Dependence of Rate on Concentration

For the relation between the corresponding pre-exponential factors A and A p,we use equations 3.1-8, and 4.2-5 and -9 to obtain 

If A and EA in the original form of the Arrhenius equation are postulated to be independent of T, then their analogues A p and EAp are not independent of T, except for a zero-order reaction. 

Applying the treatment used in the previous section to relate EA and EAp, corresponding to kip, and A and A corresponding to kip, with equation 4.2-5 replaced by equation 4.2- 

These results are similar to those in the previous section, with n — 1 replacing n, and similar conclusions about temperature dependence can be drawn, except that for a first-order reaction, ea = eAp and A = A,. The relationships of these differing Arrhenius parameters for a third-order reaction are explored in problem 4-12. 

Assessing the dependence of rate on concentration from the point of view of the rate law involves determining values, from experimental data, of the concentration parameters in equation 4.1-3 the order of reaction with respect to each reactant and the rate constant at a particular temperature. Some experimental methods have been described in Chapter 3, along with some consequences for various orders. In this section, we consider these determinations further, treating different orders in turn to obtain numerical values, as illustrated by examples. 

The rate of this reaction is observed to be proportional to the concentration of nitrogen dioxide. When the concentration of nitrogen dioxide is doubled, the rate doubles. The rate is also proportional to the concentration of fluorine doubling the concentration of fluorine also doubles the rate. 

A rate law is an equation that relates the rate of a reaction to the concentrations of reactants (and catalyst) raised to various powers. The following equation is the rate law for the foregoing reaction  

Note that in this rate law both reactant concentrations have an exponent of 1. Here k, called the rate constant, is a proportionality constant in the relationship between rate and concentrations. It has a fixed value at any given temperature, but it varies with temperature. Whereas the units of rate are usually given as mol/(L s), the units of k depend on the form of the rate law. For the previous rate law, 

As a more general example, consider the reaction of substances A and B to give D and E, according to the balanced equation 

The exponents m, n, and p are frequently, but not always, integras. They must be determined experimentally and they cannot be obtained simply by looking at the balanced equation. For example, note that the exponents in the equation Rate = k[N02][F2] have no relationship to the coefficients in the balanced equation 2NO2 + F2 - 2NO2F. 

The units of rate are given in terms of concentration per unit time or pressure per unit time, i.e. 

The conclusion follows that the rate depends in some way on [reactant] remaining, i.e. 

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The primary use of chemical kinetics in CRE is the development of a rate law (for a simple system), or a set of rate laws (for a kinetics scheme in a complex system). This requires experimental measurement of rate of reaction and its dependence on concentration, temperature, etc. In this chapter, we focus on experimental methods themselves, including various strategies for obtaining appropriate data by means of both batch and flow reactors, and on methods to determine values of rate parameters. (For the most part, we defer to Chapter 4 the use of experimental data to obtain values of parameters in particular forms of rate laws.) We restrict attention to single-phase, simple systems, and the dependence of rate on concentration and temperature. It is useful at this stage, however, to consider some features of a rate law and introduce some terminology to illustrate the experimental methods. 

Thus, the available evidence indicates that little or no adsorption of hydroquinone by silver occurs. Rabinovich s data are unacceptable because of the large experimental errors involved. The possible amount of adsorption indicated by the data of Perry, Ballard, and Sheppard does not exceed the limits of error in their analytical determination of hydroquinone and could not under any circumstances cover more than a small fraction of the silver surface. The kinetics of the reaction between hydroquinone and silver ions do not indicate adsorption of the reducing agent, although the first-order dependence of rate on concentration is not incompatible with weak adsorption. It seems unlikely, accordingly, that adsorption of hydroquinone by silver plays a role of any consequence in the silver catalysis of the reaction between hydroquinone and silver ion. 

Order of Dependence of Rate on Concentration of Initial Rate Activation Energy, Kcal./Mole... 

An early objective in a mechanistic investigation is to establish the rate law (see Chapter 3) which is an algebraic equation describing the instantaneous dependence of the rate on concentrations of compounds or other properties proportional to concentrations (e.g. partial pressures). Rate laws cannot be rehably deduced from the stoichiometry of the overall balanced chemical equation-they have to be determined experimentally. The functional dependence of rates on concentrations maybe simple or complicated, and concentrations may be of reactants, products or even materials not appearing in the overall chemical equation, as in the case of catalysis (see Chapters 11 and 12) [3-7]. 

The plot of pH versus time is linear, but the plot of [OH ] versus time is a curve. As in Worked Problem 2.4, this indicates a direct relation between loge [OH ]remaining and time, and is significant for the interpretation of the dependence of rate on concentration (see Chapter 3). 

The aim is to determine the dependence of rate on concentration for each reactant in turn. Choose one experiment and compare it with each of the others in turn. 

The field of chemistry that deals with rates of reaction, and in particular with dependence of rates on concentration, is called kinetics. Let us see what kinetics can tell us about nucleophilic aliphatic substitution. 

As indicated above, the dependence of rate on concentration can be shown to be of the general form... 

This order of events was confirmed by a study of the dependence of the observed rate at 340, 420, and 508 nm on the concentration of i-homophenyla-lanine in a series of SWSF experiments. These studies showed that in the concentration range between 0 and 1.2 mM /.-homophenylalanine, the fast processes at 420 nm and 340 nm exhibit a linear dependence of rate on concentration, giving an apparent bimolecular rate constant of 2.2 X 104 M "1 s 1 with an off-rate of nearly 15 s "1. Quinonoid formation at 508 nm exhibited a hyperbolic dependence of rate on L-homophenylalanine concentration, suggesting that quinonoid formation is closely coupled to a bimolecular binding step. These data were used to determine a for i-homophenylalanine of 110 pM, in good agreement with the K, determined by steady-state methods. 

In this Sect.4.9 we discuss Eqs. (4.156), (4.171) concerning chemical reactions in a regular linear fluids mixture (see end of Sect. 4.6), i.e. with linear transport phenomena. This model gives the (non-linear) dependence of chemical reaction rates on temperature and densities (i.e. on molar concentrations (4.288)) only (4.156), which is (at least approximately) assumed in classical chemical kinetics [132, 157]. Here, assuming additionally polynomial dependence of rates on concentrations, we deduce the basic law of chemical kinetics (homogeneous, i.e. in one fluid (gas, liquid) phase) called also the mass action law of chemical kinetics, by purely phenomenological means [56, 66, 79, 162, 163]. 

Rate Laws Experimental measurement of the rate leads to the rate law for the reaction, which expresses the rate in terms of the rate constant and the concentrations of the reactants. The dependence of rate on concentrations gives the order of a reaction. A reaction can be described as zero order if the rate does not depend on the concentration of the reactant, or first order if it depends on the reactant raised to the first power. Higher orders and fractional orders are also known. An important characteristic of reaction rates is the time required for the concentration of a reactant to decrease to half of its initial concentration, called the half-life. For first-order reactions, the half-hfe is independent of the initial concentration. 

EXERCISE 14.10 Write the rate equation, showing the dependence of rate on concentrations, for the elementary reaction... 

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