Expressing the Reaction Rate

A rate is a change in some variable per unit of time. The most common examples relate to the rate of motion (speed) of an object, which is the change in its position (that is, the distance it travels) divided by the change in time. Suppose, for instance, we measure a runner s starting position, Xi, at time t and final position, xj, at time t2- The runner s average speed is 

In the case of a chemical change, we are concerned with the reaction rate, the changes in concentrations of reactants or products per unit time reactant concentrations decrease while product concentrations increase. Consider a general 

We use square brackets, [ ], to express concentration in moles per liter. That is, [A] is the concentration of A in mol/L, so the rate expressed in terms of A is 

The rate has units of moles per liter per second (mol L s , or mol/L-s), or any time unit convenient for the particular reaction (minutes, years, and so on). 

If instead we measure the product to determine the reaction rate, we find its concentration increasing over time. That is, cone B2 is always higher than cone B. Thus, the change in product concentration, A[B], is positive, and the reaction rate for A----- B expressed in terms of B is 

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Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

In considering chemical stability of a pharmaceutical, one musf evaluate the reaction order and reaction rate. The reaction order may be the overall order (the sum of the exponents of the concentration terms of fhe rate expression), or fhe order with respect to each reactant (the exponent of the individual concentration term in the rate expression). The reaction rate expression is a description of the drug concentration with respect to time. Most commonly, zero- and first-order reactions are encountered in pharmacy. 

When a reaction occurs between gaseous species or in solution, chemists usually express the reaction rate as a change in the concentration of the reactant or product per unit time. Recall, from your previous chemistry course, that the concentration of a compound (in mol/L) is symbolized by placing square brackets, [ ], around the chemical formula. The equation below is the equation you will work with most often in this section. 

The resolution of the system of equations (5.3-B1) and (5.3-C1) to eliminate the unknown concentration and express the reaction rate only as a function of pAg is ... 

The right-hand ordinate of Figure 6 expresses the reaction rate as percent of gas-phase SO2 oxidized per hour, per unit liquid water content of the cloud. Oxidation rates in these units may be compared to clear-air oxidation rates (of order 1% h l), although this comparison should be tempered by the small fraction of the boundary layer that is occupied by clouds. 

For Karstedt s catalyst, the same calculation showed that the overall kinetic order could be either 2 or 3, both curves F (a)=f(t) presenting similar shapes. We therefore could not express the reaction rate for this system. 

The basic equation most widely accepted to express the kinetics of ammonia synthesis is that of Teml and Pyzbev (1940). It expresses the reaction rate as a function of the partial pressures of the reactants and products ... 

Express the reaction rate (if any) in terms of concentration and substitute into the differential equation. 

Express the reaction rate in moles H2 consumed per liter per second and in moles NHj produced per liter per second. 

We use for expressing the reaction rate on independent routes equation (35). Then for the reaction rates of routes I) and II) we have ... 

As win be discussed later, we formulate the design equations of chemical reactors in terms of dimensionless quantities and would like to express the reaction rate constants in terms of them. We define dimensionless temperature... 

The selection of the reference reaction rate, ro, should be done in a way that can be used for all forms of rate expressions. Furflier, since the reaction rate depends on the initial (or inlet) composition, vq should be selected such that it is independent of die specific composition of the reference state (or stream). To define ro, we express the reaction rate at any instance by... 

Equation 6.2.5 is the integral form of the design equation for an ideal, constant-volume batch reactor. Figure 6.3 shows the graphical presentation of this design equation. To solve the design equations, we have to express the reaction rate r in... 

To solve the design equation, we have to express the reaction rate r in terms of Z and, to do so we relate the species concentrations to the dimensionless extent From Eq. 6.1.11, for isothermal operations with single reactions, and when the reference state is the initial state ... 

Express the reaction rates in terms of the dimensionless extents of the independent reactions, Z s. 

To solve Eq. 7.2.2 or 7.2.4, we have to express the reaction rate r in terms of the dimensionless extent Z. To do so, we express the species concentrations in terms of Z. Selecting the inlet stream as the reference stream, Zi = 0, and for liquid-phase reactions, Eq. 7.1.11 reduces to... 

To solve Eq. 9.4.5, we have to express the reaction rates in terms of the extents of the independent reactions. We do so by expressing the local volumetric flow rate and the local molar flow rates of all reactants in terms of Z, s and calculating the local species concentrations. Using Eqs. 2.7.8 and 2.7.10, the local molar... 

The decrease in the number of A molecules and the increase in the number of B molecules with time are shown in Figure 13.2. In general, it is more convenient to express the reaction rate in terms of the change in concentration with time. Thus, for the reaction A-----B we can express the rate as... 

Wilson [666, 667] and Bauman and Maron [47] show that it is possible to express the reaction rate as a function of film thickness, diffusion constant and solubility of oxygen in the film. When the thickness of the film is reduced to less than a certain value, the chemical reaction and not the diffusion becomes the controlling factor. The activation energy of the oxidation reaction amounts to 16—35 kcal mole-1, or even more, whereas the activation energy of the diffusion of gases in polymer films [39] is of the order of only 10 kcal mole-1. Control by diffusion is facilitated at higher temperatures by the decrease of oxygen solubility in the films. 

As the reaction leading to the complex involves electron transfer it is clear that the aetivation energy AG for complex formation can be lowered or raised by an applied potential (AO). Of course, both the forward (oxidation) and well as the reverse (reduction) reaction are influenced by AO. If one expresses the reaction rate as a current flow (/ ), the above equation C2.8.11 can be expressed in terms of the Butler-Volmer equation (for a more detailed... 

In distinction from the more refined, and thus much more complicated lattice-gas model, the form of the model of the surface electronic gas provides possibilities for its application to chemisorption of gas mixtures and thus to modelling of kinetics of complex reactions. Derivation of multicomponent chemisorption isotherms based on thermodynamic approach was presented in the previous chapter. Within the framework of this model the following generalized elementary reaction A+ZI+Z=>S is considered. This reaction is written as a three-body collision, which is highly improbable, but is presented here only for illustrative purposes of how to express the reaction rate... 

We call the values of k and n the intrinsic rate constant and reaction order to distinguish them from what we may estimate from data. The typical experiment is to change the value of ca in the bulk fluid, measure the rate n as a function of ca, and then find the values of the parameters k and n that best fit the measurements. We explain this procedure in much more detail in Chapter 9. Here we show only that one should exercise caution with this estimation if we are measuring the rates with a solid catalyst. The effects of reaction, diffusion and external mass transfer may all manifest themselves in the measured rate. We express the reaction rate as... 

This equation expresses the reaction rate (molecules per unit time) in terms of N, the number of molecules, multiplied by the rate constant k E, e,). The latter is expressed in terms of a ratio of the phase space areas (note that the integral is on the constant energy surface). These phase space areas can be converted into densities of states. In fact, the denominator of Eq. (6.71) is just the density of states multiplied by the factor h" The numerator is an integral over one less dimension, so that it is a density multiplied by Thus the rate constant, k E, ,) becomes ... 

For heterogeneous reaction systems it is appropriate to express the reaction rate relative to the surface area or mass of the catalyst Equation (3.1.3-3b) ... 

These partial reactions proceed with equal reaction rates when the metal is freely corroding. In order to express the reaction rates in terms of a current, the conversion per time is multiplied by the Faraday constant according to the following equation ... 

The most practical approach, however, is to express the reaction rate relative to the mass of catalyst to give an expression for the effective reaction rate (Eq. 5-14). 

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