Chemical stoichiometric number

Since A can be obtained from thermodynamic data, and a a can be determined if V and V are known, a a plays a very important role in verifying reaction mechanism. If there is no rate determining step in a catalytic reaction, the concept of average chemical stoichiometric number can be introduced. The kinetics of reverse reaction can be determined from that of forward-reaction kinetics by crj or... 

The total enthalpy correction due to chemical reactions is the sum of all the enthalpies of dimerization for each i-j pair multiplied by the mole fraction of dimer i-j. Since this gives the enthalpy correction for one mole of true species, we multiply this quantity by the ratio of the true number of moles to the stoichiometric number of moles. This gives... 

The lepiesent formulas for the chemical species and is the stoichiometric number for species i in reaction j. Each has a magnitude and a sign ... 

The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reactionj ... 

Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions. See Amundson, N. R., Mathematical Methods in Chemical Engineering, Prentice-Hall, Englewood Cliffs, N.J. (1966, p. 50). 

STOICHIOMETRIC NUMBER Stoichiometry of elementary reactions, CHEMICAL KINETICS MOLECULARITY UNIMOLECULAR BIMOLECULAR TRANSITION-STATE THEORY ELEMENTARY REACTION STOKE S SHIFT... 

Within the Horiuti s approach, the physical meaning of the molecularity is clear. Horiuti introduced the concept of stoichiometric numbers (Horiuti numbers, v) Horiuti numbers are the numbers such that, after multiplying the chemical equation for every reaction step by the appropriate Horiuti number v, and subsequent adding, all reaction intermediates are cancelled. The equation obtained is the overall reaction. In the general case, the Horiuti numbers form a matrix. Each set of Horiuti numbers (i.e. matrix column) leading to elimination of intermediates corresponds to the specific reaction route. ... 

If we add up the chemical equations of mechanism (46) multiplied by corresponding stoichiometric numbers (i.e. 1 and 2), we obtain the equation of net reaction (47). Note how the constant term of Equation (48) and exponents in its terms correspond to the Proposition 1 (and property (36)). The only feasible root of polynomial (48) is... 

This displays the convention, tacitly assumed later, that the positive direction of a step corresponds to the advancement from left to right of the stated chemical equation. The matrix of stoichiometric coefficients for these reactions is shown in Table II. The diagonalization of the matrix in Table II gives the matrix in Table III, from which the steady-state mechanism is S + 2s2 + 2s3 + 2s4. In Horiuti s terminology the stoichiometric numbers are 1 for Sj and 2 for s2, s3, and s4. 

Let xjs be the stoichiometric coefficient of an intermediate Xj in the chemical equation of a stage s. It is understood that xjs > 0 if Xj is being formed, xjs < 0 if is being consumed in stage s, and x]s = Oif X7 does not participate in this stage. By definition, stoichiometric numbers for the route jV(p) satisfy the equation... 

If in the course of a reaction, not one, but several adsorption chemical equilibria exist, it is always possible to write the chemical equations of the equilibria in such a way that the stoichiometric number of one of these equations be 1 and that of other equations be zero. Confining ourselves to two equilibria, we have the following mechanism instead of (188) (48) ... 

When stoichiometric numbers are taken to be signed quantities, this chemical equation can be written as... 

This equation is useful for setting up the fundamental equation for consideration of a chemical reaction system described by a particular stoichiometric number matrix. 

It is hard to divide the six chemical work terms into three terms for three chemical reactions, but this can be done using equation 5.4-4 with the following stoichiometric number matrix ... 

The primes on the amounts are needed to indicate that they are amounts of reactants, which are sums of species that are pseudoisomers at specified pH. The primes on the stoichiometric number matrices and extents of reaction column matrices are needed to indicate that these matrices are for biochemical reactions written in terms reactants (sums of species). The primes are needed on the transformed chemical potentials to distinguish them from chemical potentials of species. 

Systems of biochemical reactions like glycolysis, the citric acid cycle, and larger and smaller sequential and cyclic sets of enzyme-catalyzed reactions present challenges to make calculations and to obtain an overview. The calculations of equilibrium compositions for these systems of reactions are different from equilibrium calculations on chemical reactions because additional constraints, which arise from the enzyme mechanisms, must be taken into account. These additional constraints are taken into account when the stoichiometric number matrix is used in the equilibrium calculation via the program equcalcrx, but they must be explicitly written out when the conservation matrix is used with the program equcalcc. The stoichiometric number matrix for a system of reactions can also be used to calculate net reactions and pathways. 

When equcalcrx[nt,lnkr,no] is applied to a system of R independent chemical reactions, it requires a R x N transposed stoichiometric number matrix nt, a vector of natural logarithms of the equilibrium constants of independent reactions, and a vector no of the initial concentrations. It can be used at a specified pH by using a R x N transposed stoichiometric number matrix nt, a vector lnkr of natural logarithms of the apparent equilibrium constants of independent biochemical reactions, and a vector no of the initial concentrations. 

Glycolysis involves 10 biochemical reactions and 16 reactants. Water is not counted as a reactant in writing the stoichiometric number matrix or the conservation matrix for reasons described in Section 6.3. Thus there are six components because C = N — R = 16 — 10 = 6. From a chemical standpoint this is a surprise because the reactants involve only C, H, O, N, and P. Since H and O are not conserved at specified pH in dilute aqueous solution, there are only three conservation equations based on elements. Thus three additional conservation relations arise from the mechanisms of the enzyme-catalyzed reactions in glycolysis. Some of these conservation relations are discussed in Alberty (1992a). At specified pH in dilute aqueous solutions the reactions in glycolysis are... 

Figure 6.1 Apparent stoichiometric number matrix v for the 10 reactions of glycolysis at specified pH in dilute aqueous solutions, (see Problem 6.3) [With permission from R. A. Alberty, J. Phys. Chem. B 104, 4807-4814 (2000). Copyright 2000 American Chemical Society.]...

The Vj are the stoichiometric numbers of reactants in the biochemical equation (positive for reactants on the right side of the equation and negative for reactants on the left side). The prime is needed on the stoichiometric numbers to distinguish them from the stoichiomeric numbers in the underlying chemical reactions. The standard transformed enthalpy of reaction Ar H(heat of reaction) is related to the standard transformed enthalpies of formation Af //,of the reactants by... 

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