Rate equation

A rate equation is simply a mathematical function that is found to describe the observed rate of release or uptake of a given counter-ion with respect to the resin or external solution. This is not to be confused with the rate mechanism, or to put it another way, a fit of data to a rate equation does not necessarily prove the mechanism. Many types of mathematical functions have been shown to describe 

Rate laws of the type which describe bimolecular second order chemical reactions might be expected to be a model for ion exchange reactions, and indeed this was the case for exchangers of both natural and synthetic origin. For example, the rate of ion exchange could be described by a bimolecular second order rate equation for irreversible reaction of the form  

Two further examples are worthy of comment. Firstly the second order bimolecular rate equation for reversible chemical equilibrium given by equation 6.8 was found to fit rate data for ion exchange on carbonaceous exchangers  

Secondly, unimolecular first order kinetics given by equation 6.9 is found to fit ion exchange rate data generated under film diffusion control. 

The various symbols have the same meaning as in equation 6.7 X is the equilibrium concentration in solution of the ion originally in the exchanger, and k represents the reaction rate constant for the chosen direction of exchange. The fractional attainment of equilibrium is given by X/X whilst the fractional equilibrium conversion of the resin is equal to Xja. 

A macroscopic reaction rate is defined as the derivative of the concentration of some species involved in the reaction with respect to time. 

Vx is the signed stoichiometric number for species X in the overall atom-balanced reaction 

If X is a product species, Vx is negative corresponding to X increasing as the reaction progresses, whereas Vx is positive if X is a reactant. In general, the rate R is some function of the concentrations of the species in the system, and soR = f ([A], [B], [C].). A rate equation is a specific equality between the time derivative in Equation 6.16 and the function of concentrations, R([A], [B], [C].) it is a differential equation in time. The concentrations, for example, [A] and [B], are functions of time because as a reaction progresses, concentrations change. The units for a reaction rate are the units of concentration and inverse time. 

Reactions that have rate equations with a linear dependence on only one concentration are said to be first order reactions. Reactions that depend on the concentration of only one species to the second power or that depend on the product of the concentrations of two species, each to the first power, are second order reactions. If a reaction happens to be independent of concentration, it is a zero order reaction. In general, if a reaction rate is given by 

An elementary reaction is one in which the reaction occms in a single collision. A number of elementary reaction steps may be involved in an overall reaction. The molecularity of an 

Any attempt to formulate a rate equation for solid-catalyzed reactions starts from the basic laws of chemical kinetics encountered in the treatment of homogeneous reactions. However, care has to be taken to substitute in these laws the concentrations and temperatures at the locus of reaction itsdf. These do not necessarily 

The application of Langmuir isotherms for the various reactants and products was begun by Taylor, in terms of fractional coverage, and the more convenient use of surface concentrations for complex reactions by Hougen and Watson [27], Thus, the developments below are often termed Langmuir-Hinshelwood-Hougen-Watson (L-H-H-W) rate equations. 

If both reactions are assumed to be of first order, the net rate of reaction of Al is  

Note that adsorption equilibrium constants are customarily used, rather than both adsorption and desorption constants. 

Since the overall reaction is the sum of the individual steps, the ordinary thermodynamic equilibrium constant for the overall reaction is 

The derivation of rate equations for simple monosubstrate reactions was described in Chapter 3. For bisubstrate reactions, the derivation is usually much more complex, and requires the application of a special mathematical apparatus and special mathematical procedures. 

determine whedier die rate equations for the forward and reverse rates of a reversible reaction are thermodynamically consistent  

calculate heats of reaction and equilibrium constants at various temperatures (review of thermodynamics). 

In order to design a new reactor, or analyze the behavior of an existing one, we need to know the rates of all the reactions that take place, hi particular, we must know how the rates vary with temperature, and how they depend on the concentrations of the various species in the reactor. This is the field of chemical kinetics. 

This chapter presents an overview of chemical kinetics and introduces some of the molecular phenomena that provide a foundation for the field. The relationship between kinetics and chemical thermodynamics is also treated. The information in this chapter is sufficient to allow us to solve some problems in reactor design and analysis, which is the subject of Chapters 3 and 4. In Chapter 5, we will return to the subject of chemical kinetics and treat it more fundamentally and in greater depth. 

A rate equation is used to describe the rate of a reaction quantitatively, and to express the functional dq endence of the rate on temperature and on the species concentrations. In symbolic form, 

A chemical reaction converts one or more reactants into one or more products and the rate equation describes how fast this conversion occurs. The reaction rate (r, mol/sec) is the time derivative of the number of moles of chemical species consumed or formed. 

The rate constant, k, incorporates the effects of variables such as temperature, pressure, or ionic strength on the rate. The rate constant can be thought of as a measure of the resistance to the conversion and small values of k produce small values of r. The effect of activity, concentration, or partial pressure of the reacting species on the rate appears in the rate equation as explicit terms each raised to an appropriate power. The product of these terms is the driving force for the reaction, so that large values of activity, partial pressure, or concentration produce large values of r. The exponents for the concentration, activity, or partial pressure terms are called partial orders ( ,) and the sum of those exponents for a rate equation is the overall reaction order, n.  

An elementary reaction step is a reaction that converts reactants directly to products through a single transition state (see Chapter 5). The reaction order for an elementary reaction step usually reflects the molecularity of the reaction. The molecularity of an elementary reaction step is the number of species that come together to form the activated complex. 

Composite reactions consist of multiple elementary reaction steps that occur in series, in parallel, or both. Many geochemical reactions are composites of several elementary reaction steps. This makes elucidating their reaction mechanisms very challenging because their reaction order and molecularity are not related in a simple way. Marin and Yablonsky (2011) offer extensive guidance about dealing with composite reactions. 

There are many ways to express reaction rates, and keeping track of notation for different kinds of rates along with the units of their accompanying rate equations is challenging. For a simple rate equation such as (3.1), the rate and the rate constant have units of mol/sec, which are the units expected from transition-state theory (Chapter 5). A reaction rate can also be expressed in terms of the time rate of change of concentration of a species (R, mol/kg sec = molal/sec), by dividing both sides of Eq. (3.1) by the mass of water (M) in the system. 

Differentiation of Equation (7.6b) with respect to time [37] leads to a system of differential rate equations defining auxiliary interrelated functions/](f)  

The lower limit of integration, - x , can be replaced by a finite value of time corresponding to the beginning of 

For instantaneous nucleation, the functions will have a simpler form, for instance  

The differentiation method was also used by other authors [38,39] to find sets of equations describing the conversion of melt into spherulites. The relevant equations are listed and discussed in Chapter 15. It has to be mentioned that although Equation (7.36) and Equation (7.38) create potential possibility to determine the nucleation rate or density, the practical use of such approach encounters difficulties because of the necessity of multiple differentiation and knowledge of the growth rate from independent experiments. 

Transport plays the overwhelming role in solid state kinetics. Nevertheless, homogeneous reactions occur as well and they are indispensable to establishing point defect equilibria. Defect relaxation in the (p-n) junction, as discussed in the previous section, illustrates this point, and similar defect relaxation processes occur, for example, in diffusion zones during interdiffusion [G. Kutsche, H. Schmalzried (1990)]. 

Let us first analyze the dynamic equilibrium with the help of a simple model. Atomic particles of crystal A are distributed over two sublattices, each of which has 

Instead of explicitly evaluating the equilibrium distribution by setting (9F/9/V,) = 0, let us evaluate the equilibrium condition (mass action law) from a kinetic approach. The basic kinetic equation is 

The second part of Eqn. (4.136) is true if the attempt frequency v° (Eqn. (4.131)) is independent of the composition. This kinetic steady state condition is obviously equivalent to the thermodynamic equilibrium condition. 

Equation (4.137) exists for each SE. Additionally, one has the following constraints 

At the time that the previous chapter in Volume 11 was written, the method of King and Altman (7) was the method of choice for deriving steady-state rate equations for enzymic reactions, and this is still true for any mechanism involving branched reaction pathways. The best description of this method may be found in Mahler and Cordes (8). A useful advance was made in 1975 with the introduction of the net rate constant method (9), and because it is the simplest method to use for any nonbranched mechanism, as well as for equations for isotopic exchange, positional isotopic exchange, isotope partitioning, etc., we shall present it here. 

The method involves replacing the mechanism being considered, such as the following  

In Mechanism (4), k, k, k, and kj are thus net rate constants which will produce the same rate and distribution of enzyme forms as in Mechanism (3). It is easy to demonstrate that the rate equation is then 

To determine the net rate constants for use in Mechanism (4), one starts with an irreversible step such as the release of Q in Mechanism (3). For such a step the net rate constant equals the real one  

That is, the net rate constant will be the real forward rate constant times the fraction of EQ that reacts forward as opposed to undergoing reversal to EPQ. Moving to the left again, 

The photon number inside the laser cavity and the population densities of atomic or molecular levels under stationary conditions of a laser can readily be obtained from simple rate equations. Note, however, that this approach does not take into account coherence effects (Chap. 12). 

With the pump rate P (which equals the number of atoms that are pumped per second and per cm into the upper laser level 2)), the relaxation rates RiNi (which equal the number of atoms that are removed per second and cm from the level /) by collision or spontaneous emission), and the spontaneous emission probability A21 per second, we obtain from (2.21) for equal statist - 

The loss coefficient determines the loss rate of the photon density n t) stored inside the optical resonator. Without an active medium N — N2 = 0), we obtain from (5.9c) 

A comparison with the definition (5.4) of the loss coefficient y per round-trip yields for a resonator with length d and round-trip time T = 2d/c 

Under stationary conditions we have dNi/dt = dN2/dt = dn/dr = 0. Adding (5.9a and 5.9b) then yields 

In a continuous-wave (cw) laser the pump rate equals the sum of photon loss rate fin plus the total relaxation rate N2 A2 -F R2) of the upper laser level. 

In Chapter 15 you will go on to meet a pair of mechanisms in which the polarity of the transition state is very different. You will now be prepared to expect some very significant solvent effects when such reactions take place. 

There is of course a link between the activation energy of a reaction and its rate, and the connection between them is known as the Arrhenius equation, after the Swedish chemist Svante Arrhenius (1859-1927) who formulated it and won the Nobel Prize in 1903. 

As we discussed on p. 253, the reaction between borohydride and the ketone to make an alkox-ide is only the first step of this reaction. Since ethanol likewise has to collide with the alkoxide for this second step to take place, you might very reasonably ask yourself why the rate of formation of the alcohol product does not also depend on [EtOH] why is the rate equation not 

The answer is hinted at in the energy profile diagram you saw on p. 253, which is reproduced below. The activation energy for the proton transfer step is lower than for the addition step, so it happens faster. It fact, it can happen fast whatever the concentration of ethanol, so ethanol does not appear in the rate equation. The overall rate of any reaction is determined only by what happens in the mechanistic step that is slowest, known as the rate-determining step or rate-limiting step. This is a general point about anything that happens in several 

EtOH H H / addition step EK Y ) proton transfer step EtO v 1 

The dissolution in neutral water is proportional to the interfacial area between silica and water. Since silica has low solubility in neutral water, the dissolution can be 

The dissolution rate equation for silica in nonflnoride solntions may also be expressed according to the surface complexation model described by Eq. (4.3) and Fig. 4.32 considering the contributions of different surface complexes sSiOHJ, sSiOH, =SiO-Na% and =SiO. The rate equation for the dissolution of silica in NaCl solutions at pH 2-13 at 25 °C can be described as  

According to Fig. 4.32, sSiOH is the predominant species and its concentration determines the dissolution rate at low pH. The low dissolution rates of silica in acidic solutions (Fig. 4.32) indicate that this species is most resistant to hydrolysis. The more reactive species sSiO-Na and SiO become important at higher pH at which the dissolution rate is faster as shown in Fig. 4.13. 

In acidic flnoride solutions, many etch rate equations, empirically or mechanistically derived, have been proposed. The most quoted rate equation is that of Judge, who found that the etch rate of thermal oxide in HF and NH4F solutions can be expressed as a function of HF and Hp2 concentrations  

Curve fitting indicated that the rate of attack by HF2 is about four to five times that of HF as shown in Fig. 4.36. A more general eqnation, including the effect of temperature, is 

If species A is the limiting reactant (present in least amount), the maximum extent of reaction is found from 

Either conversion or extent of reaction can be used to characterize the amount of reaction that has occurred. For industrial applications, the conversion of a feed is usually of interest, while for scientific applications, such as irreversible thermodynamics [Prigogine, 1967], the extent is often more useful. Further details are given by Boudart [1968] and Aris [1969]. 

With this rate, the change in moles of any species is, for a single reaction, 

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Consider the system of parallel reactions from Eq. (2.4) with the corresponding rate equations. " ... 

Further consideration of the reaction system reveals that the ammonia feed takes part only in the primary reaction and in neither of the secondary reactions. Consider the rate equation for the primary reaction ... 

The solution to the usual macroscopic kinetic rate equations for the reactant and product concentrations yields... 

Flere, A and B are regarded as pool chemicals , with concentrations regarded as imposed constants. The concentrations of the intemiediate species X and Y are the variables, with D and E being product species whose concentrations do not influence the reaction rates. The reaction rate equations for [X] and [Y] can be written in the following dimensionless fomi ... 

A striking feature of the images is the nonunifonnity of the distribution of the adsorbed species. The reaction between O and CO takes place at the boundaries between the surface domains and it was possible to detennine reaction rates by measuring the change in length L of the boundaries of the O islands. The kinetics is represented by the rate equation... 

The rate equation describing the kinetics of this reaction is... 

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

Using tire above equations, tire rate equation for production of batli molecules in a given quantum state due to collisions with a hot donor molecule can be written (e.g. for equation (c3.3.5))... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

When reaction rate equations can be given for the individual steps of a reaction sequence, a detailed modeling of product development over time can be made ... 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

With these reaction rate constants, differential reaction rate equations can be constructed for the individual reaction steps of the scheme shown in Figure 10.3-12. Integration of these differential rate equations by the Gear algorithm [15] allows the calculation of the concentration of the various species contained in Figure 10.3-12 over time. This is. shown in Figure 10.3-14. 

Nitric acid being the solvent, terms involving its concentration cannot enter the rate equation. This form of the rate equation is consistent with reaction via molecular nitric acid, or any species whose concentration throughout the reaction bears a constant ratio to the stoichiometric concentration of nitric acid. In the latter case the nitrating agent may account for any fraction of the total concentration of acid, provided that it is formed quickly relative to the speed of nitration. More detailed information about the mechanism was obtained from the effects of certain added species on the rate of reaction. 

Second-order rate coefficients for nitration in sulphuric acid at 25 °C fall by a factor of about 10 for every 10 % decrease in the concentration of the sulphuric acid ( 2.4.2). Since in sulphuric acid of about 90% concentration nitric acid is completely ionised to nitronium ions, in 68 % sulphuric acid [NO2+] io [HNO3]. The rate equation can be written in two ways, as follows ... 

For such a process the following rate equation holds ... 

The rate equation for the S z form is, from the steady-state approximation, as follows ... 

Use of the Bronsted rate equation gives the following expression rate = ka Q+aArnlft-... 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

The experimentally observed rates of mass transfer are often proportional to the displacement from equiHbrium and the rate equations for the gas and Hquid films are... 

Rate Equations with Concentration-Independent Mass Transfer Coefficients. Except for equimolar counterdiffusion, the mass transfer coefficients appHcable to the various situations apparently depend on concentration through thej/g and factors. Instead of the classical rate equations 4 and 5, containing variable mass transfer coefficients, the rate of mass transfer can be expressed in terms of the constant coefficients for equimolar counterdiffusion using the relationships... 

This leads to rate equations with constant mass transfer coefficients, whereas the effect of net transport through the film is reflected separately in thej/gj and Y factors. For unidirectional mass transfer through a stagnant gas the rate equation becomes... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Equation 39 can often be simplified by adopting the concept of a mass transfer unit. As explained in the film theory discussion eadier, the purpose of selecting equation 27 as a rate equation is that is independent of concentration. This is also tme for the Gj /k aP term in equation 39. In many practical instances, this expression is fairly independent of both pressure and Gj as increases through the tower, increases also, nearly compensating for the variations in Gj. Thus this term is often effectively constant and can be removed from the integral ... 

Xm are not. For unimolecular diffusion through stagnant gas = 1), and reduce to T and X and and reduce to and equation 64 then becomes equation 34. For equimolar counterdiffusion = 0, and the variables reduce tojy, x, G, and F, respectively, and equation 64 becomes equation 35. Using the film factor concept and rate equation 28, the tower height may be computed by... 

This rate equation must satisfy the boundary conditions imposed by the equiUbrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to 2ero at equiUbrium. It maybe a linear driving force expression of the form... 

If a self-sustained oxidation is carried out under limiting rate conditions, the hydroperoxide provides the new radicals to the system (by reaction 4 or analogues) and is maintained at a low concentration (decomposition rate = generation rate). For these circumstances, the rate equation 9 holds, where n = average number of initiating radicals produced (by any means) per molecule of ROOH decomposed and / = fraction of RH consumed which disappears by ROO attack (25). 

Droplet trajectories for limiting cases can be calculated by combining the equations of motion with the droplet evaporation rate equation to assess the likelihood that drops exit or hit the wall before evaporating. It is best to consider upper bound droplet sizes in addition to the mean size in these calculations. If desired, an instantaneous value for the evaporation rate constant may also be used based on an instantaneous Reynolds number calculated not from the terminal velocity but at a resultant velocity. In this case, equation 37 is substituted for equation 32 ... 

Activation Parameters. Thermal processes are commonly used to break labile initiator bonds in order to form radicals. The amount of thermal energy necessary varies with the environment, but absolute temperature, T, is usually the dominant factor. The energy barrier, the minimum amount of energy that must be suppHed, is called the activation energy, E. A third important factor, known as the frequency factor, is a measure of bond motion freedom (translational, rotational, and vibrational) in the activated complex or transition state. The relationships of yi, E and T to the initiator decomposition rate (kJ) are expressed by the Arrhenius first-order rate equation (eq. 16) where R is the gas constant, and and E are known as the activation parameters. 

The defects generated in ion—soHd interactions influence the kinetic processes that occur both inside and outside the cascade volume. At times long after the cascade lifetime (t > 10 s), the remaining vacancy—interstitial pairs can contribute to atomic diffusion processes. This process, commonly called radiation enhanced diffusion (RED), can be described by rate equations and an analytical approach (27). Within the cascade itself, under conditions of high defect densities, local energy depositions exceed 1 eV/atom and local kinetic processes can be described on the basis of ahquid-like diffusion formalism (28,29). 

The Rate Law The goal of chemical kinetic measurements for weU-stirred mixtures is to vaUdate a particular functional form of the rate law and determine numerical values for one or more rate constants that appear in the rate law. Frequendy, reactant concentrations appear raised to some power. Equation 5 is a rate law, or rate equation, in differential form. 

If the same measurement is repeated for different [BJ it should be possible to extractjy by plotting log vs log [BJ. This should be a straight line with slopejy. In a similar manner, can be obtained by varying [CJ. At the same time the assumption that x equals 1 is confirmed. Ideally, a variety of permutations should be tested. Even if xis not 1, and the integrated rate equation is not a simple exponential, a usefiil simplification stiU results from flooding all components except one. 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

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