Algebraic acid-base equations

The arsenous acid-iodate reaction is a combination of the Dushman and Roebuck reactions [145]. These reactions compete for iodine and iodide as intermediate products. A complete mathematical description has to include 14 species in the electrolyte, seven partial differential equations, six algebraic equations for acid-base equilibriums and one linear equation for the local electroneutrality. 

Many real-world applications of chemistry and biochemistry involve fairly complex sets of reactions occurring in sequence and/or in parallel. Each of these individual reactions is governed by its own equilibrium constant. How do we describe the overall progress of the entire coupled set of reactions We write all the involved equilibrium expressions and treat them as a set of simultaneous algebraic equations, because the concentrations of various chemical species appear in several expressions in the set. Examination of relative values of equilibrium constants shows that some reactions dominate the overall coupled set of reactions, and this chemical insight enables mathematical simplifications in the simultaneous equations. We study coupled equilibria in considerable detail in Chapter 15 on acid-base equilibrium. Here, we provide a brief introduction to this topic in the context of an important biochemical reaction. 

T raditionally, titration curve calculations are described in terms of equations that are valid only for parts of the titration. Equations will be developed here that reliably describe the entire curve. This will be done first for acid-base titration curves. In following chapters, titration curves for other reaction systems (metal complexation, redox, precipitation) will be developed and characterized in a similar fashion. For all, graphical and algebraic means of locating the endpoints will be described, colorimetric indicators and how they function will be explained, and the application of these considerations to (1) calculation of titration errors, (2) buffo design and evaluation, (3) sharpness of titrations, and finally, (4) in Chapter 18, the use of titration curve data to the determination of equilibrium constants will be presented. 

A variety of kinetic experiments is used to deduce this information. The algebraic form of the rate equation as a function of substrate concentrations limits the kinetic mechanism, wh inhibition patterns for products or dead-end inhibitors versus the various substrates pin it down, and often help to determine the rate-limiting step. Isotope exchange and partitioning studies complete the analysis of kinetic mechanism. The chemical mechanism is deduced by studying the pH variation of the kinetic parameters, which identifies the acid-base catalysts, and necessary protonation states of the substrate for binding and catalysis, and by certain kinetic isotope effect studies. 

One may question the need for a four parameter enthalpy equation, i.e., whether describing an acid or base by two parameters each is redundant. The following simple matrix algebra shows the conditions whereby a four parameter model reverts to a less redundant two parameter equation. Letting A be the transformation matrix, E and C represent the parameters for the four parameter model, and a represent the acid parameters for the two parameter model, the following equation results ... 

In performing a calculation based on an acid or base ionization constant expression such as Eqs. (13-7) or (13-8), there are often many unknowns. Remember that in an algebraic problem involving multiple unknowns, one needs as many equations as there are unknowns. The equilibrium constant expression itself is one equation, and the Kw expression is always available. Two other types of equation are often useful equations expressing... 

We recognize that both NaCH3COO and NaCN are salts of strong bases and weak acids. The anions in such salts hydrolyze to give basic solutions. As we have done before, we first write the appropriate chemical equation and equihbrium constant expression. Then we complete the reaction summary, substimte the algebraic representations of equilibrium concentrations into the equilibrium constant expression, and solve for the unknown concentration(s). 

In this section, we present the use of MATLAB to model chemical reactors. We focus on two examples, a semibatch reactor and a fixed-bed reactor. Both represent common cases of study such as the production of polymethyl methacrylate (PMMA) in suspension and the oxidation of SO2 to SO3 that is one of the steps in the production of sulfuric acid via heterogeneous method. The models are based on explicit algebraic equations and differential equations. Thus, we use ODEXX function in MATLAB to solve the concentration, temperature, and/or pressure profiles along the operation of such equipment. 

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