Rate Laws with Explicit Solutions

There is a limited number of reaction mechanisms with sets of ODEs that can be integrated analytically, i.e. for which there are explicit formulas to calculate the concentrations of the reacting species as a function of time. 

This set includes all reaction mechanisms that contain only first order reactions, as well as very few mechanisms with second order reactions. Any textbook on chemical kinetics or physical chemistry supplies a list. A few examples for such mechanisms are given below  

Integration of the ODEs results in the concentration profiles for all reacting species as a function of the reaction time and the initial concentrations. The explicit solutions for the ODEs above (3.75) are given below (3.76). We list the equations for one concentration only. The remaining concentrations can be calculated from the closure principle, which is nothing else but the law of conservation of mass (e.g. in the first example [B]=[A]o-[A], where [A]o is the concentration of A at time zero). Only in example d) two concentrations need 

Solutions for the integration of ODE s, such as the ones given in equations (3.76), are not always readily available. For non-specialists it is difficult to determine if there is an explicit solution at all. The symbolic toolbox (which is not contained in the standard Matlab) provides very convenient means to integrate systems of differential equations and also to test whether there is an explicit solution. As an example the reaction 2A —-— B  

Note that Matlab s symbolic toolbox demands lower case characters for species names. 

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Integration of the appropriate differential equations for the reaction scheme is straightforward, the resulting equations for the concentrations of A, B, and C as a function of time (see Chapter 3.4.2, Rate Laws with Explicit Solutions) are ... 

In the most general case, when there exists no restriction with respect to the relative magnitude of rate constants k, and the relative concentration of E and A, and the particular time t for which we want to establish the rate law, the explicit solution is not possible. The four differential Eqs. (3.10)-(3.13) plus the conservation Eq. (3.14) provide us with four unknowns and they can only be integrated with the help of a computer. A typical solution, obtained with a KINSIM computer program (Barshop et al, 1983), is shown in Fig. 3. 

There are many analytical chemistry textbooks that deal with the chemical equilibrium in fairly extensive ways and demonstrate how to resolve the above system explicitly. However, more complex equilibrium systems do not have explicit solutions. They need to be resolved iteratively. In kinetics, there are only a few reaction mechanisms that result in systems of differential equations with explicit solutions they tend to be listed in physical chemistry textbooks. All other rate laws require numerical integration. 

The methods developed in the preceding section, but not the explicit equations, are applicable for reactions that are not second-order. We start with an example of the reaction between Fe(IIl) and Sn(II), as studied in solutions of HCIO4, HC1, and LiC104. With these components both [H+] and [Cr ] could be varied at constant p. We consider conditions in which the major species are Fe3 and Sn2q+. The rate law is... 

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