Equations, stoichiometric balancing

Figure 3-2. Two reaction equations showing two completely different uses for the (+) symbol a) giving a fully balanced single reaction, b) combining two parallel reactions into a single equation that is not stoichiometrically balanced.

Stoichiometric Balances The amounts of aU participants in a group of reactions can be expressed in terms of a number of key components equal to the number of independent stoichiometric relations. The independent rate equations will then involve only those key components and will be, in principle, integrable. 

These equations will have to he solved numerically for A, B, and C as functions of time then D and E can he found hy algehra. Alternatively, five differential equations can he written and solved directly for the five participants as functions of time, thus avoiding the use of stoichiometric balances, although these are really involved in the formulation of the differential equations. 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

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. 

Alternatively, five differential equations can be written and solved simultaneously for the five participants as functions of time, thus avoiding the making of the stoichiometric balances. 

By stoichiometric balances all concentrations can be expressed in terms of C For the evaluation of the constants from measurements of data of (C, t), the three equations require somewhat different techniques. Some of the methods are summarized on the graphs of Figure 3.1. 

A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59 62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63 69]. 

Despite its widely recognized limitations, flux balance analysis has resulted in a large number of successful applications [35, 67, 72 74], including several extensions and refinements. See Ref. [247] for a recent review. Of particular interest are recent efforts to augment the stoichiometric balance equations with thermodynamic constraints providing a link between concentration and flux in the constraint-based analysis of metabolic networks [74, 149, 150]. For a more comprehensive review, we refer to the very readable monograph of Palsson [50]. 

And these must be written to obtain an electronically and stoichiometrically balanced equation ... 

An extension of isomerism from molecules to ensembles of molecules (EM) leads to new perspectives in chemistry. The left and right hand sides of a stoichiometrically balanced reaction equation are isomeric EM. Any chemical reaction may be regarded as an isomerization, i.e. the conversion of an EM into an isomeric EM. Let A = A1(... A be a finite collection of atoms with the empirical formula A. Any EM that contains each atom of A exactly once is an EM (A). The family of all isomeric EM (A), the FIEM(A) contains the complete chemistry ofA = At,... A . The FIEM(A) is closed and finite. It has well-defined limitations and invariancies. Accordingly, the logical structure of the chemistry of an FIEM(A) is much easier to elucidate than the logical structure of chemistry without the above restrictions [9],... 

It tells us about stoichiometric ratios. First, make sure that the equation is balanced Then you can see that 1 mole not Ibm or kg) of heptane will react with 11 moles of oxygen to give 7 moles of carbon dioxide plus 8 moles of water. These may be lb mol, g mol, kg mol, or any other type of mole, as shown in Fig. 1.15. One mole of CO2 is formed from each mole of C7H16. Also, 1 mole of H2O is formed... 

In a balanced equation, the stoichiometric coefficients are chosen so that the equation contains an equal number of each type of atom on each side. In our example, there are four H atoms and two 0 atoms on both sides. Therefore, the equation is balanced with respect to atoms. 

The product mixture does not contain equal numbers of moles of the dichloroethanes so we do not show a stoichiometrically balanced equation. Because reactions of saturated hydrocarbons with chlorine can produce many products, the reactions are not always as useful as might be desired. 

The numbers a, b. .. andp,q. .. ensure that the equation is balanced and so are known as balancing coefficients. In practice they are usually chosen to have their smallest possible integer values, and they must be positive. Writing a chemical reaction in this way allows a quantity called the stoichiometric number to be introduced. It is given the symbol Vy (v is the Greek letter nu and Vy is pronounced nu Y ) where the subscript Y represents a given species (reactant or product) in the reaction. The stoichiometric number is then defined so that for... 

The molar concentrations are raised to a power equal to the stoichiometric coefficients of the stoichiometrically balanced equation. 

If the stoichiometric balance is to be expressed in the form of a mathematical equation, it is necessary to discriminate between continuous and discontinuous processes. The following equation is valid for a discontinuous process ... 

This is the case of, for example, externally catalyzed reactions. If the stoichiometric balance is exact, then Equation 3.8 can be written as... 

In practice, p = I is rarely achieved, nor is it a perfect stoichiometric balance. The consequences of this are shown in Figure 2.2 where is plotted as (a) a function of / , calculated from Equation 2.2, and (b) as a function of the stoichiometric ratio for... 

Expanding on the mercuric sulfate - sulfuric acid technology, it was known as early as 1912 that addition of a single equivalent of acetic acid to acetylene in the presence of mercuric sulfate and sulfuric acid resulted in the generation of vinyl acetate (a mere laboratory curiosity at the time) whereas addition of two or more equivalents of acetic acid provided ethylidene diacetate (1,1-diacetoxyethane, EDA). EDA was known to evolve acetaldehyde upon heating in the presence of acids. Upon complete removal of a mole of acetaldehyde, an equimolar amount of acetic anhydride could then be distilled from the residue. If one balances the sequential processes as shown in equations [3] through [6], one finds that the process is in stoichiometric balance for conversion of acetylene and air to cellulose acetate without by-products. (In practice there is some net acetic acid production since cellulose is never completely dry.)... 

A mathematical model was used to calculate the concentrations of all components and the pH at every equilibrium stage. The pH and concentrations of the components at every equilibrium stage were predicted with reasonable accuracy. This model is based on dissociation and reaction equilibria of the compounds, stoichiometric balances and an electroneutrality equation. Precipitation of 6-aminopenicillanic acid, which was observed at a combination of low pH and high 6-APA concentration in the aqueous phase, is not taken into account in the model. 

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