C Negative Reaction Orders

C.1 What Are Negative Reaction Orders?

In most cases, increasing the concentration of a reactant increases the reaction rate; this corresponds to positive reaction orders. In some reactions, however, increasing the concentration of a particular species slows down the reaction. This unusual behaviour manifests as a negative reaction order.

Mathematically, when a species X appears with a negative order \(n\) in a rate law:

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

or equivalently:

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

increasing \([\mathrm{X}]\) decreases the reaction rate. The species X effectively acts as an inhibitor rather than a promoter of the reaction.

C.2 Physical Origins of Negative Orders

Negative reaction orders always signal the presence of a complex, multi-step mechanism. They cannot arise from simple, elementary processes but instead emerge from:

  1. Competitive inhibition - where a species competes with productive reaction pathways
  2. Product inhibition - where reaction products interfere with the forward mechanism
  3. Site blocking - in surface reactions and enzyme systems where species compete for limited active sites
  4. Reverse reactions - where a species promotes the backward reaction

The key insight is that these species participate in the overall mechanism but in ways that hinder rather than help the net forward reaction.

C.3 Example 1: Chain Reaction Inhibition in H2 + Br2 → 2HBr

The formation of hydrogen bromide demonstrates product inhibition leading to negative orders. The empirical rate law for this reaction is:

\[\nu = \frac{k[\mathrm{H_2}][\mathrm{Br_2}]^{1/2}}{1 + k'[\mathrm{HBr}]/[\mathrm{Br_2}]}\]

At high concentrations of HBr, this simplifies to show negative dependence on the product:

\[\nu \approx \frac{k[\mathrm{H_2}][\mathrm{Br_2}]^{3/2}}{k'[\mathrm{HBr}]}\]

Here, HBr has an effective order of −1. The very product we are trying to make actually inhibits its own formation!

C.3.1 Why Does This Happen?

The H2 + Br2 reaction proceeds through a chain mechanism involving reactive radical intermediates. The key steps include:

  • Chain propagation: \(\mathrm{Br}^{\bullet} + \mathrm{H_2} \rightarrow \mathrm{H}^{\bullet} + \mathrm{HBr}\)
  • Chain propagation: \(\mathrm{H}^{\bullet} + \mathrm{Br_2} \rightarrow \mathrm{Br}^{\bullet} + \mathrm{HBr}\)
  • Chain inhibition: \(\mathrm{H}^{\bullet} + \mathrm{HBr} \rightarrow \mathrm{Br}^{\bullet} + \mathrm{H_2}\)

The crucial point is the final step: HBr can react with the reactive H\(^{\bullet}\) radicals that drive the chain reaction forward. This consumes the very intermediates needed for product formation. The more HBr present in the system, the more this reverse reaction competes with productive chain propagation, effectively slowing down the net rate of HBr formation.

This illustrates a general principle: in chain reactions, anything that consumes or deactivates the chain-carrying intermediates will show negative kinetic behaviour.

C.4 Example 2: Competitive Adsorption in Surface Reactions

Surface-catalysed reactions provide another important class of systems that show negative orders. Consider a reaction A + B → products occurring on a catalyst surface.

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 between different species.

For a typical surface reaction following the Langmuir-Hinshelwood mechanism:

  1. \(\mathrm{A}_{(\mathrm{g})} + * \rightleftharpoons \mathrm{A}*\) (A adsorbs on empty site *)
  2. \(\mathrm{B}_{(\mathrm{g})} + * \rightleftharpoons \mathrm{B}*\) (B adsorbs on 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. However, if one species (say B) adsorbs much more strongly than the other, or if its gas-phase pressure becomes very high, it can monopolise the surface sites.

C.4.1 The Site-Blocking Effect

When B saturates the surface, there are fewer sites available for A to adsorb, even though we need both species present for the reaction to occur. Under these conditions, increasing the pressure of B actually decreases the reaction rate because:

  • B blocks sites that A needs
  • The surface coverage of A decreases
  • Fewer A-B pairs can form on adjacent sites
  • The overall reaction rate falls

This leads to negative apparent orders in B at high pressures, despite B being an essential reactant.

This behaviour is commonly observed in industrial catalysis. For example:

  • In ammonia synthesis, high hydrogen pressures can inhibit the reaction by preventing nitrogen adsorption
  • In CO oxidation, excess CO can poison the catalyst by blocking oxygen adsorption sites
  • In hydrogenation reactions, too much hydrogen can prevent the organic substrate from accessing the catalyst surface

C.5 Recognising Negative Orders

Negative orders often appear in disguised forms within rate expressions:

  1. Denominator terms: Any concentration appearing in the denominator has a negative effective order, whether directly (e.g., \(k[\mathrm{A}]/[\mathrm{X}]\)) or within inhibition factors (e.g., \(k[\mathrm{A}]/(1 + k'[\mathrm{X}])\))
  2. Inhibition factors: Terms like \((1 + k[\mathrm{X}])\) in the denominator indicate negative effects
  3. Saturation behaviour: Rates that plateau or decrease with increasing concentration suggest competitive effects

C.6 Practical Implications

Understanding negative orders has important practical consequences:

Process Optimisation: Simply increasing all reactant concentrations won’t necessarily speed up a reaction. There may be optimal concentration ratios that maximise the rate.

Catalyst Design: Recognising competitive adsorption helps in designing catalysts with appropriate site distributions or selectivities.

Product Removal: For reactions showing product inhibition, continuously removing products can maintain higher reaction rates.

Industrial Control: Many industrial processes must carefully control reactant ratios to avoid operating in negative-order regimes where increasing a reactant concentration actually decreases productivity.