Stoichiometric coefficient, defined

This set of relations between reaction orders and stoichiometric coefficients defines what we call an elementary reaction, one whose kinetics are consistent with stoichiometry. We later wiU consider another restriction on an elementary reaction that is frequently used by chemists, namely, that the reaction as written also describes the mechanism by which the process occurs. We will describe complex reactions as a sequence of elementary steps by which we will mean that the molecular collisions among reactant molecules cause chemical transformations to occur in a single step at the molecular level. 

P = total system pressure, typically 1.01325 x 10 Pa P = stoichiometric coefficient, defined by... 

Here N is the number of reagents and products involved in the reaction and a/ are a new set of stoichiometric coefficients, defined so that the product values are positive but the reagent values are negative. For the reaction defined by equation (13.6), N = 3 and the new stoichiometric coefficients are ... 

Stoichiometry The compounds and stoichiometric coefficients defining the bioreaction. 

These points may need some additional emphasis. For solid-solid reactions of the type designated by Eqs. (5.4.1a) and (5.4.1b) or (5.4.34) and (5.4.35), the stoichiometry of the system is not immediately defined and recourse must be made to the earlier defined dimensionless quantity y, which, together with the stoichiometric coefficient, defines the proportion of the solid reactants that will result in the complete consumption of both species. 

Thermodynamics defines the relationship governing the concentrations at equilibrium. The exponents of the equilibrium concentrations in the expression for the equilibrium constant are the same as the stoichiometric coefficients in the net equation, as in... 

The careful worker explicitly states the usage chosen for v. This choice defines the rate constant, which may otherwise be uncertain by a small numerical factor. When the definition is obvious, as in a reaction in which all the stoichiometric coefficients are unity, the definition of the numerical factor can be omitted. 

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... 

Now we work from completion to equilibrium. The equilibrium constant for this reaction is very large, but the partial pressure of NO cannot be zero at equilibrium. We define y to be the change in NO pressure on going from completion to equilibrium. Then the stoichiometric coefficients and Equation give -FO.5 y for the change in pressure of O2 and -y for the change in NO2. ... 

Most chemical reactions do not progress completely from reactants to products. Instead, the net reaction stops in the forward direction when equilibrium is established. Analysis of the contents of the reaction vessel would show a constant concentration of monomers and polymer once equilibrium is reached. This situation is actually a dynamic equilibrium, where the monomers are forming polymers at the same rate as the polymers depolymerize to monomer. Therefore, at equilibrium, the net concentrations of any one species remains constant. The amount of monomer converted into polymer will be defined by the equilibrium constant, K. This constant is the ratio of the concentration of the products to the reactants, with each concentration raised to the stoichiometric coefficients in the balanced equation. For Eq. 3.5 ... 

One of the challenges in kinetic studies is how to define the rate of a reaction. Typically, we have two options we can define the rate in terms of how quickly reactant molecules are depleted from the reaction vessel, or we can monitor the rate at which product molecules are formed during the reaction. Either way, we are attempting to define the rate for the same reaction, so we ought to get the same answer. But, often, the stoichiometric coefficients for the individual reactant and product species are different. For example, for the general reaction shown in Eq. 4.1... 

The generalized stoichiometric coefficients are defined as positive quantities for the products of the reaction and as negative quantities for the reactants. The coefficients of species that are neither produced nor consumed by the indicated reaction are taken to be zero. Equation 1.1.2 has been written in inverted form with the zero first to emphasize the use of this sign convention, even though this inversion is rarely used in practice. 

The reaction rate is properly defined in terms of the time derivative of the extent of reaction. It is necessary to define k in a similar fashion in order to ensure uniqueness. Definitions in terms of the various rt would lead to rate constants that would differ by ratios of their stoichiometric coefficients. 

Attempts to define operationally the rate of reaction in terms of certain derivatives with respect to time (r) are generally unnecessarily restrictive, since they relate primarily to closed static systems, and some relate to reacting systems for which the stoichiometry must be explicitly known in the form of one chemical equation in each case. For example, a IUPAC Commission (Mils, 1988) recommends that a species-independent rate of reaction be defined by r = (l/v,V)(dn,/dO, where vt and nf are, respectively, the stoichiometric coefficient in the chemical equation corresponding to the reaction, and the number of moles of species i in volume V. However, for a flow system at steady-state, this definition is inappropriate, and a corresponding expression requires a particular application of the mass-balance equation (see Chapter 2). Similar points of view about rate have been expressed by Dixon (1970) and by Cassano (1980). 

The matrices defining the Model and the p-values are contained in the upper part of the spreadsheet. The given total concentrations of M and L are collected in the columns A and B, row 8 downwards. Initially guessed values for the component concentrations [M and [L] for each solution are in the respective rows of columns D and E. The next two entries, [ML] and [ML2], are calculated from the component concentrations and the respective formation constant, according to equation (3.23). Next, the calculated total concentrations are computed, making sure the stoichiometric coefficients are incorporated correctly, equation (3.30). The task is to juggle the initially estimated component concentrations [M and [L] in columns D and E until the calculated total concentrations collected in the columns I and J match the known concentrations in the columns A and B. The reader is encouraged... 

The reason for following this complex notation will become apparent shortly. The law of mass action, which is confirmed experimentally, states that the rate of disappearance of a chemical species i, defined as RRit is proportional to the product of the concentrations of the reacting chemical species, where each concentration is raised to a power equal to the corresponding stoichiometric coefficient that is,... 

B. LAW OF MASS ACTION. Using the conventional notation, we will define an overall reaction rate 31 as the rate of change of moles of any component per volume due to chemical reaction divided by that component s stoichiometric coefficient... 

The approach of specific interactions, developed primarily by Pitzer (1973) and Whitfield (1975a,b), considers all salts, from a purely formal point of view, as completely dissociated, and embodies the effects of specific interactions into particular activity coefficients, defined as total activity coefficients or stoichiometric activity coefficients, with symbol y. For instance, for ion /,... 

During the course of a chemical reaction, the number of moles of a particular reactant will change in a manner consistent with the stoichiometric relationships defined by the stoichiometric coefficients a, b, p, and q) ... 

We will consider the rate r as a single positive quantity describing the rate of a particular reaction. Note that the rate can now only be defined after we write the chemical reaction. In our two ways of writing the NO formation reaction previously, the rate would be smaller by a factor of two when the stoichiometric coefficients are multiphed by a factor of two. 

A batch reactor is defined as a closed spatially uniform system which has concentration parameters that are specified at time zero. It might look as illustrated in Figure 2-4. This requires that the system either be stirred rapidly (the propeller in Fig. 24) or started out spatially uniform so that stirring is not necessary. Composition and temperature are therefore independent of position in the reactor, so that the number of moles of species in the system Nj is a function of time alone. Since the system is closed (no flow in or out), we can write simply that the change in the total number of moles of species j in the reactor is equal to the stoichiometric coefficient Vj multiplied by the rate multiphed by the volume of the reactor. 

The elementary reactions in Eqs. (1) are not necessarily linearly independent, and, accordingly, let Q denote the maximum number of them in a linearly independent subset. This means that the set of all linear combinations of them defines a 0-dimensional vector space, called the reaction space. In matrix language 0 is the rank of the S x A matrix (2) of stoichiometric coefficients which appear in the elementary reactions (1) ... 

Note that reactions 2.14, 2.15, and 2.23 involve fractional stoichiometric coefficients on the left-hand sides. This is because we wanted to define conventional enthalpies of formation (etc.) of one mole of each of the respective products. However, if we are not concerned about the conventional thermodynamic quantities of formation, we can get rid of fractional coefficients by multiplying throughout by the appropriate factor. For example, reaction 2.14 could be doubled, whereupon AG° becomes 2AG, AH° = 2AH , and AS° = 2ASf, and the right-hand sides of Eqs. 2.21 and 2.22 must be squared so that the new equilibrium constant K = K2 = 1.23 x 1083 bar-3. Thus, whenever we give a numerical value for an equilibrium constant or an associated thermodynamic quantity, we must make clear how we chose to define the equilibrium. The concentrations we calculate from an equilibrium constant will, of course, be the same, no matter how it was defined. Sometimes, as in Eq. 2.22, the units given for K will imply the definition, but in certain cases such as reaction 2.23 K is dimensionless. 

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