Reaction Rates Near Equilibrium

In thermodynamics we learned how to describe the composition of molecules in chemical equilibrium. For the generalized single reaction 

At chemical equilibrium at constant temperature and pressure the Gibbs fiee ener of the system is a niinimum and AG = 0. Therefore, we have 

dividing the preceding equation by RT and taking exponentials on both sides, we obtain 

Since we define vy/r = AG, the Gibbs fi ce energy change of the reaction in the 

We define the standard state of a liquid as ay = 1 and for gases as an ideal gas pressure of 1 bar, Pj = I- For ideal liquid solutions (activity coefficients of unity), we write ay = Cy so at chemical equilibrium 

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Substituting (1.20) into (1.16) and setting r] = 0 we get the forward reaction rate near equilibrium... 

Ljunggren, S., and Lamm. (1958). A relaxation method for the determination of moderately rapid reaction rates near chemical equilibrium. Acta Chem. Scand. 12, 1834-1850. 

Here is the forward rate constant, aj is the activity of species j in the rate-determining reaction, mj and are constants, and R and T are the gas constant and absolute temperature, respectively. The sign of the rate indicates whether the reaction goes forward or backward. The relationship of this equation to transition state theory and irreversible kinetics has been discussed in the literature (Lasaga, 1995 Alekseyev et al., 1997 Lichtner, 1998 Oelkers, 2001b). The use of this equation with = 1 is generally associated with a composite reaction in which all the elementary reactions are near equilibrium except for one step which is ratedetermining. This step must be shared by both dissolution and precipitation. 

The effectiveness factors of reactions I and III both decrease through the reactor, despite the decrease in the rate of reaction III. When the reaction is near equilibrium, the reaction rate is even more sensitive with respect to changes in... 

There are three important points to note. First, for purely catalystic eflFectors (i.e., when the eflFector does not participate in the reaction catalyzed) the value of r is zero and = r. Second, the above treatment has been applied to a very simple reaction to illustrate the approach but it can be applied to any reaction or process no matter how complex the rate equation. Third, r is always calculated by assuming the reaction to be nonequilibrium if the reaction is near-equilibrium in vivo, the eflFects of nearness to equilibrium (i.e., reversibility) are incorporated into the intrinsic sensitivity at a later stage (see below). 

A reaction is nonequilibrium if the rate of the reverse component of the reaction is much less than the rate of the forward component, and a reaction is near-equilibrium if the rates of the forward and reverse components of the reaction are similar. In the following examples, where the numbers refer to the actual rates in either direction, the reactions Ei and E3 are considered to be nonequilibrium, whereas E2 is considered to be near-equilibrium. 

Bokhoven and Rayen [97] measured reaction rates on 0.5-0.7 mm and 2.4-2.8 mm particles at 1, and 30 atm and 325-550 °C. The effective diffusion coefficient of NH3 was calculated from the results of O2 diffusion measurements on a catalyst with the same surface area and porosity as the catalyst used in the activity measurements. The authors approximated the reaction rate by a pseudo first-order reaction suggested by Wagner [103] to calculate the effectiveness factor. Good agreement between measured and calculated data was obtained. However, the approximation above is only good, if the reaction is near equilibrium. 

FIGURE 13.21 The equilibrium constant for a reaction is equal to the ratio of the rate constants for the forward and reverse reactions, (a) A forward rate constant (A) that is relatively large compared with the reverse rate constant means that the forward rate matches the reverse rate when the reaction has neared completion and the concentration of reactants is low. (b) Conversely, if the reverse rate constant (A ) is larger than the forward rate constant, then the forward and reverse rates are equal when little reaction has taken place and the concentration of products is low. 

When a 1 1 mixture of NO and NO2 (i.e., NO2/NOx=0,5) is fed to the SCR reactor at low temperature (200 °C) where the thermodynamic equilibrium between NO and NO2 is severely constrained by kinetics, the NO2 conversion is much greater than (or nearly twice) the NO conversion for all three catalysts. This observation is consistent with the following parallel reactions of the SCR process [6] Reaction (2) is the dominant reaction due to its reaction rate much faster than the others, resulting in an equal conversion of NO and NO2. On the other hand, Reaction (3) is more favorable than Reaction (1), which leads to a greater additional NO2 conversion by Reaction (3) compared with the NO conversion by Reaction... 

Figure 26.7. Only after the mineral has almost disappeared does the silica concentration begin to decrease. Since the surface area and rate constant for cristobalite are considerably greater than those of quartz, the fluid remains near equilibrium with cristobalite until it in turn nearly disappears. Finally, after several hundred thousand years of reaction, the fluid approaches saturation with quartz and hence thermodynamic equilibrium.

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