Stoichiometry of multiple reactions

Multiple reactions have an extent of reaction, ei, for each reaction. If there are N components and M reactions. 

Equation 2.48 is a generalization to M reactions of the stoichiometric constraints of Equation 2.45. If the vector e is known, the amounts of all components that are consumed or formed by the reaction can be calculated. 

In this formulation,. S ) is the M x 1 vector of reaction rates. See Equation 

Numerical solutions are usually necessary. Once solved, the values for e can be used to calculate the N composition variables using Equation 2.48. These compositions can in turn be used in an equation of state to calculate the volume for variable-volume problems. 

Apply the reaction coordinate method to the reactions in Example 2.12. 

Consider a system with N chemical components undergoing a set of M reactions. Obviously, N M. Define the A x M matrix of stoichiometric coefficients as 

Note that the matrix of stoichiometric coefficients devotes a row to each of the N components and a column to each of the M reactions. We require the reactions to be independent. A set of reactions is independent if no member of the set can be obtained by adding or subtracting multiples of the other members. A set will be independent if every reaction contains one species not present in the other reactions. The student of linear algebra will understand that the rank of v must equal M. 

Using v, we can write the design equations for a batch reactor in very compact form  

Example 2.12 Consider a constant-volume batch reaction with the following set of reactions  

These reaction rates would be plausible if B were present in great excess, say as water in an aqueous reaction. Equation (2.38) can be written out as 

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A The first step will be rate-limiting. It will determine the rate for the entire reaction because it is slower than the other steps. This step is a unlmolecular process with the rate given by answer A. Choice C would be correct if the reaction as a whole were one elementary step instead of three, but the stoichiometry of a reaction composed of multiple elementary steps cannot be used to predict a rate law. 

One of the tasks closely related to documentation is simple calculations that have to be performed to prepare an experiment. The number of calculations performed, for instance, in the organic synthesis laboratory is quite small, but those calculations required are very important. The calculations associated with conversion of the starting materials to the product are based on the assumption that the reaction will follow simple ideal stoichiometry. In calculating the theoretical and actual yields, it is assumed that all of the starting material is converted to the product. The first step in calculating yields is to determine the limiting reactant. The limiting reactant in a reaction that involves two or more reactants is usually the one present in lowest molar amount based on the stoichiometry of the reaction. This reactant will be consumed first and will limit any additional conversion to product. These calculations, which are simple rules of proportions, are subject to calculation errors due to their multiple dependencies. 

The reactor design equations in this book can be applied to all components in the system, even inerts. When the reaction rates are formulated using Equation 2.8, the solutions automatically account for the stoichiometry of the reaction. This is the simplest and preferred approach, but it has not always been followed in this book. Several examples have ignored product concentrations when they do not affect reactions rates and when they are easily found from the amount of reactants consumed. Also, some of the analytical solutions have used stoichiometry to ease the algebra. The present section formalizes the use of stoichiometric constraints. We begin with a matrix formulation for the reaction rates of the components in multiple reactions. The presentation is rather elegant from a mathematical viewpoint and does have some practical utility. 

In order to quantify the rate of a chemical transformation, we need to introduce some definitions. First, we distinguish between different types of reactions based on the form used to describe eventual chemical transformation, as single or multiple reactions. Usually this can be done from material balance after examining the stoichiometry between reacting materials and products. If a single stoichiometric equation can present the transformation, this is a single reaction. If more than one equation is necessary to present all observed components and their transformations, this it is a case of multiple reactions. The examples are as following ... 

In order to analyze a biological process with a biomass, the starting point is the knowledge of the nutrient consumption rate, of the product generation rate and, especially, of the biomass growth rate. These different rates are related to each other through the stoichiometry of the reactions that take place inside the cells and, in a simple approach, they can be expressed by means of yield factors. Since the biomass growth is related to cell multiplication, its rate (per unit volume, for instance) can be set proportional to the biomass concentration X ... 

A Case of Multiple Reactions with Slightly More Complex Stoichiometry... 

Chapter 1 begins with a review of the stoichiometry of chemical reactions, which leads into a discussion of various definitions of the reaction rate. Both homogeneous and heterogeneous systems are treated. The material in this chapter recurs throughout the book, and is particularly useful in Chapter 7, which deals with multiple reactions. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

Stoichiometry (from the Greek stoikeion—element) is the practical application of the law of multiple proportions. The stoichiometric equation for a chemical reaction states unambiguously the number of molecules of the reactants and products that take part from which the quantities can be calculated. The equation must balance. 

If the reaction in the cell proceeds to unit extent, then the charge nF corresponding to integral multiples of the Faraday constant is transported through the cell from the left to the right in its graphical representation. Factor n follows from the stoichiometry of the cell reaction (for example n = 2 for reaction c or d). The product nFE is the work expended when the cell reaction proceeds to a unit extent and at thermodynamic equilibrium and is equal to the affinity of this reaction. Thus,... 

At least for ethylene hydrogenation, catalysis appears to be simpler over oxides than over metals. Even if we were to assume that Eqs. (1) and (2) told the whole story, this would be true. In these terms over oxides the hydrocarbon surface species in the addition of deuterium to ethylene would be limited to C2H4 and C2H4D, whereas over metals a multiplicity of species of the form CzH D and CsHs-jD, would be expected. Adsorption (18) and IR studies (19) reveal that even with ethylene alone, metals are complex. When a metal surface is exposed to ethylene, selfhydrogenation and dimerization occur. These are surface reactions, not catalysis in other words, the extent of these reactions is determined by the amount of surface available as a reactant. The over-all result is that a metal surface exposed to an olefin forms a variety of carbonaceous species of variable stoichiometry. The presence of this variety of relatively inert species confounds attempts to use physical techniques such as IR to char-... 

Equations (2.7) and (2.8) can be added directly because the number of electrons produced equals the number of electrons consumed. If this is not the case, equalization must be done as the preliminary step. After multiplication with four and realizing that H+ + OH- —> H20, the final equation showing the stoichiometry of the total redox reaction is as follows ... 

For multiple reactions, material balances must be made for each stoichiometry. An example is the consecutive reactions, A = B = C, for which problem P4.04.52 develops a closed form solution. Other cases of sets of first order reactions are solvable by Laplace Transform, and of course numerically. 

In this chapter we will discuss some aspects of the carbonylation catalysis with the use of palladium catalysts. We will focus on the formation of polyketones consisting of alternating molecules of alkenes and carbon monoxide on the one hand, and esters that may form under the same conditions with the use of similar catalysts from alkenes, CO, and alcohols, on the other hand. As the potential production of polyketone and methyl propanoate obtained from ethene/CO have received a lot of industrial attention we will concentrate on these two products (for a recent monograph on this chemistry see reference [1]). The elementary reactions involved are the same formation of an initiating species, insertion reactions of CO and ethene, and a termination reaction. Multiple alternating (1 1) insertions will lead to polymers or oligomers whereas a stoichiometry of 1 1 1 for CO, ethene, and alcohol leads to an ester. 

This is the first time we have encountered multiple reactions. For these in general, if it is necessary to write N stoichiometric equations to describe what is happening, then it is necessary to follow the decomposition of N reaction components to describe the kinetics. Thus, in this system following C, or Cr, or Q alone will not give both ki and k2. At least two components must be followed. Then, from the stoichiometry, noting that + Cr + Q is constant, we can find the concentration of the third component. 

The key to optimum design for multiple reactions is proper contacting and proper flow pattern of fluids within the reactor. These requirements are determined by the stoichiometry and observed kinetics. Usually qualitative reasoning alone can already determine the correct contacting scheme. This is discussed further in Chapter 10. However, to determine the actual equipment size requires quantitative considerations. 

These rates are the rates of production of species A, B, and C (rj = Vjr) so these rates are written as negative quantities for reactants and positive quantities for products. This notation quickly becomes cumbersome for complex reaction stoichiometry, and the notation is not directly usable for multiple reaction systems. 

Each system considered in this section has a space of overall reactions whose dimension exceeds one. In many industrial reactions involving organic substances a major product is formed, but a side reaction contributes to loss in selectivity or yield of the desired product. Such cases may be said to exhibit a multiple overall reaction, unless the ratio of desired product to by-product remains constant over a range of operating conditions, so that a simple chemical equation might be employed to express the stoichiometry. 

The stoichiometry of reaction (14) is reminiscent of the electroreduction of 02 to H20. It is of interest that oxygenation of vanadium(III) gives oxovanadi-um(V), whereas multiple one-electron redox centers have been considered to be essential for the incorporation of 02. An explanation for this can be found in... 

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