Stoichiometry stoichiometric coefficients

The respiratory quotient (RQ) is often used to estimate metabolic stoichiometry. Using quasi-steady-state and by definition of RQ, develop a system of two linear equations with two unknowns by solving a matrix under the following conditions the coefficient of the matrix with yeast growth (y = 4.14), ammonia (yN = 0) and glucose (ys = 4.0), where the evolution of C02 and biosynthesis are very small (o- = 0.095). Calculate the stoichiometric coefficient for RQ =1.0 for the above biological processes ... 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

Sometimes we need to know how much product to expect from a reaction, or how much reactant we need to make a desired amount of product. The quantitative aspect of chemical reactions is the part of chemistry called reaction stoichiometry. The key to reaction stoichiometry is the balanced chemical equation. Recall from Section H that a stoichiometric coefficient in a chemical equation tells us the relative amount (number of moles) of a substance that reacts or is produced. Thus, the stoichiometric coefficients in... 

There are some exotic chemical names here, but they should not distract you from the basic principles of reaction stoichiometry. The stoichiometric coefficients state that one mole of each reactant will produce one mole of each product. A flowchart summarizes the steps used to convert the mass of geraniol into the mass of geranyl formate. 

All changes are related by stoichiometry. Each ratio of changes in amount equals the ratio of stoichiometric coefficients in the balanced equation. In the example above, the changes in amounts for H2 and N2 are in the ratio 3 1, the same as the ratio for the coefficients of H2 and N2 in the balanced equation. 

The problem asks for a yield, so we identify this as a yield problem. In addition, we recognize this as a limiting reactant situation because we are given the masses of both starting materials. First, identify the limiting reactant by working with moles and stoichiometric coefficients then carry out standard stoichiometry calculations to determine the theoretical amount that could form. A table of amounts helps organize these calculations. Calculate the percent yield from the theoretical amount and the actual amount formed. 

Figure 15-5 also shows that each species has its own rate, but the individual rates are linked by the stoichiometric coefficients. The reaction generates one molecule of N2 O4 for every two molecules of NO2 that are destroyed. That is, the 1 2 stoichiometry of this reaction results in a 1 2 relationship between the rate of disappearance of NO2 and the rate of appearance of N2 O4. The ratio of rates for different species is always equal to the ratio of their stoichiometric coefficients. 

The task now is to select the linear combinations that will most probably correspond to independent parts of the reaction network with easily interpretable stoichiometry. A simplification of the data in the matrix can be achieved by such a rotation that the axes go through the points in Fig. A-2 (this is equivalent to some zero-stoichiometric coefficients) and that the points of Fig. A-3 are in the first quadrant (this corresponds to positive reaction extents) if possible. Rotations of the abscissa through 220° and the ordinate through 240° lead to attaining both objectives. The associated rotation matrix is ... 

After rotating and scaling to make the largest stoichiometric coefficients unity, the matrices of stoichiometry and reaction extents are ... 

This example shows that the method discussed can deal with the difficulties frequently met in real situations. One of the products (D) was difficult to measure and another one (F) not accurately analyzed. So the balance could not close and conventional methods of determining stoichiometry via balancing could fail. The standard error in determination of species (C) was in the range of 6-14 % of the measured value in the first period of the experiment . Despite these difficulties, two simple reactions were found with stoichiometry that can adequately represent the reactions. The final representation of the chemical system is not unique but the final stoichiometric coefficients are within 10 % of the original ones. This indicates that the proposed methodology can yield reasonable approximations. 

The choice is a matter of personal convenience. The essential point is that the ratios of the stoichiometric coefficients are unique for a given reaction i.e., vC0/v02 = ( —2/—1) = [ —1/( —1/2)] = 2. Since the reaction stoichiometry can be expressed in various ways, one must always write down a stoichiometric equation for the reaction under study during the initial stages of the analysis and base subsequent calculations on this reference equation. If a consistent set of stoichiometric coefficients is used throughout the calculations, the results can be readily understood and utilized by other workers in the field. 

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). 

However, if reaction is not of a simple stoichiometry but involves different number of moles of reactants or products, the rate should be divided by corresponding stoichiometric coefficient in the balanced chemical equation for normalizing it and making it comparable. For example, for a general reaction aA + bB — cC + dD... 

The overfired batch conversion process, as well as the combustion process, of wood fuels is shown to be extremely dynamic. The dynamic ranges for the air factor of the conversion system is 10 1 and for the stoichiometric coefficients is CHs.iOiCHoOo during a batch for a constant volume flux of primary air. The dynamics of the stoichiometry indicates the dynamics of the molecular composition of the conversion gas during the course of a run. From the stoichiometry it is possible to conclude that... 

We repeat that the procedure we foUow is first to write the reaction steps with a consistent stoichiometry and then to express the rate of each reaction to be consistent with that stoichiometry. Thus, if we wrote a reaction step by multiplying each stoichiometric coefficient by two, the rate of that reaction would be smaller by a factor of two, and if we wrote the reaction as its reverse, the forward and reverse rates would be switched. 

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. 

The stoichiometric coefficients of all species are unity. These stoichiometries require that, if we feed pure A, it must react to form either B or C. Therefore, the loss of A is equal to the gain in B and C,... 

The definitions in the previous section are simple for simple stoichiometry, but they become more comphcated for complex reaction networks. In fact, one frequently does not know the reactions or the kinetics by which reactants decompose and particular product form. The stoichiometric coefficients (the v,y) in the preceding expressions are complicated to write in general, but they are usually easy to figure out for given reaction stoichiometry. Consider the reactions... 

In general, concentrations of the products are divided by the concentrations of the reactants. In the case of gas-phase reactions, partial pressures cire used instead of molar concentrations. Multiple product or reactant concentrations are multiplied. Each concentration is raised to an exponent equal to its stoichiometric coefficient in the balanced reaction equation. (See Chapters 8 and 9 for details on balanced equations and stoichiometry.)... 

The order of a reaction cannot in general be predicted from the chemical equation a rate law is an empirical law. That is, a rate law is an experimentally determined characteristic of the reaction and cannot in general be written down from the stoichiometry of the chemical equation for the reaction. For instance, both the decomposition of N205 and that of N02 have a stoichiometric coefficient of 2 for the reactant, but one reaction is first order and the other is second order. The decomposition of ammonia also has a stoichiometric coefficient of 2 for the reactant, but its rate law is zero order. 

For our present purposes, we use the term reaction mechanism to mean a set of simple or elementary chemical reactions which, when combined, are sufficient to explain (i) the products and stoichiometry of the overall chemical reaction, (ii) any intermediates observed during the progress of the reaction and (iii) the kinetics of the process. Each of these elementary steps, at least in solution, is invariably unimolecular or bimolecular and, in isolation, will necessarilybe kinetically first or second order. In contrast, the kinetic order of each reaction component (i.e. the exponent of each concentration term in the rate equation) in the observed chemical reaction does not necessarily coincide with its stoichiometric coefficient in the overall balanced chemical equation. 

The formation constants and associated stoichiometries have been reported for a number of different aluminium oxalic acid complexes (Sjoberg and Ohman, 1985), and these are shown in Table 5.1. From the stoichiometric coefficients the reaction for the formation of A1L2 can be deduced and expressed more fully as... 

Chemical reactions change the molecular structure of matter, thus resulting in the destruction of some chemical species (reactants) and in the formation of different ones (products). The relevant quantities of reactants and products involved in the reaction are strictly determined by stoichiometry, which states a law of proportionality deriving from the mass conservation of the single elements. Often, the stoichiometric coefficients are imposed to be constant during the reaction however, this is not true in most real systems. When variable stoichiometric coefficients are observed, the system cannot be described by a single reaction. 

In the usual case, t and ain will be known. Equation (1.49) is an algebraic equation that can be solved for aout. If the reaction rate depends on the concentration of more than one component, versions of Equation (1.49) are written for each component and the resulting set of equations is solved simultaneously for the various outlet concentrations. Concentrations of components that do not alfect the reaction rate can be found by writing versions of Equation (1.49) for them. As for batch and piston flow reactors, stoichiometry is used to relate the rate of formation of a component, say Sl-c, to the rate of the reaction SI, using the stoichiometric coefficient vc, and Equation (1.13). After doing this, the stoichiometry takes care of itself. 

Even if the stoichiometric coefficients of the half-reactions are not the same, do not multiply ° values, as they do not depend on stoichiometry. 

The conversion is defined with reference to a reactant and could be different for different reactants. As a matter of fact, in a general reaction the conversions of reactants A and B will only be equal when the initial (or feed) ratio of moles of A to B is equal to the stoichiometry of the reaction ajb, where a and b are the stoichiometric coefficients of A and B, respectively. 

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