The Rate Law and Its Components

The centerpiece of any kinetic study is the rate law (or rate equation) for the reaction in question. The rate law expresses the rate as a function of reactant concentrations, product concentrations, and temperature. Any hypothesis we make about how the reaction occurs on the molecular level must conform to the rate law because it is based on experimental fact. 

In this discussion, we generally consider reactions for which the products do not appear in the rate law. In these cases, the reaction rate depends only on reactant concentrations and temperature. First, we look at the effect of concentration on rate for reactions occurring at a fixed temperature. For a general reaction, 

Aside from the concentration terms, [A] and [B], the other parameters in Equation 16.3 require some definition. The proportionality constant k, called the rate constant, is specific for a given reaction at a given temperature it does not change as the reaction proceeds. (As you ll see in Section 16.5, k does change with temperature and therefore determines how temperature affects the rate.) The exponents m and n, called the reaction orders, define how the rate is affected by reactant concentration. Thus, if the rate doubles when [A] doubles, the rate depends on [A] raised to the first power, [A], so m = 1. Similarly, if the rate quadruples when [B] doubles, the rate depends on [B] raised to the second power, [B], so /I = 2. In another reaction, the rate may not change at all when [A] doubles in that case, the rate does not depend on [A], or, to put it another way, the rate depends on [A] raised to the zero power, [A]°, so w = 0. Keep in mind that the coefficients a and b in the general balanced equation are not necessarily related in any way to these reaction orders m and n. 

A key point to remember is that the components of the rate law—rate, reaction orders, and rate constant—must be found by experiment they cannot be deduced from the reaction stoichiometry. Chemists take an experimental approach to finding these components by 

Using concentration measurements to find the initial rate 2 Using initial rates from several experiments to find the reaction orders 

Using concentration measurements to find the initial rate 

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The Rate Law and Its Components Reaction Order Terminology Determining Reaction Orders Experimentally Determining the Rate Constant... 

The Rate Law and Its Components Laboratory Methods for Determining Initial Rate Determining Reaction Orders... 

When the stoichiometric coefficients, va, vy, etc., are included in the rate law, as in Equation 3.5, the reaction has a unique rate constant (k) under specified conditions regardless of whether the rate is measured by monitoring the changing concentration of A, B or C. It also follows from Equation 3.5 that (except for zero-order reactions) the instantaneous rate of a reaction changes as the reaction proceeds, as will be illustrated later in Fig. 3.1. Thus, k is the parameter which measures whether the reaction (imprecisely expressed) is fast or slow . In any case, it follows that any property of a reacting system which relates (preferably directly) to the concentration of any component in the chemical reaction maybe monitored to measure the rate and, hence, to investigate the rate law and quantify the rate constant. 

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. 

If the rate law depends on the concentration of more than one component, and it is not possible to use the method of one component being in excess, a linearized least squares method can be used. The purpose of regression analysis is to determine a functional relationship between the dependent variable (e.g., the reaction rate) and the various independent variables (e.g., the concentrations). 

The advection-dispersion equation follows directly from the transport laws already presented in this chapter, and the divergence principle. The latter states that the time rate of change in the concentration of a component depends on how rapidly the advective and dispersive fluxes change in distance. If, for example, more of component i moves into the control volume shown in Figure 20.1 across its left and front faces than move out across its right and back, the component is accumulating in the control volume and its concentration there increasing. The time rate of... 

It should be noted that for shear thinning and shear thickening behaviour the shear stress-shear rate curve passes through the origin. This type of behaviour is often approximated by the power law and such materials are called power law fluids . Using the negative sign convention for stress components, the power law is usually written as... 

Geochemical kinetics is stiU in its infancy, and much research is necessary. One task is the accumulation of kinetic data, such as experimental determination of reaction rate laws and rate coefficients for homogeneous reactions, diffusion coefficients of various components in various phases under various conditions (temperature, pressure, fluid compositions, and phase compositions), interface reaction rates as a function of supersaturation, crystal growth and dissolution rates, and bubble growth and dissolution rates. These data are critical to geological applications of kinetics. Data collection requires increasingly more sophisticated experimental apparatus and analytical instruments, and often new progresses arise from new instrumentation or methods. 

Answer, (a) Since the reaction is elementary and reversible, and it occurs in the gas phase, the rate law should be constructed via partial pressures instead of molar densities, particnlarly if the forward kinetic rate constant has dimensions of mol/volume time (atm)". The order of the reaction with respect to each component is eqnivalent to the magnitude of its stoichiometric coefficient. Reactant partial pressnres appear in the forward rate, and product partial pressures are used for the backward rate. The backward kinetic rate constant is rewritten in terms of the forward rate constant and the equilibrium constant based on gas-phase partial pressures. In agreement with all these statements,... 

We do experiments on systems in the real world but analyze the results on models of experiments done on models of the systems. We have in mind a model of the system which incorporates current hypotheses of its structure, rate laws, and values of some of the parameters. An experiment involves adding inputs and making measurements (outputs). So we are concerned with models of the experiments. Let x be the vector of state variables of the model. The inputs in the experiment are often described as the product of a matrix B and a vector of possible inputs, u. The inputs are combinations of the components of the vector u, i.e., Bu. For given initial conditions and input to the model, the time course of change in the vector of state variables is usually given by a set of differential equations. 

And is called kinetic equation or reaction rate law. Here r. is rate of reactions normalized over volume, C.,. is molar concentrations of reac-tants, k. is constant value characterizing the rate reactions at reactants concentration equal to 1, which is called reaction rate constant or intrinsic reaction rate, v.. is stoichiometric coefficient of the component i usually called partial order of reaction. Sum of one reaction partial order determines order of the reaction overall or order of its rate law. Elementary reactions (acts) dominate, which are subject to the rate law of zero, first and second order. For instance, for an elementary direct reaction... 

The method of initial rates might not reveal the full rate law. for the products may participate in the reaction and affect the rate. For example, products participate in the synthesis of HBr. where the full rate law depends on the concentration of HBr. To avoid this difficulty, the rate law should be fitted to the data throughout the reaction. The fitting may be done, in simple cases at least, by using a proposed rate law to predict the concentration of any component at any time, and comparing it with the data. 

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