F Negative Apparent Orders

In Lecture 1, we defined the order of a reaction with respect to a given species as the exponent in the rate law. For a simple power-law rate law \(\nu = k[\mathrm{A}]^a[\mathrm{B}]^b\), the orders \(a\) and \(b\) are well defined. In more complex rate laws — particularly those arising from multi-step mechanisms — a species can appear in both the numerator and the denominator. When this happens, the rate law cannot be written as a simple power law and the order with respect to that species is not defined. We have already seen this pattern in the Lindemann mechanism (Lecture 7), the Michaelis–Menten equation (Lecture 8), and the H₂ + Br₂ rate law (Lecture 9).

Although these rate laws have no well-defined order, they can exhibit striking behaviour: increasing the concentration of a species can slow down the reaction. In limiting cases, the rate law simplifies to a power-law form in which the species appears with a negative exponent — a negative apparent order. This appendix explores how this behaviour arises from underlying mechanisms. The material is not examinable, but it illustrates an important theme of the course: macroscopic rate laws encode information about microscopic mechanisms.

F.1 When Increasing a Concentration Slows the Reaction

If a rate law can be written as a simple power law with a negative exponent:

\[\nu = \frac{k[\mathrm{A}]^a[\mathrm{B}]^b}{[\mathrm{X}]^n}\]

then X has a well-defined negative order \(-n\), and increasing \([\mathrm{X}]\) always decreases the rate. Such a rate law cannot arise from a single elementary step — it always signals a complex, multi-step mechanism in which the species interferes with one or more productive reaction pathways.

More commonly, the species appears within a denominator of the form \((1 + \alpha[\mathrm{X}])\), where \(\alpha\) is some combination of rate constants (not necessarily an equilibrium constant). This is the pattern in the Michaelis–Menten and Lindemann rate laws. The order with respect to X is then undefined: it varies continuously from 0 (when \(\alpha[\mathrm{X}] \ll 1\)) to \(-1\) (when \(\alpha[\mathrm{X}] \gg 1\)). It is only in this high-concentration limit that we can speak of a negative apparent order.

F.2 Example 1: Product Inhibition in H₂ + Br₂ \(\rightarrow\) 2HBr

The formation of hydrogen bromide — a chain reaction discussed in detail in Lecture 9 — provides a classic example of product inhibition. The empirical rate law is:

\[\nu = \frac{k[\mathrm{H}_2][\mathrm{Br}_2]^{1/2}}{1 + k'[\mathrm{HBr}]/[\mathrm{Br}_2]}\]

Because HBr appears in a \((1 + \ldots)\) denominator term, the order with respect to HBr is not well defined. Early in the reaction, when very little HBr has formed, the denominator is close to 1 and HBr has essentially no effect on the rate. As HBr accumulates, the denominator grows and the rate decreases. In the limit \(k'[\mathrm{HBr}]/[\mathrm{Br}_2] \gg 1\), the rate law simplifies to:

\[\nu \approx \frac{k[\mathrm{H}_2][\mathrm{Br}_2]^{3/2}}{k'[\mathrm{HBr}]}\]

In this limiting form, HBr has an apparent order of \(-1\). The very product we are trying to make inhibits its own formation.

The origin lies in the chain mechanism. The key propagation steps produce HBr via reactive H\(^{\bullet}\) radicals (Lecture 9), but HBr can also react with these same radicals:

\[\mathrm{H}^{\bullet} + \mathrm{HBr} \rightarrow \mathrm{Br}^{\bullet} + \mathrm{H}_2\]

This chain inhibition step consumes the very intermediates needed for product formation. The more HBr present, the more this reverse reaction competes with productive chain propagation, slowing the net rate. This is a general principle in chain kinetics: anything that consumes or deactivates chain-carrying intermediates can produce inhibition.

F.3 Example 2: Competitive Adsorption in Surface Reactions

Surface-catalysed reactions provide another class of systems where increasing a reactant’s pressure can decrease the rate, closely related to the Langmuir adsorption model discussed in Lecture 8.

Unlike homogeneous reactions, surface processes require that reactants first adsorb onto the catalyst before they can react. The catalyst surface has a limited number of active sites, creating the potential for competition. For a typical surface reaction following the Langmuir–Hinshelwood mechanism:

1. \(\mathrm{A}_{(\mathrm{g})} + {*} \rightleftharpoons \mathrm{A}{*}\) (A adsorbs on an empty site \({*}\))
2. \(\mathrm{B}_{(\mathrm{g})} + {*} \rightleftharpoons \mathrm{B}{*}\) (B adsorbs on an empty site \({*}\))
3. \(\mathrm{A}{*} + \mathrm{B}{*} \rightarrow \text{products} + 2{*}\) (reaction between adsorbed species)

The rate depends on having both A and B adsorbed on adjacent sites. Using the competitive Langmuir isotherm, the coverages are \(\theta_\mathrm{A} = K_\mathrm{A} p_\mathrm{A}/(1 + K_\mathrm{A} p_\mathrm{A} + K_\mathrm{B} p_\mathrm{B})\) and similarly for B. The rate is proportional to \(\theta_\mathrm{A} \theta_\mathrm{B}\), which is not a simple power law in either pressure — the order with respect to each species is undefined, just as in the Michaelis–Menten and HBr examples.

If B adsorbs much more strongly, or is present at very high pressure, it can monopolise the surface sites. In the limit \(K_\mathrm{B} p_\mathrm{B} \gg 1 + K_\mathrm{A} p_\mathrm{A}\), the rate becomes approximately proportional to \(p_\mathrm{A}/p_\mathrm{B}\): the apparent order in B approaches \(-1\), despite B being an essential reactant. Increasing the pressure of B decreases the reaction rate by blocking the sites that A needs.

This site-blocking effect is commonly observed in industrial catalysis. For example, in CO oxidation on platinum, excess CO can poison the catalyst by blocking oxygen adsorption sites. The mathematical framework from Lecture 8 — where the Langmuir isotherm describes surface coverage as a function of pressure — provides the quantitative basis for understanding these effects.

F.4 Recognising Inhibition in Rate Laws

The phenomenon of increasing a species’ concentration slowing the reaction appears in several forms:

• Species in the denominator of a \((1 + \ldots)\) term: The order with respect to that species is undefined, varying continuously from 0 at low concentration to an apparent \(-1\) at high concentration. This is the most common pattern, seen in the Michaelis–Menten equation, the Lindemann mechanism, the HBr rate law, and competitive Langmuir adsorption.
• Species in the denominator of a simple ratio (e.g., \(k[\mathrm{A}]/[\mathrm{X}]\)): This represents a well-defined negative order of \(-1\). Such forms typically arise as limiting cases of the more general \((1 + \ldots)\) pattern.

The common thread is that these behaviours arise when a species participates in the overall mechanism in ways that hinder rather than help the net forward reaction — whether by consuming reactive intermediates, blocking active sites, or driving reverse reactions.