One-plus rate equations

Rate equations of product formation usually contain additive terms in the denominator if the pathway or network includes reverse steps. The number of phenomenological coefficients can then be reduced by one if numerator and denominator are divided by one of the terms. The result is a "one-plus" rate equation, with a "1" as the leading term in the denominator. (Exception This procedure is superfluous if all terms in the denominator consist only of coefficients, or of coefficients multiplied with the same concentration or concentrations, so that they can be combined to give a true power-law rate equation.)... 

One-plus rate equations play a key role in network elucidation. Perhaps the most difficult step in that endeavor is the translation of a mathematical description of experimental results into a correct network of elementary reaction steps. The observed behavior can usually be fitted quite well by a traditional power law with empirical, fractional exponents, at least within a limited range of conditions. This has indeed been standard procedure in times past. However, such equations are highly unlikely to result from a combinations of elementary steps. Their acceptance may be expedient, but as far as network elucidation is concerned they are a dead... 

If fitting a power law requires fractional exponents, a one-plus rate equation with integer exponents should be tried instead. 

Moreover, being more likely to reflect the true mechanism, the one-plus rate equation is also more likely to remain valid upon extrapolation to still unexplored ranges of conditions. 

There is one important exception Certain types of chain reactions and reactions involving dissociation produce exponents of one half or integer multiples of one half in power-law or one-plus rate equations (see Sections 5.6, 9.2, and 10.3.1). Such exponents should be accepted if found not to vary with conversion and if there is good reason to believe that a mechanism of this kind may be operative. 

For an example of the simplest type of one-plus rate equation, let us return to the reaction... 

Another, already encountered example of a one-plus rate equation is that of olefin hydroformylation (see Example 6.2 in Section 6.3). Here, the rate equation after cancellations but before reduction was... 

Establishment of one-plus rate equations from experimental data. 

The principle of establishing a one-plus rate equation and the values of its phenomenological coefficients is very simple. If the reaction is irreversible and found to be of an order between zero and one with respect to a participant i, the simplest one-plus equation contains the respective concentration Cj (or p ) as a factor in the numerator and in some but not all terms of the denominator. More generally, if the order is between n (positive integer) and n + 1, the simplest equation contains the factor Cin+1 in the numerator and Q in some but not all terms of the denominator. Many other combinations are possible, but less likely. For instance, an order between zero and plus one might also result from a numerator with factor Cj2 and a denominator with Cj in some terms and Cf in the others. Occam s razor suggests the best policy to try the simplest option first. 

As an example, the four simplest one-plus rate equations for an irreversible reaction of first order in A and orders between zero and plus one in B and C are ... 

Example 7.3. One-plus rate equation for hydrocarbonyl-catalyzed hydrogenation of aldehyde [7]. Homogeneous liquid-phase hydrogenation of aldehydes to alcohols... 

A fit to the experimental results may require a one-plus rate equation with three or more terms in the denominator (e.g., see eqns 7.5). If so, the coefficients can be determined by linear regression or, long-hand, by cross-plotting. Say, the one-plus equation is... 

A one-plus rate equation requires a pathway with at least one reverse step. 

Example 7.4. Pathway elucidation of hydrocarbonyl-catalyzed aldehyde hydrogenation [7,9]. In Example 7.3 in the previous section, a one-plus rate equation for hydrocarbonyl-catalyzed aldehyde hydrogenation was established ... 

The conventional procedure of fitting a rate equation to experimental data is to use a power law reflecting the observed reaction orders. However, while fractional reaction orders may provide an acceptable fit, they cannot be produced by reasonable mechanisms. A better way is to fit the data to "one-plus" rate equations, that is, equations containing concentrations with integer exponents only, but with denominators composed of two or more additive terms of which the first is a "one." Such equations behave much like power laws with fractional exponents but, in contrast to these, can arise from reasonable mechanisms and therefore are more likely to hold over wide ranges of conditions. As an exception, rate equations with constant exponents of one half or integer (positive or negative) multiples of one half can result from chain reactions and reactions initiated by dissociation, and are acceptable if such a mechanism is probable or conceivable. 

The one-plus rate equation 8.22 is of the same algebraic form as the Michaelis-Menten equation 8.18, only the physical significance of the coefficients is different [instead of the constant K, the expression kAX /(k + kXP) now appears]. Accordingly, the behavior is the same as for Michaelis-Menten kinetics, and that name is often used for Briggs-Haldane kinetics as well. 

Reactions orders between zero and one in accordance with one-plus rate equations are very common in enzyme catalysis, even if the cycle is more complex and involves additional reactants or products. The plots just described thus are more broadly applicable. On the other hand, straight lines in such plots are only evidence of saturation kinetics, not an indication that the catalyst cycle has only one intermediate. 

In general terms, the determination of coefficients in power-law and one-plus rate equations has been described in Sections 3.3 and 7.2, respectively. Here, some detail will be added. 

The phenomenological coefficients in a one-plus rate equation derived for a specific mechanism are composites of individual rate coefficients of steps. Activation energies for them can be established from their temperature dependence with the Arrhenius equation. With regard to their activation energies, two types of phenomenological coefficients can be distinguished those consisting entirely of products, ratios, or ratios of products of individual rate coefficients and those which also involve additive terms. 

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