Mean stoichiometric number

Fig. 7-11. Potential energy curves for a mialtistep reaction of (a) single rate-determining step and (b) multiple rate-determining steps v = stoichiometric number of a single rate-determining step v = mean stoichiometric number of multiple rate-determining steps.

Ihis number v is the mean stoichiometric number of the rate-determining multiple steps. The mean stoichiometric number is thus represented by the energy average (afiiniiy average) of the stoichiometric niunbers v weighed with the step affinity 4gi in the respective rate-determining steps. 

The reaction affinity - AG and the ratio v /vj of the forward to the backward rate can be estimated, regardless of whether the reaction rate is determined by a single step or multiple steps. Thus, Eqn. 7-51 can be used to determine the mean stoichiometric number of the multiple rate-determining steps. 

In the reaction kinetics, the mean stoichiometric number is important to elucidate the reaction mechanism. 

We consider a simple reaction composed of two elementaiy steps in series with the affinity distribution ratio m - AgilAg and the stoichiometric niunbers vj and vj. The mean stoichiometric number v is then given by Eqn. 7-52 ... 

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. ... 

For a further verification of this conclusion, we determined equilibrium constant, K, by combining the results of kinetic measurements in the region of r+ > r and in the region of r+ < r, having assumed that v = 1. The agreement of the obtained value with the value known from thermodynamic data (K = 0.0364 atm2 at 530°Q has confirmed that v = 1. This result means that stoichiometric numbers of all nonequilibrium stages of the reaction equal unity. 

The enthalpy of reaction, AHr (or reaction enthalpy ), is closely related to the quantity AH that appears in a thermochemical equation, but its units are kilojoules per mole (kj-mol-1) rather than kilojoules. We interpret the per mole to mean per number of moles of each substance as indicated by its stoichiometric coefficient in the chemical equation. For example, the oxidation of methane described by reaction A signifies that 890 kj of heat is released per mole of CFI4 molecules consumed or per... 

It means that we consider only mono-, bi- and (rarely) termolecular reactions. The coefficients stoichiometric coefficients and stoichiometric numbers observed in the Horiuti-Temkin theory of steady-state reactions. The latter indicate the number by which the elementary step must be multiplied so that the addition of steps involved in one mechanism will provide a stoichiometric (brutto) equation containing no intermediates (they have been discussed in Chap. 2). 

Equation 5.1-10 provides the means for calculating a basis for the stoichiometric number matrix that corresponds with the conservation matrix. Similarly the transposed stoichiometric number matrix provides the means for calculating a basis for the transposed conservation matrix. This is done by using the following equation, which is equivalent to equation 5.1-10 ... 

In Equation (18b), the activity quotient is separated into the terms relating to the silver electrode and the hydrogen electrode. We assume that both electrodes (Ag+/Ag and H+/H2) operate under the standard condition (i.e. the H+/H2 electrode of our cell happens to constitute the SHE). This means that the equilibrium voltage of the cell of Figure 3.1.6 is identical with the half-cell equilibrium potential E°(Ag+l Ag) = 0.80 V. Furthermore, we note that the activity of the element silver is per definition unity. As the stoichiometric number of electrons transferred is one, the Nemst equation for the Ag+/Ag electrode can be formulated in the following convenient and standard way ... 

We see from Eq. (15.3) that either the v, s or e must be expressed in molest and that the other quantity must be a pure number. As a matter of convenience we choose to express the reaction coordinate e in moles. This allows one to speak of a mole of reaction, meaning that e has changed by a unit amount, i.e., by one, mole. When Ae = 1 mol, the reaction proceeds to such an extent that the chaise in mole number of each reactant artd product is equal to its stoichiometric number. When two or more independent reactions proceed simultaneously, we let subscript j be the reaction index, and associate a separate reaction coordinate with each reaction. The stoichiometric numbers are doubly subscripted to identify , their association with both a species and a reaction. Thus designates th stoichiometric number of species i in reaction j. Since the number of moles of i species n,- may change because of several reactions, the general equation analogous to Eq. (15.3) includes a sum ... 

CxN) NxR) = CxR. Equation 7.1-8 is useful because it makes it possible to calculate a stoichiometric matrix from a conservation matrix. This operation is called taking the null space of A, and the Mathematica operation for doing this is called NullSpace. The use of NullSpace yields a basis for the stoichiometric number matrix. We will see what this means and how it is handled. 

Suppose that oiu data consist of the concentrations of all reactants and products at a series of times under a wide variety of initial conditions together with the corresponding volumes of the reaction system. From these data, we can estabUsh whether or not the disappearance of a given number of moles of reactant A is always accompanied by the disappearance of a constant number of moles of reactant B and the appearance of a constant number of moles of the products P and Q irrespective of the extent to which the reaction has taken place and under all conditions of temperature, pressure etc. For this purpose, we calculate for each set of concentration-time determinations a series of mean stoichiometric ratios at different extents of reaction, thus... 

The first route consists of steps 1), 2), and 3) with stoichiometric numbers = 2, a" = 2, and = L The second route consists of steps 3) and 4) with stoichiometric numbers = —3 and = 1. The third elementary step participates in both routes with different stoichiometric numbers. The minus sign for means that elementary step 3) in the second route is proceeding in the reverse direction. On the light side of the mechanism (5) are given the stoichiometric numbers for both routes. At the end of mechanism (5) are given stoichiometric equations for both routes. 

The route N1 2 above cannot be obtained by multiplication of N(1) with a certain number, which means that the routes N( 11 are N(2j different, although their overall equations are identical. Denoting the stoichiometric number of a stage of a route N<3 in the following way... 

If besides there exists an identifiable rate-determining step with stoichiometric number s, the value of n has a definite physical meaning. Indeed, nesir equilibrium, as was just seen ... 

Dobereiner showed (mentioning Dalton) that atmospheric air is not a compound. He pointed out that the equivalent of strontiiun (42 5) is the arithmetic mean of those of calcium (20) and barium (65), and in 1829 he extended this relation (afterwards called the law of triads ) to many other groups of three analogous elements ( Trias ). In 1817 he pointed out that widely distributed elements have small atomic weights (stoichiometric numbers), and that some elements (Fe, Co, Ni, Cr, Mn) have nearly the same atomic weights. He established the identity of chromic acid, which had been questioned by... 

Consider now relations elucidating the physical meaning of effective stoichiometric numbers. Comparing formulas (75), (69), and (82), we can easily obtain... 

Thus, p evaluated in this way has the physical meaning of an average of stoichiometric numbers of the constituent steps, each weighted in the affinity values associated with them. 

Therefore, we have four rates of reaction R, Rg, Rc and Rj). Does this mean that the single reaction is defined by four different rates of reactions Of course not The four rates are related by the stoichiometric numbers. Intuitively, we can easily note that... 

Geometric average (mean) of stoichiometric numbers of an electrolyte [dimensionless]... 

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