Christiansen matrix

If X2 is the macs, the Christiansen matrix retains only its third row ... 

Whenever the forward and reverse X coefficients of a step are much smaller than all others, the denominator terms involving those coefficients become negligible (see Section 4.1). Say, the step Xj — Xj+1 (co-reactants or -products not shown) is rate-controlling. The terms involving XjJ+1 and Xj+1 j will be then small compared with those which, instead, involve Xjj, and Xj+1J+2, respectively. This makes all terms but one disappear in each row of the Christiansen matrix. In the general case of a k-membered cycle, this reduces the number of terms from k2 to k. 

Assume the second step in the four-membered cycle 8.34 is irreversible, so that X21 = 0. The Christiansen matrix then reduces to... 

The approximations in this section can be combined in many different ways. Significant further simplification may result. For example, for a cycle 8.34 with X as the macs and irreversible step X, + B — X2, the only remaining row of the Christiansen matrix is the second, and that row has but one single element, A qAq, (see reduced matrix 8.45). In this case, the rate equation reduces to... 

Reaction orders in cycles with macs [47]. The rules for reaction orders in simple pathways do not apply to catalytic cycles with arbitrary distribution of catalyst material over the cycle members. In the general case, any cycle with more than two or three members gives rise to a Christiansen matrix with a profusion of terms, many of which are apt to involve different combinations of reactant and product concentrations as co-factors. This makes it impossible to formulate general rules for such cycles. All that can be said is that a distribution of catalyst material over several species makes for fractional and varying reaction orders as concentrations tend to appear in the numerator and some but not all terms of the denominator. 

As a comparison shows, the reaction orders derived above with the Christiansen matrix could equally well have been predicted with the rules set forth earlier in this section. The example confirms the usefulness of the rules. It also demonstrates how profoundly kinetics can be affected by the distribution of catalyst material. 

For competitive inhibition with single-step external pathway, eqn 8.65 is directly applicable, except that the inhibitor concentration replaces that of the free ligand. For noncompetitive inhibition by reaction with Xj5 A must be replaced by Djj, the sum of the elements of row j +1 of the Christiansen matrix. Both cases are covered by ... 

This matrix, if we omit the stoichiometric numbers of steps, is reduced to the so-called Christiansen matrix , named after the famous Dannish scientist Jens Anton Christiansen. 

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