equilibrium constant K5- 5 calculated from thermodynamic properties for N03 measured by Schott and Davidson and Ray and Ogg. These results, obtained in a nitrogen pentoxide system, pertain to the nitrate radical, not the pernitrite radical. Guillory and Johnston176 reported an equilibrium constant based on estimated... 

Notwithstanding the possibility of variation of an intrinsic barrier within a reaction series, for comparisons between different reactions it is often convenient to assume that an unmodified Marcus expression applies. This approximation is justified partly by the high intrinsic barriers and small amounts of curvature characteristic of most reactions at carbon, including reactions of carbocations. The Marcus relationship then provides a common framework for comparisons between reactions based on the measurement of even a single combination of rate and equilibrium constants. Thus, calculation... 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

The composition at equilibrium can be determined from the equilibrium constant based on molar fractions Kx = Ka/Kr This is, in turn, calculated from the equilibrium-constant-based activities IQ and activity coefficients Kr Figure 8.3 shows the variation over the range 350-475 K [1, 2], IQ takes values between 10 and 50, but the correction by IQ is important bringing IQ between 2 and 10. Above 430 K equilibrium conversion over 80% should be expected. If the water is continuously removed by distillation, then full conversion may be achieved. 

Note the use of activities, as well as of an equilibrium constant based on activities. The kinetic constants for autocatalyzed and catalyzed reactions, k and k, were determined from initial reaction rates with liquid activity coefficients calculated by UNIQUAC. Near chemical equilibrium the fCT is about 6, while Kx is about 5. Table 8.7 gives activation energies and pre-exponential factors obtained by nonlinear regression. The simulation shows tbat the autocatalysis effect is neghgible below 150 °C, but it might increase to 20% at 180 °C. 

Miller and Kusch (3 ) determined the molecular composition of KI vapor by measurement of the velocity distribution of the molecules in the beam produced as the vapor effused through a small slit in a source. The analysis was based on an assumption that the velocity distribution within the oven is Maxwellian and that the vapor effuses through the ideal slit of kinetic theory. The velocity distributions of potassium and thallium atomic beams were found to be in excellent agreement with the theoretical distributions so the determination of the molecular composition of KI beams was tried. Using the derived equilibrium constants, we calculate the enthalpy change of the dissociation reaction by the 2nd and 3rd law methods. The results are presented in the following table. 

The details of the calculation of the equilibrium constant based on the methods of Chapter 17 follows. Consider the equilibrium between hydrogen atoms and their component charged particles ... 

The equilibrium constant obtained when all components are present at their equilibrium partial pressures is designated Kp, the equilibrium constant based on pressures. In many cases, Kp has a value different from K, but if you know one, you can calculate the other by noting the change in amount (mol) of gas, from... 

Almost all the equilibrium constants were calculated by our program based on the equation... 

Here we have represented concentration by molality, m, (instead of number of moles, rii) and include an activity coefficient correction, 7. As with equilibrium constant-based equilibrium calculations, the activity coefficients can be computed from successive estimates of the concentrations. 

The principle of microscopic reversibility allows one to express the backward rate constant in terms of the forward rate constant divided by Kp, which is the equilibrium constant based on gas-phase partial pressures. Kp has units of pressure to the power 5, where 5 is the sum of the stoichiometric coefficients (i.e., 8 = —2 for this problem). Handbook values for standard-state free energies of formation at 298 K are used to calculate the Gibbs free-energy change for reaction at 298 K (i.e., 29s) thi to calculate a dimensionless... 

Based on JANAF or equivalent thermochemical data, equilibrium constants were calculated for the gas-phase equilibria ... 

A series of Pfeiffer-active systems was prepared, using racemic -[Ni(phen)3]Cl2 (8) and /cvo-malic acid, which differ in concentrations of the complex and the environment substance as well as in the ratios of the complex to the environment substance. For each system the equilibrium shift was observed experimentally, and the equilibrium constant was calculate (9), based on the optical rotations observed for the system and corrected for the presence of the optically active environment substance. 

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