Rate laws exponents

The balanced equation for the overall reaction is equal to the sum of all the individual steps, including any steps that might follow the rate-determining step. We emphasize again that the rate-law exponents do not necessarily match the coefficients of the overall balanced equation. 

The exponents n and m in Eq. VII-1 are expected to be 1 and 3, respectively, under some conditions. Assuming spheres and letting Vr = V/Vq, derive the rate law dVrjdt =f(Vr). Expound on the peculiar nature of this rate law. 

The rate of a process is expressed by the derivative of a concentration (square brackets) with respect to time, d[ ]/dt. If the concentration of a reaction product is used, this quantity is positive if a reactant is used, it is negative and a minus sign must be included. Also, each derivative d[ ]/dt should be divided by the coefficient of that component in the chemical equation which describes the reaction so that a single rate is described, whichever component in the reaction is used to monitor it. A rate law describes the rate of a reaction as the product of a constant k, called the rate constant, and various concentrations, each raised to specific powers. The power of an individual concentration term in a rate law is called the order with respect to that component, and the sum of the exponents of all concentration terms gives the overall order of the reaction. Thus in the rate law Rate = k[X] [Y], the reaction is first order in X, second order in Y, and third order overall. 

A mechanism is a series of simple reaction steps which, when added together, account for the overall reaction. The rate law for the individual steps of the mechanism may be written by inspection of the mechanistic steps. The coefficients of the reactants in the chemical equation describing the step become the exponents of these concentrations in the rate law for... 

Results may be reported for any component. The functional form of the rate law and the exponents x,j, w,... are not affected by such an arbitrary choice. The rate constants, however, may change in numerical value. Similarly, the stoichiometric chemical equation may be written in alternative but equivalent forms. This also affects, at most, the numerical value of rate constants. Consequentiy, one must know the chemical equation assumed before using any rate constant. 

The goal of a kinetic study is to establish the quantitative relationship between the concentration of reactants and catalysts and the rate of the reaction. Typically, such a study involves rate measurements at enough different concentrations of each reactant so that the kinetic order with respect to each reactant can be assessed. A complete investigation allows the reaction to be described by a rate law, which is an algebraic expression containing one or more rate constants as well as the concentrations of all reactants that are involved in the rate-determining step and steps prior to the rate-determining step. Each concentration has an exponent, which is the order of the reaction with respect to that component. The overall kinetic order of the reaction is the sum of all the exponents in the... 

The order of a reaction (this is the common parlance more precisely,2 the order of a rate law) is the sum of the exponents of the concentration factors in the rate law. One can also refer to the order with respect to a particular species. Consider the reaction in Eq. (1-11), with the rate law given by Eq. (1-12) ... 

To construct an overall rate law from a mechanism, write the rate law for each of the elementary reactions that have been proposed then combine them into an overall rate law. First, it is important to realize that the chemical equation for an elementary reaction is different from the balanced chemical equation for the overall reaction. The overall chemical equation gives the overall stoichiometry of the reaction, but tells us nothing about how the reaction occurs and so we must find the rate law experimentally. In contrast, an elementary step shows explicitly which particles and how many of each we propose come together in that step of the reaction. Because the elementary reaction shows how the reaction occurs, the rate of that step depends on the concentrations of those particles. Therefore, we can write the rate law for an elementary reaction (but not for the overall reaction) from its chemical equation, with each exponent in the rate law being the same as the number of particles of a given type participating in the reaction, as summarized in Table 13.3. 

This equation is known as the rate law for the reaction. The concentration of a reactant is described by A cL4/df is the rate of change of A. The units of the rate constant, represented by k, depend on the units of the concentrations and on the values of m, n, and p. The parameters m, n, and p represent the order of the reaction with respect to A, B, and C, respectively. The exponents do not have to be integers in an empirical rate law. The order of the overall reaction is the sum of the exponents (m, n, and p) in the rate law. For non-reversible first-order reactions the scale time, tau, which was introduced in Chapter 4, is simply 1 /k. The scale time for second-and third-order reactions is a bit more difficult to assess in general terms because, among other reasons, it depends on what reactant is considered. 

Here, the equation for the crack growth rate Aa/AN is defined in terms of the maximum energy release rate Tmax. the power-law exponent F R), and the point at which the crack growth rate data converge. The power-law exponent F R) for the Mars-Fatemi model is of the form... 

The exponents in a rate law depend on the reaction mechanism rather than on the stoichiometry of the overall reaction. The order of reaction often differs from the stoiehiometrie eoejfieient. Consequently, a rate law must always be determined by conducting experiments it can never be derived from the stoichiometry of the overall chemical reaction. 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

The coefficients of the balanced overall equation bear no necessary relationship to the exponents to which the concentrations are raised in the rate law expression. The exponents are determined experimentally and describe how the concentrations of each reactant affect the reaction rate. The exponents are related to the ratedetermining (slow) step in a sequence of mainly unimolecular and bimolecular reactions called the mechanism of the reaction. It is the mechanism which lays out exactly the order in which bonds are broken and made as the reactants are transformed into the products of the reaction. 

Flere, As is the mineral s surface area (cm2) and k+ is the intrinsic rate constant for the reaction. The concentrations of certain species Aj, which make up the rate law s promoting and inhibiting species, are denoted m , and P- are those species exponents, the values of which are derived empirically. Qg is the activity product for Reaction 16.1 (Eqn. 3.41). In the absence of promoting and inhibiting species, the units of the rate constant are mol cm-2 s-1, and in any case these are the units of the product of k+ and the n term. 

Autocatalysis is a special type of molecular catalysis in which one of the products of reaction acts as a catalyst for the reaction. As a consequence, the concentration of this product appears in the observed rate law with a positive exponent if a catalyst in the usual sense, or with a negative exponent if an inhibitor. A characteristic of an autocat-alytic reaction is that the rate increases initially as the concentration of catalytic product increases, but eventually goes through a maximum and decreases as reactant is used up. The initial behavior may be described as abnormal kinetics, and has important consequences for reactor selection for such reactions. 

If a reactant concentration is doubled, and the reaction rate increases by a factor of 8, the exponent for that reactant in the rate law should be... 

C—The 2 exponent means this is a second-order rate law. Second-order rate laws give a straight-line plot for 1 /[A] versus t. 

The values of the exponents in the rate law equation establish the order of the reaction. If a given reactant is found to have an exponent of 1, the reaction is said to he first order in this reactant. Similarly, if the exponent of a reactant is 2, the reaction is second order in this reactant. The sum of the exponents (m + n) is known as the overall reaction order. For example, the rate law equation helow represents a reaction that is first order in A, second order in B, and third order (1 + 2) overall. 

For the decomposition reaction of hydrogen iodide, the value of the exponent of [HI] in the rate law equation is the same as its molar coefficient in the balanced chemical equation. This is not always the case. The values of the exponents in a rate law equation must be determined by experiment... 

As you have learned, the values of the exponents in a rate law equation must be determined experimentally. Chemists determine the values of m and n by carrying out a series of experiments. Each experiment has a different, known set of initial concentrations. All other factors, such as temperature, remain constant. Chemists measure and compare the initial rate of each reaction. Thus, this method is called the initial rates method. 

It must be found by experiment. Elementary reactions are the exception to this rule. For an elementary reaction, the exponents in the rate law equation are the same as the stoichiometric coefficients for each reactant in the chemical equation. Table 6.3 shows how rate laws correspond to elementary reactions. 

What are the exponents m and n, and the rate constant, in the following general rate law equation ... 

The size distribution for the large diameter fraction of the H2O oxidizer combustion products is shown in Fig. 8.8, along with the size distribution of the aluminum powder fuel. The mean particle size of the unburned fuel fraction in the combustion products is about 10.7 pm, while the mean size of the fuel particles is 17.4 pm. Most sources report that burning aluminum particles follow a rate law of the form d = do" — Pt, where / is a constant and the exponent n is between 1.5 and 2.0. In that case, the size distribution of the unburned fraction of the combustion products would be expected to be larger than that of the fuel. A size distribution of unburned aluminum smaller than that of the parent fuel is more consistent with particles that never ignited, since the larger particles would probably be undersampled. On the other hand, it seems unlikely that any particle could... 

The overall order of a chemical reaction is equal to the sum of the exponents of all the concentration terms in the differential rate expression for a reaction considered in one direction only. For example, if the empirical rate law for a particular chemical reaction was... 

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