Number stoichiometric algebraic

Remember that in the laws of physical chemistry, stoichiometric numbers are algebraic coefficients. The general equation that is obtained as a result can equally be applied to a half-reaction occurring either in the direction of oxidation (Ve = -i-n) or reduction (Vg = - n) R provided that the reaction rate is also taken in algebraic terms following the direction of the redox reaction. 

This equation uses stoichiometric algebraic numbers positive for a reaction product and negative for a reactant. 

For mechanisms that are more complex than the above, the task of showing that the net effect of the elementary reactions is the stoichiometric equation may be a difficult problem in algebra whose solution will not contribute to an understanding of the reaction mechanism. Even though it may be a fruitless task to find the exact linear combination of elementary reactions that gives quantitative agreement with the observed product distribution, it is nonetheless imperative that the mechanism qualitatively imply the reaction stoichiometry. Let us now consider a number of examples that illustrate the techniques used in deriving an overall rate expression from a set of mechanistic equations. 

Another important aspect that we have so far ignored is that reactions almost never actually proceed as represented by a stoichiometric equation (eg, Eq 1). Usually a stoichiometric equation is the algebraic sum of a number of steps called elementary reactions. Frequently one of these elementary reactions is much slower than the others, and thus con-... 

We believe that it is not necessary to consider the overall kinetic order of steps above three in mechanism (4). We have analyzed comprehensively [97, 102, 103] all the possible versions for mechanism (4) assuming that the stoichiometric coefficients n, m, p, and q can amount to 1 or 2, p + q < 3, and k 3 = 0. The principal results of this analysis are listed in Table 2. By using the method of general analysis and the Sturm and Descartes theorem concerning the number of positive roots in the algebraic polynomial (ref. 219, pp. 248 and 255), we could show that there exist four detailed mechanisms providing the possibility of obtaining three steady states with a non-zero catalytic reaction... 

This does not correspond with reactions 5.1-28 to 5.1-32, but it is equivalent because the row-reduced form of equation 5.1-34 is identical with the row-reduced form of the stoichiometric number matrix for reactions 5.1-28 to 5.1-32 (see Problem 5.2). The application of matrix algebra to electrochemical reactions is described by Alberty (1993d). 

Systems of biochemical reactions can be represented by stoichiometric number matrices and conservation matrices, which contain the same information and can be interconverted by use of linear algebra. Both are needed. The advantage of writing computer programs in terms of matrices is that they can then be used with larger systems without change. 

Calculate the algebraic sum of the stoichiometric numbers for the reaction Asdiscussed in Example 4.4, stoichiometric numbers for the components in a given reaction are numerically equal to the stoichiometric coefficients, but with the convention that the reactants get minus signs. In the present case, the stoichiometric numbers for H20, H2, and 02 are, respectively, —1, +1, and +V2-Their sum is + V2 ... 

Relate equilibrium mole fractions to the equilibrium constants. By definition, K = YUf1, where the a,- are the activities of the components with the mixture, and the v, are the stoichiometric numbers for the reaction (see Example 4.4). The present example is at relatively large reduced temperatures and relatively low reduced pressures, so the activities can be represented by the equilibrium mole fractions y,-. For Reaction 4.11, Kk = yco,Vch4 A> co)2( Vh, )2- Substituting the value for K from step 2 and the values for the y, from the last column of the table in step 3 and algebraically simplifying,... 

To use linear algebra as much as possible, pseudo-first order rate coefficients are introduced for steps with co-reactants. Each such coefficient is the product of the actual, higher-order rate coefficient and the concentration of the respective co-reactant (or concentrations of the co-reactants, if there are several) raised to the power that corresponds to the respective stoichiometric number. For example, for a step... 

As a rule, a reduction to a single, explicit rate equation (plus algebraic equations for stoichiometric constraints and yield ratios) is not achieved. Rather, the equations for the end members of the piecewise simple network portions must be solved simultaneously. Nevertheless, The concentrations of all trace-level intermediates that do not react with one another have been eliminated by this procedure and, in many cases of practical interest, the reduction in the number of simultaneous rate equations and their coefficients is substantial. 

The number of independent reactions is an index of the stoichiometric complexity of the system. As Piret and Trambouze (1959) have pointed out, the expressions r, or /, may be found to contain fewer than the m concentrations. In this case it is possible to work with fewer than m equations, but integral, rather than algebraic, relations will be needed to recover the other concentrations. This observation has also been made by Horn (1961). 

In this section, we describe two types of volumetric calculations. The first involves computing the molarity of solutions that have been standardized against either a primary-standard or another standard solution. The second involves calculating the amount of analyte in a sample from titration data. Both types are based on three algebraic relationships. Two of these are Equations 13-1 and 13-3, both of which are based on millimoles and milliliters. The third relationship is the stoichiometric ratio of the number of millimoles of the analyte to the number of millimoles of titrant. 

The fact that chemical reactions are expressed as linear homogeneous equations allows us to exploit the properties of such equations and to use the associated algebraic tools. Specifically, we use elementary row operations to reduce the stoichiometric matrix to a reduced form, using Gaussian elimination. A reduced matrix is defined as a matrix where all the elements below the diagonal (elements 1,1 2,2 3,3 etc.) are zero. The number of nonzero rows in the reduced matrix indicates the number of independent chemical reactions. (A zero row is defined as a row in which all elements are zero.) The nonzero rows in the reduced matrix represent one set of independent chemical reactions (i.e., stoichiometric relations) for the system. 

In this investigation ten flux equations and ten dusty gas model equations have to be solved. From the number of the stoichiometric equations and the number of the reactants and products it is clear that the values of the fluxes of four components can be expressed by means of the value of the remaining six components. Simple algebraic manipulation gives the following relations ... 

The term stoichiometric coefficients for reaction j, and since the stoichiometric coefficients for products are positive while those of the reactants are negative, there is a fair degree of cancelling inherent in f7 i= ii However, AU j = AH j only when the number of kilogram-moles of products formed by the reaction equals the number of kilogram-moles of reactants used up in doing so, i.e. when the reaction exhibits neither a... 

Here N is the number of moles of species i initially present, N is the total number of moles initially present, and Oy = E, v y is the algebraic smn of stoichiometric coefficients for reaction j. 

For reaction-equilibrium computations, we have discussed only stoichiometric methods, in which the elemental balances are imposed explicitly through F sets of stoichiometric coefficients. For one-phase systems, these formulations require us to solve only F algebraic equations for F extents of reaction therefore, they require us to identify F independent reactions. Such stoichiometric methods appear to be most effective when the number of species C is not much greater than the number of elements (C nig). Otherwise, when C m, nonstoichiometric methods may be more computationally efficient [16,18], though this comment probably depends on the particular algorithms being compared. 

Comments The elimination process can be done in many different ways and the surviving rows may contain different coefficients from ones obtained above, but the number of nonzero rows will be the same. This method comes from linear algebra and amounts to the determination of the rank of the matrix of the stoichiometric coefficients. 

This example shows that the usuai form of writing the balanced reaction uses absolute values for the stoichiometric numbers. In some books, one finds a more mathematical form of these equations, in which algebraic stoichiometric coefficients are used directly ... 

Taking these definitions into account, when a redox half-reaction is written in the oxidation direction then the algebraic stoichiometric number of electrons corresponds to the positive number (n > 0) of the electrons exchanged ... 

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