Differential rate expression

Equation (8) is the differential rate expression for a first-order reaction. The value of the rate constant, k, could be calculated by determining the slope of the concentration versus time curve at any point and dividing by the concentration at that point. However, the slope of a curved line is difficult to measure accurately, and k can be determined much more easily using integrated rate expressions. 

If c0 is the initial concentration of the reactant and c the concentration of reactant at any time t, the differential rate expression may be given as... 

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... 

Caution 1. In the special case where reactants are introduced in their stoichiometric ratio, the integrated rate expression becomes indeterminate and this requires taking limits of quotients for evaluation. This difficulty is avoided if we go back to the original differential rate expression and solve it for this particular reactant ratio. Thus, for the second-order reaction with equal initial concentrations of A and B, or for the reaction... 

Another method to obtain kinetic data is to use a differential reactor in which the concentration does not change much from the initial concentration Cao- this case the differential rate expression... 

This differential rate expression shows that the reaction rate is directly dependent on the concentration of A—the greater is [A], the faster is A converted to B. The reaction is said to be first order with respect to A because the exponent of [A] is 1. 

Only a relatively few kinetic schemes for complex reactions have differential rate expressions which can be integrated straightforwardly. Examples are the following. 

This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)—(3.66). 

Equation (2.23)—(2.25) show the three steps in this cycle and their corresponding differential rate expressions. 

The Use of Differential Rate Expressions According to this method, which was devised by van t Hoff, the rate of an nth-order reaction is given by... 

The differential rate expression for the different types of radicals in the aqueous phase are as follows ... 

Although the rate law derived from this mechanism is completely integrable, it was simpler to test the data in terms of the differential rate expression. This expression, assuming Reactions 1 and 2 to be at the low pressure limit, may be put in the following form ... 

At sufficiently high concentrations of arene so that Reaction 1 is complete at the end of the pulse, and at sufficiently low pulse intensity so that the rate of Reaction 3 is negligible, the differential rate expression for the decay of the radical anion is adequately represented (2) by ... 

The general differential rate expressions then have the form... 

If the rate of Reaction 18.5 is measured while the reaction goes to a significant degree of completion under conditions where [A]o is of the same order of magnitude as [R]o, then the method is called a second-order method and the exact differential-rate expression (Eqn. 18.6) must be employed in analyzing the data. Note also that only when the reaction mechanism is virtually irreversible can the reverse reaction be ignored in Equation 18.6. Furthermore, for the special case [R]o = [A]o, a modified form of the calculation of initial concentrations must be used. 

For termination by combination, a sharper distribution results, but a general derivation including various types of termination is more easily obtained using kinetic instead of probabilistic arguments. In the kinetic approach, differential rate expressions are written for each polymer species and the QSS approximation is used. The derivation is based on the kinetic scheme listed in Table 4.6. 

If reaction occurs by collision and interaction of two molecules A and B, the rate of reaction will be proportional to the number of collisions. The number of collisions in unit volume is seen from simple kinetic considerations to be proportional to the product of the concentrations of A and B. Hence we may write as the differential rate expression for this second-order reaction... 

The present pure calculations illustrate the significance of non-Boltzmann rate coefficients for hot atom reaction systems, bxnce i (t) is a strongly varying function, Eq. 22 cannot be approximated by a first order linear differential rate expression. The mean hot atom reactive lifetime is given by Eq. 26. 

Solution The differential rate expression for a second order reaction of A with B is... 

With this relation, we substitute for [B] in the product fc[A][B] in the differential rate expression and obtain an expression involving only the concentration variable [A]. We can also substitute for [A] and obtain an expression for [B]. 

The second reaction is simply the reverse of the first. However, the rates for the two reactions are intrinsically different. The first reaction is spontaneous and first order. The second reaction is bimolecular and second order. A convention is to refer to one reaction as the forward reaction, and to the other as the reverse reaction. The forward reaction can be the one we intuitively take as the most favorable under the initial conditions of the system. Using k as the rate constant for the first reaction and k, for the second reaction in this simple example, the differential rate expression is... 

Find the differential rate expression for the change in the concentration of A. 

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