Solutions to Rate Expressions

Two of the more common conversion and conversion-related terms will now be discussed in this section. The symbol N is used to represent the number of moles of a species at time t at initial conditions t = 0), a subscript, o, can be included with N. In line with the definition of conversion the symbol a is employed to represent the 

There are numerous conversion variables that can be employed in developing solutions to rate equations. These as well as other conversion-related terms will be addressed in the next chapter. 

Cases 2 (Equation 4.21) and 3a (Equation 4.22) in the previous section are reviewed in order to present analytical solutions to the rate equation using a and X as the conversion variables. Several other rate expressions are also examined— some of which appeared in the previous section. 

The following problem-solving approach is employed. The initial moles or concentration of the reacting species is noted above the stoichiometric equation. The moles or concentration at time t is placed below the symbols. 

Case 3a Although more complex than Case 2, this is a more representative chemical reaction system from an analytical solution viewpoint. For this reaction, one first writes  

---

Rate vs Equilibrium Considerations Representation of Rate Expressions Solutions to Rate Expressions Reaction Rate Theories... 

SOLUTIONS TO RATE EXPRESSIONS 63 If this equation is integrated between time 0 and t, then... 

Upon addition of a solution of sulfuric acid in D20 the reaction of A-acetoxy-A-alkoxyamides obeys pseudo-unimolecular kinetics consistent with a rapid reversible protonation of the substrate followed by a slow decomposition to acetic acid and products according to Scheme 5. Here k is the unimolecular or pseudo unimolecular rate constant and K the pre-equilibrium constant for protonation of 25c. Since under these conditions water (D20) was in a relatively small excess compared with dilute aqueous solutions, the rate expression could be represented by the following equation ... 

For the nth order reaction with respect to one reactant, the general solution for rate expression is ... 

Before elosing this ehapter, it is important to emphasize the eontext in whieh the transition rate expressions obtained here are most eommonly used. The perturbative approaeh used in the above development gives rise to various eontributions to the overall rate eoeffieient for transitions from an initial state i to a final state f, these eontributions inelude the eleetrie dipole, magnetie dipole, and eleetrie quadrupole first order terms as well eontributions arising from seeond (and higher) order terms in the perturbation solution. 

Use of Mass-Transfer-Rate Expression Figure 14-3 shows a section of a packed absorption tower together with the nomenclature that will be used in developing the equations which follow. In a differential section dh, we can equate the rate at which solute is lost from the gas phase to the rate at which it is transferred through the gas phase to the interface as follows ... 

The UCKRON AND VEKRON kinetics are not models for methanol synthesis. These test problems represent assumed four and six elementary step mechanisms, which are thermodynamically consistent and for which the rate expression could be expressed by rigorous analytical solution and without the assumption of rate limiting steps. The exact solution was more important for the test problems in engineering, than it was to match the presently preferred theory on mechanism. 

Molecular bromine is believed to be the reactive brominating agent in uncatalyzed brominations. The brominations of benzene and toluene are first-order in both bromine and the aromatic substrate in trifluoroacetic acid solution, but the rate expressions become more complicated when these reactions take place in the presence of water. " The bromination of benzene in aqueous acetic acid exhibits a first-order dependence on bromine concentration when bromide ion is present. The observed rate is dependent on bromide ion concentration, decreasing with increasing bromide ion concentration. The detailed kinetics are consistent with a rate-determining formation of the n-complex when bromide ion concentration is low, but with a shift to reversible formation of the n-complex... 

Here, we shall examine a series of processes from the viewpoint of their kinetics and develop model reactions for the appropriate rate equations. The equations are used to andve at an expression that relates measurable parameters of the reactions to constants and to concentration terms. The rate constant or other parameters can then be determined by graphical or numerical solutions from this relationship. If the kinetics of a process are found to fit closely with the model equation that is derived, then the model can be used as a basis for the description of the process. Kinetics is concerned about the quantities of the reactants and the products and their rates of change. Since reactants disappear in reactions, their rate expressions are given a... 

Consider the solution of Equation 6-170 for eaeh of the four types of rate expressions to determine the optimum temperature progression at any given fraetional eonversion X. ... 

We shall now consider a simplifying approximation for the system A 2P. The reaction proceeds at a rate expressed in terms of S by Eq. (3-33). If the shift resulting from the concentration jump is small, the term AK l82 is negligible in comparison to (1 + 4 l [P](,)S. In that case, the solution is... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

State the assumptions made in the penetration theory for the absorption of a pure gas inlo a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the bottom of the liquid. Starting with Hick s second law ol diffusion obtain an expression for (i) the concentration, and (ii) the muss transfer rate at a time t and a depth v below the surface. 

The times t are in minutes and the dye concentrations [dye] are in milliliters of stock dye solution per 100 ml of the reactant mixture. The stock dye solution was 7.72 x 10 molar. Use these data to fit a rate expression of the form... 

The orders of reaction, U , ivith respect to A, B and AB are obtained from the rate expression by differentiation as in Eq. (11). In the rare case that we have a complete numerical solution of the kinetics, as explained in Section 2.10.3, we can find the reaction orders numerically. Here we assume that the quasi-equilibrium approximation is valid, ivhich enables us to derive an analytical expression for the rate as in Eq. (161) and to calculate the reaction orders as ... 

In the former, the solute diffuses from the bulk of the solution to the external surface of crystals. Molecules of the solute already adsorbed on the surface are then integrated into the crystal lattice. The rate of both steps can be expressed by the equation ... 

For a batch system, with no inflow and no outflow, the total mass of the system remains constant. The solution to this problem, thus involves a liquid-phase, component mass balance for the soluble material, combined with an expression for the rate of mass transfer of the solid into the liquid. 

Since both hydrogen in the solution and the product A are weakly adsorbed species, equilibrium constants ka and k, are very small, which leads to KACA 1 and kh CH 1. Thus, the rate expression for the debenzylation can be simplified as a conventional Langmuir-Hinshelwood model. 

The reaction takes place in aqueous solution. Equimolal concentrations of the ester and the phenolate are used. These concentrations are equal to 30 moles/m3. By the time the samples are brought to thermal equilibrium in the reactor and efforts made to obtain data on ester concentrations as a function of time, some saponification has occurred. At this time the concentration of ester remaining is 26.29 moles/m3, and the concentration of phenol present in the reactant mixture is 7.42 moles/m3. The rate expression for the reaction is believed to be of the form... 

For mechanisms that are more complex than the above, the task of showing that the net effect of the elementary reactions is the stoichiometric equation may be a difficult problem in algebra whose solution will not contribute to an understanding of the reaction mechanism. Even though it may be a fruitless task to find the exact linear combination of elementary reactions that gives quantitative agreement with the observed product distribution, it is nonetheless imperative that the mechanism qualitatively imply the reaction stoichiometry. Let us now consider a number of examples that illustrate the techniques used in deriving an overall rate expression from a set of mechanistic equations. 

Thus mechanism B, which consists solely of bimolecular and unimolecular steps, is also consistent with the information that we have been given. This mechanism is somewhat simpler than the first in that it does not requite a ter-molecular step. This illustration points out that the fact that a mechanism gives rise to the experimentally observed rate expression is by no means an indication that the mechanism is a unique solution to the problem being studied. We may disqualify a mechanism from further consideration on the grounds that it is inconsistent with the observed kinetics, but consistency merely implies that we continue our search for other mechanisms that are consistent and attempt to use some of the techniques discussed in Section 4.1.5 to discriminate between the consistent mechanisms. It is also entirely possible that more than one mechanism may be applicable to a single overall reaction and that parallel paths for the reaction exist. Indeed, many catalysts are believed to function by opening up alternative routes for a reaction. In the case of parallel reaction paths each mechanism proceeds independently, but the vast majority of the reaction will occur via the fastest path. 

The first thing to note is that stoichiometric quantities of reactants were used in this investigation. Because the reaction rate expression simplifies when stoichiometric quantities of reactants are used, the equations developed earlier in this chapter cannot be applied directly in the solution of this problem. Thus we will have to derive appropriate relations in the course of our analysis. 

If it is assumed that in more concentrated solutions the rate of the forward reaction continues to follow this rate expression, what forms of the reverse rate are thermodynamically consistent in concentrated acid solution Equilibrium is to be established with respect to equation A when written in the N204 form. It may be assumed that the dependence on N02 and N204 concentrations may be lumped together by equation C. 

In this case + nx differs from m2 + n2 and there are a variety of possible forms that the rate expression may take. We will consider only some of the more interesting forms. In this case elimination of time as an independent variable leads to the same general result as in the previous case (equation 5.2.50). As before, in order to obtain a closed form solution to this equation, it is convenient to restrict our consideration to a system in which A0 = B0. In this specific case equation 5.2.50 becomes... 

In this subsection we have treated a variety of higher-order simple parallel reactions. Only by the proper choice of initial conditions is it possible to obtain closed form solutions for some of the types of reaction rate expressions one is likely to encounter in engineering practice. Consequently, in efforts to determine the kinetic parameters characteristic of such systems, one should carefully choose the experimental conditions so as to ensure that potential simplifications will actually occur. These simplifications may arise from the use of stoichiometric ratios of reactants or from the degeneration of reaction orders arising from the use of a vast excess of one reactant. Such planning is particularly important in the early stages of the research when one has minimum knowledge of the system under study. 

One may clearly extend the technique to include as many reactions as desired. The irreversibility of the reactions permits one to solve the rate expressions one at a time in recursive fashion. If the first reaction alone is other than first-order, one may still proceed to solve the system of equations in this fashion once the initial equation has been solved to determine A(t). However, if any reaction other than the first is not first-order, one must generally resort to numerical methods to obtain a solution. 

---