Second-order reactions process

The units for the reaction rate constant k when the reaction is of order n (different from the n power of T) will be [(cone)"-1 (time)]-1. Thus, for a first-order reaction the units of k are in reciprocal seconds (s-1), and for a second-order reaction process the units are in cubic centimeter per mol per second (cm3 mol-1 s-1). Thus, as shown in Appendix C, the most commonly used units for kinetic rates are cm3molkJ, where kilojoules are used for the activation energy. 

In the sections that follow, we will delve deeply into the atomistic world of reaction kinetics and learn how to predict the rates of a number of fairly simple zero, first, and second-order reaction processes. While this chapter will focus mostly on simple gas-phase chemical reaction processes, the principles learned here will apply just as well to the solid-state materials kinetic examples that we will confront later in the textbook. This is because bond-breaking and bond-forming processes are remarkably similar at the atomistic level whether they happen between molecules in the gas phase or between atoms in a solid. Thus, most reaction processes can be described using a common set of approaches. Toward the end of the chapter, in preparation for later solid-state applications of reaction kinetic principles, we will examine how reaction rates can be affected by a catalyst or a surface, and we will learn how to model several gas-solid surface reaction processes relevant to materials science and engineering. 

The mathematical form of the reaction rate law for a specific chemical reaction depends on the order of the reaction, which is itself dictated by the reaction mechanism. Reaction order cannot be determined from a simple inspection of a stoichiometric chemical reaction it must be determined empirically from experiment or from detailed knowledge about the underlying reaction mechanism. Analytical rate law expressions are available for zero-, first-, and second-order reaction processes. First- and second-order reaction processes are the most common in everyday occurrence. 

When the overflow clarity is independent of overflow rate and depends only on detention time, as in the case for high soHds removal from a flocculating suspension, the required time is deterrnined by simple laboratory testing of residual soHd concentrations in the supernatant versus detention time under the conditions of mild shear. This deterrnination is sometimes called the second-order test procedure because the flocculation process foUows a second-order reaction rate. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

Second law of thermodynamics A basic law of nature, one form of which states that all spontaneous processes occur with an increase in entropy, 457 Second order reaction A reaction whose rate depends on the second power of reactant concentration, 289,317q gas-phase, 300t... 

There have been comparatively few kinetic studies of the decompositions of solid malonates [1103]. The sodium and potassium salts apparently melt and non-isothermal measurements indicate second-order rate processes with high values of E (962 125 and 385 84 kJ mole-1, respectively). The reaction of barium malonate apparently did not involve melting and, from the third-order behaviour, E = 481 125 kJ mole-1. 

Some progress has been made in developing theoretical expressions for rj(6) for deactivation processes such as coking. Deactivation by loss of active sites can be modeled as a chemical reaction proceeding in parallel with the main reaction. It may be substantially independent of the main reaction. Site sintering, for example, will depend mainly on the reaction temperature. It is normally modeled as a second-order reaction ... 

If the pressure for the process is lowered, the reaction (R3) will shift from a first-order reaction (high-pressure limit) to a second-order reaction (low-pressure limit). If (R3) is now considered a second-order reaction and assuming that the other pressure dependent reactions do not shift regime, determine expressions for d[C2H6]/dt, d[CH3]/dt, d[C2Hs]/dt and d[H]/dt. 

Ion-exchange processes are strictly heterogeneous reactions that involve two phases. They, however, mostly comply with homogeneous, second-order reaction kinetics. An exchange reaction represented by... 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

There are two other limiting forms of these equations that are also of interest. If k 1 k2, the first step is very rapid compared to the second, so that it is essentially complete before the latter starts. The reaction may then be treated as a simple irreversible second-order reaction with the second step being rate limiting. On the other hand, if k2 ku the first step controls the reaction so the kinetics observed are those for a single second-order process. However, the analysis must take into account the fact that in this case 2 moles of species A will react for each mole of B that is consumed. 

The overall reaction, A+B+C+D => ABCD, is assumed to proceed by three second order reversible processes,... 

Reactant A decomposes according to a second order reaction. Two streams are to be processed, the first with Ca0 = 1 and = 1 the other with Ca0 - 2... 

A second order reaction with kC0 = 1.5 is to be processed under... 

The rate of a batch slurry reaction is controlled by diffusion from the bulk liquid to the surface of the catalyst and by a second order reaction on the surface. Equations for the two processes are rd = 0.25(C-Cs)... 

We will only consider the derivation of the simplest case of a second-order reaction, where the concentrations of the two reacting species are the same. Being second order, the rate law has the form rate = k2[A]2. The subscript 2 on k indicates a second-order process. Again, by rate we mean the rate of change of the concentration of reactant A. 

More complicated reactions that combine competition between first- and second-order reactions with ECE-DISP processes are treated in detail in Section 6.2.8. The results of these theoretical treatments are used to analyze the mechanism of carbon dioxide reduction (Section 2.5.4) and the question of Fl-atom transfer vs. electron + proton transfer (Section 2.5.5). A treatment very similar to the latter case has also been used to treat the preparative-scale results in electrochemically triggered SrnI substitution reactions (Section 2.5.6). From this large range of treated reaction schemes and experimental illustrations, one may address with little adaptation any type of reaction scheme that associates electrode electron transfers and homogeneous reactions. 

When a product, formed in a second order reaction, acts as a catalyst or effects the rate of reaction, the reaction is known as autocatalytic reaction. For example, the acid catalysed hydrolysis of various esters and similar compounds and various biochemical processes. 

In the unimolecular reactions which are also of the first order, only one molecule takes part in the reaction. The process of activation in unimolecular reactions, if caused by collisions should ordinarily lead to second order reactions. How then the observed rate of reaction could be of first order. If however, the activation is by absorption of the radiant energy, this problem can be avoided. But many unimolecular reactions take place under conditions where there is no absorption of radiant energy. For example... 

Very rarely are measurements themselves of much use or of great interest. The statement "the absorption of the solution increased from 0.6 to 0.9 in ten minutes", is of much less use than the statement, "the reaction has a half-life of 900 sec". The goal of model-based analysis methods presented in this chapter is to facilitate the above translation from original data to useful chemical information. The result of a model-based analysis is a set of values for the parameters that quantitatively describe the measurement, ideally within the limits of experimental noise. The most important prerequisite is the model, the physical-chemical, or other, description of the process under investigation. An example helps clarify the statement. The measurement is a series of absorption spectra of a reaction solution the spectra are recorded as a function of time. The model is a second order reaction A+B->C. The parameter of interest is the rate constant of the reaction. 

It should be noted that the unfolding kinetics can sometimes involve quite complex unfolding schemes of different substates in equilibrium with the native state. Staphylococcal nuclease is an example of such behavior, known to unfold with three different substates that exhibit an equilibrium that does not appear to shift with temperature.49 Irreversible aggregation processes of proteins have been known to involve first- or second-order reactions.132141 The mechanism of recombinant human interferon-y aggregation is an example where thermodynamic and kinetic aspects of the reaction provided a powerful tool for understanding the pathway of instability and permitted a rationale for screening excipients that inhibited the process.141... 

Suppose that an irreversible reaction between A and B leading to >95% product or products, designated D, is examined in the usual way. One of the reactants, B, is held in excess and the loss of A monitored. It is likely that the loss will be a first-order process (rate constant k). At low concentrations of B (but still > [A]), the value of k may be proportional to the concentration of B. At higher concentrations of B however this direct proportionality may disappear and eventually k will become independent of [B]. Obviously, a second-order reaction at low reactant concentrations has lost its simplicity at higher reactant concentrations and eventually turned over to first-order in A alone. Such a situation is accommodated by a rate law of the form... 

To consider the early and late mixing of a microfluid, consider the two flow patterns shown in Fig. 11.17 for a reactor processing a second-order reaction. In (a) the reactant starts at high concentration and reacts away rapidly because n> l. n b) the fluid drops immediately to a low concentration. Since the rate of reaction drops more rapidly than does the concentration you will end up with a lower conversion. Thus, for microfluids... 

The biperoxy radical produced by the ceric ion oxidation of 2,5-di-methylhexane-2,5-dihydroperoxide decays rapidly with first-order kinetics [k = ioio.e exp( -11,500 1000)/RT sec.1 = 180 sec."1 at 30°C. (30)]. After the first-order decay has run to completion, there is a residual radical concentration (—4% of the initial hydroperoxide concentration) which decays much more slowly by a second-order process. The residual second-order reaction cannot be eliminated or changed even by repeated recrystallization of the dihydroperoxide. This suggests that a small fraction of the biperoxy radicals react intermolecularly rather than by an intramolecular process and thus produce monoperoxy radicals. The bimolecular decay constant for this residual species of peroxy radical is similar to that found for the structurally similar radical from 1,1,3,3-tetra-methylbutyl hydroperoxide. Photolysis of the dihydroperoxide gave radicals with second-order decay kinetics which are presumed to be 2,5-hydroperoxyhexyl-5-peroxy radicals. 

A number of unsteady state problems in diffusion have been considered, in which chemical reactions are occurring (F6, G8). Sherwood and Pigford (S9, pp. 332-337) have studied the unsteady state absorption of a substance A which diffuses into the solvent S and undergoes an infinitely fast, irreversible, second-order reaction with a solute B (that is A + B— AB). It is assumed that Fick s second law adequately describes the diffusion process and that because of the infinitely fast reaction of A and B there will be a plane parallel to the liquid surface at a distance a from it, which separates the region containing no A from that containing no B. The distance s is a function of t, inasmuch as the boundary between A and B retreats as B is used up in the chemical reactions. 

In Chapter 8, we addressed proton transfer reactions, which we have assumed to occur at much higher rates as compared to all other processes. So in this case we always considered equilibrium to be established instantaneously. For the reactions discussed in the following chapters, however, this assumption does not generally hold, since we are dealing with reactions that occur at much slower rates. Hence, our major focus will not be on thermodynamic, but rather on kinetic aspects of transformation reactions of organic chemicals. In Section 12.3 we will therefore discuss the mathematical framework that we need to describe zero-, first- and second-order reactions. We will also show how to solve somewhat more complicated problems such as enzyme kinetics. 

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