The definition of reaction rate

Very fast reactions can be studied by flash photolysis, in which the sample is exposed to a brief flash of light that initiates the reaction and then the contents of the reaction chamber are monitored spectrophotometrically. Biological processes that depend on the absorption of Ught, such as photosynthesis and vision, can be studied in this way. Lasers can be used to generate nanosecond flashes routinely, picosecond flashes quite readily, and flashes as brief as a few femtoseconds in special arrangements. Spectra are recorded at a series of times following the flash, using instrumentation described in Chapter 12. 

In a relaxation technique the reaction mixture is initially at equilibrium but is then disturbed by a rapid change in conditions, such as a sudden increase in temperature. The equilibrium composition before the application of the perturbation becomes the initial state for the return of the system to its equi-hbrium composition at the new temperature, and the return to equilibrium— the relaxation of the system—is monitored spectroscopically. Relaxation techniques are described in more detail in In the laboratory 7.1. 

The concepts introduced here for the description of reaction rates are used whenever we expiore such bioiogicai processes as enzymatic transformations, eiectron transfer reactions in metaboiism, and the transport of moiecuies and ions across membranes. The average rate of a reaction is defined in terms of the rate of change of the concentration of a designated species  

3 The rate of a chemical reaction is the slope (without the sign) of the tangent to the curve showing the variation of concentration of a species with time. This graph is a plot of the concentration of a reactant, which is consumed as the reaction progresses. The rate of consumption decreases in the course of the reaction as the concentration of reactant decreases. 

In general, the various reactants in a given reaction are consumed at different rates, and the various products are also formed at different rates. However, these rates are related by the stoichiometry of the reaction. For example, in the decomposition of urea, (NH2)2CO, in acidic solution 

---

It is prudent at this point to note that dilatometry does not fulfil the initial requirement of a reaction at constant volume (see Sections 3.2.1 and 3.5.2.1), so the definition of reaction rate needs to be revised to take account of the possibility that the volume may change as the reaction proceeds. Thus, for a first-order reaction, for example,... 

In practice, conditions in a reactor are usually quite different than the ideal requirements used in the definition of reaction rates. Normally, a reactor is not a closed system with uniform temperature, pressure, and composition. These ideal conditions can rarely if ever be met even in experimental reactors designed for the measurement of reaction rates. In fact, reaction rates cannot be measured directly in a closed system. In a closed system, the composition of the system varies with time and the rate is then inferred or calculated from these measurements. 

The definition of reaction rate developed in these two examples may be generalized to cover any reaction. 

Why must coefficients be included in the definition of reaction rate ... 

The simplified analysis of kinetics given here is only valid if the back reaction can be neglected. For example, as reaction 1.98 proceeds, the product C accumulates and may begin to dissociate back to A and B. (Eventually, once the back reaction rate equals that of the forward reaction, steady-state or equilibrium is achieved.) For this reason, kinetic studies are typically done in the early stages of a reaction before back reactions begin to invalidate the definition of reaction rate as given by equation 1.92. 

Notice in the definition of reaction rate, we are taking the reac- tion stoichiometry literally. We are postulating that these collision and transformation events are taking place at the molecular level. These literal reactions are known as elementary reactions. We delay a more complete discussion of elementary reactions and reaction mechanisms until Chapter 5. We will also see that for complex reacting systems, it may be difficult to know whether or not a reaction is an elementary reaction. But that is a separate issue, which we take up later, and that issue does not prevent us from defining the reaction rate. 

When analyzing kinetic data or designing a chemical reactor, it is important to state clearly the definitions of reaction rate, conversion, yield, and selectivity. For a homogeneous reaction, the reaction rate is defined either as the amount of product formed or the amount of reactant consumed per unit volume of the gas or liquid phase per unit time. We generally use moles (g mol, kg mol, or lb mol) rather than mass to define the rate, since this simplifies the material balance calculations. 

For reactions with two fluid phases, the definition of reaction rate is arbitrary. When a reactant gas is bubbled through a liquid in a tank or column, the rate could be expressed per unit volume of clear liquid or per unit volume of gas-liquid mixture, and these volumes may differ by 5-30%. Unless the reactor is made of glass or has several measuring probes, the froth height is unknown, and the original or clear liquid volume may have to be used to express the rate. Unfortunately, many literature sources do not state the basis for calculation when reporting kinetic data for gas-liquid systems. 

From the definition of reaction rate in a closed system... 

In order to compute ARs in mass fraction residence time space, we must also be able to express o as a component in the rate vector. From the definition of reaction rate, we know... 

Of course many alternative definitions of specific rates for homogeneous and heterogeneous reactions are conceivable. Many of these are in use but conversion to the standard forms (1.3.2) and (1.3.12) should be attempted in every case where enough information is available. Whatever the choice of the definition of reaction rate adapted to a particular situation, it must be proportional to the derivative with respect to time of the extent of reaction. 

The use of volume (usually a variable) in the definition of reaction rate is an unfortunate tradition in chemical kinetics. So is the use of concentrations (moles per unit volume). Alternative, more desirable schemes have not gained favor so far. See S. S. Penner, Chemical Reactions in Flow Systems, Butterworths, London, 1955. For a recent discussion of r vs ry, see Max S. Peters and Edward J. Skorpin-ski, J. Chem. Ed., 42, 329 (1965). 

Therefore, a high ratio of VJV reduces the problem of correcting the equation for the definition of reaction rates, but at the same time it may prolong the time needed to achieve measurable conversion. 

Determine the slope of the curve using numerical, analytical, or graphical methods e.g., (dcjdt) = for DCSTR. Table 4.4 lists the definitions of reaction rates in different reactor operations. 

Reaction rate. According to the recommendation of International Union of Pure and Applied Chemistry (lUPAC) in 1981, the definition of reaction rate is ... 

The definition of reaction rate can be nsed to interpret kinetic data with respect to liquid phase or with respect to catalyst volume or mass (Levespiek 1999). In the case of packed-bed reactors, the definition of reaction rate based on mass or volume of catalyst is useful, and superficial velocity can be used instead of intrinsic velocity. Thus, in the balance equations using superficial velocity the intrinsic kinetic rate constant is better related to the apparent rate constant by anploying the wetting efficiency factor (/J as follows ... 

Attempts to determine how the activity of the catalyst (or the selectivity which is, in a rough approximation, the ratio of reaction rates) depends upon the metal particle size have been undertaken for many decades. In 1962, one of the most important figures in catalysis research, M. Boudart, proposed a definition for structure sensitivity [4,5]. A heterogeneously catalyzed reaction is considered to be structure sensitive if its rate, referred to the number of active sites and, thus, expressed as turnover-frequency (TOF), depends on the particle size of the active component or a specific crystallographic orientation of the exposed catalyst surface. Boudart later expanded this model proposing that structure sensitivity is related to the number of (metal surface) atoms to which a crucial reaction intermediate is bound [6]. 

A definition of instantaneous fractional yield is based on the ratio of reaction rates... 

In homogeneous systems the volume of fluid in the reactor is often identical to the volume of reactor. In such a case V and Vj. are identical and Eqs. 2 and 6 are used interchangeably. In heterogeneous systems all the above definitions of reaction rate are encountered, the definition used in any particular situation often being a matter of convenience. 

In an alternate definition of reaction rates, one writes a rate as the rate offormation of each species. In that notation one would define, rg, and rc, with the definitions... 

In the relevant literature, many definitions of reaction rates can be found, especially in the case of catalytic systems. Depending on the approach followed, a catalytic reaction rate can be based on catalyst volume, surface, or mass. Moreover, in practical applications, rates are often expressed per volume of reactor. Each definition leads to different manipulations and special attention is required when switching from one expression to another, hi the following, the various forms of catalytic reaction rates and their connection is going to be presented. Stalling from the fundamental rate defined per active site, the reader is taken step -by step to the rate based on the volume of the reactor and the concept of the overall rate in two- and three-phase systems. 

The determination of rate of reaction in a flow system requires knowledge both of the feed rate, v, of a given reactant and of the fraction converted, x. The definition of feed rate as the amount of reactant fed per unit time to the inlet of the reactor is consistent with 1.5.1. The rate of reaction is then given by... 

Before designing a chemical reactor, one must know the reaction(s) rate. Rates of reaction can be written in intrinsic form or in terms of a specific reactant of interest. An intensive measure, based on a unit volume of fluid, is normally used for homogeneous reacting systems. Thus, the general definition of reaction rate can be written as... 

This is the general definition of reaction rate. Only in special cases is this equal to dC/dt (C = concentration). 

It is obvious that such a definition of solvent polarity cannot be measured by an individual physical quantity such as the relative permittivity. Indeed, very often it has been found that there is no correlation between the relative permittivity (or its different functions such as l/sr, (sr — l)/(2er + 1), etc.) and the logarithms of rate or equilibrium constants of solvent-dependent chemical reactions. No single macroscopic physical parameter could possibly account for the multitude of solute/solvent interactions on the molecular-microscopic level. Until now the complexity of solute/solvent interactions has also prevented the derivation of generally applicable mathematical expressions that would allow the calculation of reaction rates or equilibrium constants of reactions carried out in solvents of different polarity. 

This definition of reaction rate assumes that the volume remains constant. A more general definition of reaction rate is the change in the number of mols per unit of volume per unit of time. 

Slight variations in the definition of reaction order may be found in the literature. For example, for some reactions there may be reason to believe that one species (for example, nitrogen) is inert, in the sense that changes in its concentration do not influence the rate of the reaction at constant pressure, and in such cases rij may be defined by considering simultaneous changes in the concentrations of species j and of the inert, with the total pressure held constant. In general, the resulting value of rij diflers from that defined above therefore it is important to ascertain the specific definition employed. An empirical formula that is often useful—for example, in the presence of an inert—is... 

Thus far, the discussion of reaction rate has been confined to homogeneous reactions taking place in a closed system of uniform composition, temperature, and pressure. However, many reactions are heterogeneous they occur at the interface between phases, for example, the interface between two fluid phases (gas-liquid, liquid-liquid), the interface between a fluid and solid phase, and the interface between two solid phases. In order to obtain a convenient, specific rate of reaction it is necessary to normalize the reaction rate by the interfacial surface area available for the reaction. The interfacial area must be of uniform composition, temperature, and pressure. Frequently, the interfacial area is not known and alternative definitions of the specific rate are useful. Some examples of these types of rates are ... 

In the study of reaction rates of organic acids with lead (Turnbull and Frey, 6), a solid lead cylinder was rotated at high speeds in a solvent containing the acid. The liquid space in the vessel and velocity of rotation were chosen to yield Reynold s numbers of the order of 12,000 to 60,000, thus definitely creating turbulent flow in the medium. The reaction rates measured in this system were of the order of 10 moles/sec./cm. metal surface. While the bulk of the liquid medium was doubtless in turbulent... 

The amount of each substance present can be given by its concentration, usually expressed as molarity (mol/L) and designated by brackets. The rate at which the reaction proceeds can be described in terms of the rate at which one of the reactants disappears, —A[A]/At or —A[B]/Ar, or the rate at which one of the products appears, A[C]/Ar or A[D]/At. The reaction rate must be positive because it describes the forward (left-to-right) reaction, which consumes A and B. The concentrations of reactants A and B decrease in the time interval At. Thus, A[A]/Ar and A[B]/Ar would be negative quantities. The purpose of a negative sign in the definition of a rate of reaction is to make the rate a positive quantity. 

For a batch reactor the very definition of reaction rate gives the equation... 

---