Single-Reaction Equilibrium

Example 6 Single-Reaction Equilibrium The hydrogenation of benzene to produce cyclohexane by the reaction 

A feed stream containing 3 mol H2 for each 1 mol CeHe is the basis of calculation, and for this single reaction, Eq. (4-361) becomes rij = rij, + v j, yielding 

Assume first that the equilibrium mixture is an ideal gas, and apply Eq. (4-372), written for a single reaction, with subscript omitted and v = — 3  

the assumption of ideal gases leads to a calculated conversion of 81.5 percent. 

An alternative assumption is that the equilibrium mixture is an ideal solution. This requires application of Eq. (4-371). However, in the case of an ideal solution Eq. (4-218) indicates that = ([), in which case Eq. (4-371) for a single reaction becomes 

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This result means in general for single-reaction equilibrium between two species A and B that two degrees of freedom exist, and that pressure as well as temperature must be specified to fix the equilibrium state of the system. However, here, the specification that the gases are ideal removes the pressure dependence, which in the general case appears through the, s. 

Example 2 Single-Reaction Equilibrium Consider the equilibrium state at 1,000 K and atmospheric pressure for the reaction... 

Single reactions. For single reactions, a good initial setting is 95 percent conversion for irreversible reactions and 95 percent of the equilibrium conversion for reversible reactions. Figure 2.9 summarizes the influence of feed mole ratio, inert concentration, temperature, and pressure on equilibrium conversion. ... 

For multiple reactions in which the byproduct is formed in series, the selectivity decreases as conversion increases. In this case, lower conversion than that for single reactions is expected to be appropriate. Again, the best guess at this stage is to set the conversion to 50 percent for irreversible reactions or to 50 percent of the equilibrium conversion for reversible reactions. 

This situation is called a substrate titration. That is, the change in rate with [H+] is the sole consequence of an equilibrium incidental to the main event. It is customary to display pH-dependent rates by plots of (v/[A]t) versus pH that is, by log versus pH. Two common patterns are shown in Fig. 6-1, for cases in which there is a single protonation equilibrium. The case in Fig. 6-la corresponds to Eq. (6-81) we return later to Fig. 6-1 b. The line bends down, as do all instances of substrate titration. The apparent order of the reaction with respect to [H+] is +1 in the limit of low [H+] and 0 at high. 

The second chapter is by Aogaki and includes a review of nonequilibrium fluctuations in corrosion processes. Aogaki begins by stating that metal corrosion is not a single electrode reaction, but a complex reaction composed of the oxidation of metal atoms and the reduction of oxidants. He provides an example in the dissolution of iron in an acidic solution. He follows this with a discussion of electrochemical theories on corrosion and the different techniques involved in these theories. He proceeds to discuss nonequilibrium fluctuations and concludes that we can again point out that the reactivity in corrosion is determined, not by its distance from the reaction equilibrium but by the growth processes of the nonequilibrium fluctuations. ... 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

The reaction coordinate defined in Section 2.8 provides an algebraic method for calculating equilibrium concentrations. For a single reaction. 

Equilibrium for a single reaction in the liquid-phase. A significant proportion of fine chemistry processes occur in the liquid phase. The equilibrium constant is expressed by Eqn. (5.4-8), which can be rewritten as ... 

Problems with the determination of chemical equilibria in multiphase systems are solved in practice by assuming that the reaction takes place in any phase and all components are also equilibrated between phases. Accordingly, for a single reaction any of Eqns. (5.4-32) and (5.4-33) must be solved, while the relationships of Eqn. (5.4-31) must also be fulfilled. Since vapour-liquid equilibrium coefficients are functions of the compositions of both phases, the search for the solution is an iterative procedure. Equilibrium compositions are assumed, vapour-liquid equilibrium coefficients are then estimated, and new equilibrium compositions are evaluated. If the new equilibrium compositions are close to those assumed initially one may consider the assumed values to be the solution of the problem. Otherwise the evaluated compiositions are taken as the start for repetition of the procedure until a reasonable agreement between tissumed and evaluated comfiositions has been reached. 

Thermodynamic equilibrium in the sample has to be ensured. That requires that the sample is isothermal, otherwise the heat flux cannot be attributed to the single temperature indicated at the sensor. Furtheron, the sample has to be in reaction equilibrium, so there should be no subcooling of the sample. 

Single reactions. For single reactions, a good initial setting is 95% conversion for irreversible reactions and 95% of the equilibrium conversion for reversible reactions. 

In the preceding chapter, the choice of reactor type was made on the basis of the most appropriate concentration profile as the reaction progressed, in order to minimize reactor volume for single reactions or maximize selectivity (or yield) for multiple reactions for a given conversion. However, there are still important effects regarding reaction conditions to be considered. Before considering reaction conditions, some basic principles of chemical equilibrium need to be reviewed. 

The method just described can only be applied in the simplest cases, where a single reaction is present. The equivalent of equation 11.20 for the general case of i equilibrium reactions inside the calorimetric vessel is... 

Single-temperature equilibrium constant values may also yield quantitative information about reaction enthalpies, provided that the entropy term can be estimated. Take, for example, reaction 14.32, which involves hydrogen transfer between two substituted phenols (ArOH and Ar OH see examples in figure 14.4). Note that Kc = Km in this case. 

It should be stressed that the condition ArX 0 is not required to derive a series of relative values of reaction enthalpies from single-temperature equilibrium constants. We only need to ensure that Arremains constant. Consider, for example, the case of reaction 14.33 (see figure 14.5) ... 

A final word of caution regarding the use of single-temperature equilibrium constants. Although this is a rather expeditious method to derive reaction enthalpies, the obtained values may be quite inaccurate. For instance, a small 10 J K I mol-1 error in the estimated Ar,Vy. yields a 3 kJ mol-1 error in Ar at 298.15 K. 

Although it has been shown that thermodynamic models which imply phase separations can create difficulties with uniqueness in solving the reaction equilibrium equations (10, 25, 24), there proved to be only one solution to equation (26) under the conditions studied. It is conceivable that more than one critical point could be found for some reacting mixtures at certain reaction extents (two critical points are indeed indicated in Figure 1 for some C02 - CO mixtures), in which case F(e) will not be a single-valued function. This possibility was not explored. 

The potential of a mixed electrode at which a coupled reaction of charge transfer proceeds is called the mixed electrode potential , this mixed electrode potential is obviously different from the single electrode potential at which a single reaction of charge transfer is at equilibrium. For corroding metal electrodes, as shown in Fig. 11—2, the mixed potential is often called the corrosion potential, E . At this corrosion potential Eemt the anodic transfer current of metallic ions i, which corresponds to the corrosion rate (the corrosion current ), is exactly balanced with the cathodic transfer current of electrons for reduction of oxidants (e.g. hydrogen ions) i as shown in Eqn. 11-4 ... 

The single-site equilibrium binding of a small molecule ligand S with its receptor E can be expressed as the chemical reaction shown here ... 

As exemplified in figure 2.10, in the attainment of new equilibrium a single reaction may proceed in opposite directions (i.e., forward and backward). In hydrolytic equilibria, the dissolution process is conventionally defined as moving in... 

If equilibrium in a system is perturbed for some reason, attainment of the new equilibrium requires a certain amount of time t, depending on the rates of the single reactions, and a certain amount of energy spent in activating the reaction process. The relationship between reaction rate k and activation energy has the exponential form... 

In this chapter we deal with single reactions. These are reactions whose progress can be described and followed adequately by using one and only one rate expression coupled with the necessary stoichiometric and equilibrium expressions. For such reactions product distribution is fixed hence, the important factor in comparing designs is the reactor size. We consider in turn the size comparison of various single and multiple ideal reactor systems. Then we introduce the recycle reactor and develop its performance equations. Finally, we treat a rather unique type of reaction, the autocatalytic reaction, and show how to apply our findings to it. 

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

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