Time-independent stoichiometry

Note that rB and v can also be defined on the basis of partial pressure, number concentration, surface concentration, etc., with analogous definitions. If necessary differently defined rates of reaction can be distinguished by a subscript, e.g. vp = vB 1dpB/dt, etc. Note that the rate of reaction can only be defined for a reaction of known and time-independent stoichiometry, in terms of a specified reaction equation also the second equation for the rate of reaction follows from the first only if the volume V is constant. The derivatives must be those due to the chemical reaction considered in open systems, such as flow systems, effects due to input and output processes must also be taken into account. 

Thus, after 2 000 s of reaction the magnitudes of the changes in the concentrations of reactants and products in Reaction 3.1 are the same, although there is a decrease for reactants and an increase for products. In fact, this type of result would have been obtained irrespective of the time period selected. This means that the stoichiometry of Reaction 3.1 applies throughout the whole course of reaction that is it has time-independent stoichiometry. 

It might be tempting to conclude that intermediates are not present for a reaction that has time-independent stoichiometry. However, this is not the case. Time-independent stoichiometry simply means that, within the accuracy of the chemical analysis used, intermediates cannot be detected and so they do not affect the stoichiometric relationship between reactants and products. In fact. Reaction 3.1 is thought to be composite with a three-step mechanism in which case intermediates must be involved. 

The discussion that resulted in Equation 3.3 can be applied to any chemical reaction that has time-independent stoichiometry. For example, nitrogen dioxide (NO2) decomposes in the gas phase at temperatures in the region of 300 °C to give nitric oxide (NO) and oxygen... 

It is important to remember, however, that this definition only holds for a reaction with time-independent stoichiometry. If, for example, intermediates build up to measurable quantities during the course of a reaction then there are no simple relationships between the rates of change of concentrations of reactants and products. 

For the following two reactions express the rate of reaction, /, in terms of the rate of change of concentration of each reactant and each product. (Assume that both reactions have time-independent stoichiometry.)... 

If the same stoichiometry for a reaction applies throughout the whole course of a reaction then the reaction is said to have time-independent stoichiometry. 

The rate of a chemical reaction (strictly at constant volume) for a reaction with time-independent stoichiometry is defined by... 

The thermal decomposition of dinitrogen pentoxide (N2O5) in the gas phase has time-independent stoichiometry... 

The gas-phase thermal decomposition of oxygen difluoride (F2O) in the temperature range 230 °C to 310 °C has the following time-independent stoichiometry... 

The equations you have written down provide a mathematical description of the reaction mechanism. In fact, if we focus on the rate of production of the product P then all the equations are seen to be interlinked the formation of P depends on the decomposition of Y, which in turn depends on the decomposition of X, which in turn depends on the decomposition of A. In principle, the equations can be solved to find how the concentrations of all of the species present (A, X, Y and P) vary with time but, overall, it is not possible to find a simple form of the rate equation. In fact, these comments need not be specific to the model mechanism we have chosen to discuss in general, they will apply to any form of mechanism. However, we know from experimental investigations that many composite reactions which have time-independent stoichiometry have relatively simple rate equations. This must mean that it is legitimate to simplify the analysis of reaction mechanisms. But how A clue to one approach can be taken from the unlikely source of an article in the Daily News (London) 26 December 1896 ... 

The idea of a bottleneck in a reaction mechanism is an important one. In particular, for reactions with time-independent stoichiometry it can be used to simplify the analysis of a reaction mechanism and, in the process, derive a predicted rate equation which can be compared with experiment. We shall not look at this in general, but we will consider specific cases. 

The situation may seem intractable. However, the fact that Step 2 is relatively slow must mean that the reaction intermediate X will be formed in Step 1 more rapidly than it can be consumed in Step 2 and so during the initial stages of the reaction its concentration will increase. However, it must be remembered that the reaction has time-independent stoichiometry and so this increase in the concentration of X must be moderated by another process. The key point is that as the concentration of X builds up, the rate of a new process, the reverse of Step 1, becomes significant. We represent this reverse process as fc-i... 

The discussion, so far, has focused on a particular form of two-step reaction mechanism reactant A reacts to give a reactive intermediate X which then further reacts with reactant B to give product . It is possible to envisage different variations on this theme. Nonetheless, if the reaction in question has time-independent stoichiometry and the second step in the reaction mechanism is taken to be rate-limiting then... 

For a reaction with time-independent stoichiometry and a proposed reaction mechanism in which the first step is rate-limiting then the overall rate of reaction is determined by this first step and is not influenced by any of the following steps. 

Appreciate that the determination of the stoichiometry of a chemical reaction, including whether it is time-independent or not, is an important step in any chemical kinetic investigation. (Question 3.1)... 

Reaction 3.1 has a known stoichiometry, which is time-independent, and the reaction essentially goes to completion. Although concise, this statement summarizes key factors to be considered in the kinetic investigation of any chemical reaction. 

Sometimes, but not always, it is possible to define a quantity, known as the rate of reaction, which is independent of the reactants and products. This can only be done if the reaction is of known stoichiometry for some reactions there are numerous minor products and the stoichiometry is uncertain. Another condition for defining a rate of reaction is that the stoichiometric equation must remain the same throughout the course of reaction for some reactions intermediates are formed in significant amounts, and the stoichiometry varies as the reaction proceeds. If these two conditions are satisfied (i.e., if the stoichiometry is known and is time independent), the rate of reaction is given by any of the expressions that appear in Eq. (3). In other words, the rate of reaction is the rate of consumption or formation divided by the appropriate coefficient that appears in the stoichiometric equation. In the case of products these coefficients are called the stoichiometric coefficients in the case of reactants the stoichiometric coefficients are the negatives of the coefficients in the rate equation. As seen in Eq. (3), this division has made the four rates equal to one another, so that the rate of reaction is unique for the reaction under the particular conditions of the experiment. 

The integer n refers to the number of linearly independent reactions in the mechanism. Reactions that are linearly dependent may be expressed as linear combinations of n independent mechanistic reactions, just as the observed stoichiometric relation may be expressed as a linear combination of

A, for instance, are linearly dependent, since the second is —1 times the first. In general, forward and reverse reactions of a reversible mechanistic step are dependent, hence the rule given in Example 2. 

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). 

Note that Eq. (6) includes thermodynamic equilibrium (v° = 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions. 

If the concentration of methanol is monitored with time, it will decrease twice as fast as carbon monoxide and four times as fast as oxygen as shown in Figure 2.8. However, by adjusting the change in concentration with time for the stoichiometry of the reaction, the determined rate of the reaction will be identical, no matter which of the reagents is monitored during the course of the reaction. Thus, the determined rate of reaction is independent of stoichiometry. 

In spite of the many approximations used in deriving this set of equations, the mathematics of solving them is formidable. The set of equations is 2s + 1 in number, where s is the number of species that occur within the flame. Of these, s + r — 1 equations (r < s) form a set of independent equations since the quantities x, and G, are fractions such that Y,iXi = 1 and Y.i Gi = 1, and stoichiometry reduces the number of independent species by some number (s — r), (H3). Part of the mathematical complexity is due to the presence of intermediate species whose concentrations are low in the hot and cold gases but which may pass through a maximum within the flame zone. Since the rates of appearance and disappearance must be comparable for this to occur, it is approximately true near the maximum that Kt = 0 for these species. This fact results in an indeterminancy of the system of equations, although, at the same time, it provides a method which permits an approximate solution to be obtained. 

Figure 6.6 Evolution of tan 8 during polyurethane synthesis at 110°C, at different angular frequencies,

The form of the dependence of / on will be determined by the order of the reaction with respect to A and B. Provided the reaction does indeed follow the overall stoichiometry in the above equation (in this particular case we have ru = vb = -1) then, once we know the initial concentrations of A and B, the concentration of B at any time can be determined uniquely if the concentration of A is known, i.e., these concentrations are not independent, and so the rate can be formally expressed as a function solely of one concentration or extent of reaction variable. 

Efflux of lithium occurs via sodium-lithium countertransport. It is ouabain insensitive, blocked by phloretin, independent of ATP, and exhibits saturation. A 1 1 stoichiometry occurs and the maximum affinity for both cations occurs at the internal surface, that for lithium being 20-30 times higher than for sodium (120). Chronic administration of lithium in humans eventually reduces lithium-sodium countertransport in erythrocytes (121). 

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