Lecture 7 Complex Reaction Mechanisms: Understanding Third-Order Reactions

7.1 The Lindemann-Hinshelwood Mechanism

Many reactions that appear simple at first glance actually proceed through complex mechanisms involving multiple elementary steps. A classic example is the thermal decomposition of azomethane:

\[\mathrm{CH_3N_2CH_3} \rightarrow \mathrm{C_2H_6} + \mathrm{N_2}\]

At high pressures, this reaction exhibits first-order kinetics:

\[-\frac{\mathrm{d}[\mathrm{CH_3N_2CH_3}]}{\mathrm{d}t} = k[\mathrm{CH_3N_2CH_3}]\]

This simple rate law presents us with a fundamental mechanistic puzzle when we examine the molecular-level implications. We observe two seemingly contradictory behaviours:

1. The reaction shows first-order kinetics (\(\nu = k[\mathrm{A}]\)), suggesting each molecule reacts independently.

2. The rate increases with temperature, indicating that energy is required to overcome an activation barrier. But how does each molecule acquire this activation energy?

If the activation energy came from molecular collisions (the usual mechanism for thermal activation), we would expect the rate to depend on the frequency of these collisions. Since collision frequency is proportional to the concentration of both colliding species, this would lead to second-order kinetics (\(\nu = k[\mathrm{A}]^2\)). Yet we observe first-order behaviour.

This puzzle suggests that a more complex mechanism must be at work - one that can explain both the thermal activation and the first-order kinetics.

7.1.1 The Lindemann Mechanism

In 1921, Frederick Lindemann proposed a three-step mechanism to explain this behaviour:

1. Excitation by collision: \[\mathrm{A} + \mathrm{M} \xrightarrow{k_1} \mathrm{A^*} + \mathrm{M}\]

2. Deactivation by collision: \[\mathrm{A^*} + \mathrm{M} \xrightarrow{k_{-1}} \mathrm{A} + \mathrm{M}\]

3. Unimolecular decomposition: \[\mathrm{A^*} \xrightarrow{k_2} \mathrm{P}\]

Here, M represents any collision partner (often the reactant molecule itself), and A* represents an energetically excited molecule.

7.1.2 Mathematical Analysis

We can apply the steady-state approximation to the excited intermediate A*:

\[\frac{\mathrm{d}[\mathrm{A^*}]}{\mathrm{d}t} = k_1[\mathrm{A}][\mathrm{M}] - k_{-1}[\mathrm{A^*}][\mathrm{M}] - k_2[\mathrm{A^*}] = 0\]

Solving for [A*]:

\[[\mathrm{A^*}] = \frac{k_1[\mathrm{A}][\mathrm{M}]}{k_{-1}[\mathrm{M}] + k_2}\]

The overall reaction rate is determined by the decomposition of A*:

\[\frac{\mathrm{d}[\mathrm{P}]}{\mathrm{d}t} = k_2[\mathrm{A^*}] = \frac{k_1k_2[\mathrm{A}][\mathrm{M}]}{k_{-1}[\mathrm{M}] + k_2}\]

For the specific case of azomethane decomposition, where M = CH₃N₂CH₃:

\[\frac{\mathrm{d}[\mathrm{C_2H_6}]}{\mathrm{d}t} = \frac{k_1k_2[\mathrm{CH_3N_2CH_3}]^2}{k_{-1}[\mathrm{CH_3N_2CH_3}] + k_2}\]

7.1.3 High-Pressure Limit

At high pressures, collisional deactivation is much faster than unimolecular decomposition (\(k_{-1}[\mathrm{CH_3N_2CH_3}] \gg k_2\)). Under these conditions:

\[\frac{\mathrm{d}[\mathrm{C_2H_6}]}{\mathrm{d}t} = \frac{k_1k_2}{k_{-1}}[\mathrm{CH_3N_2CH_3}]\]

This simplifies to first-order kinetics with an effective rate constant \(k = \frac{k_1k_2}{k_{-1}}\), explaining the observed behaviour.

7.1.4 Low-Pressure Limit

Conversely, at low pressures where \(k_{-1}[\mathrm{CH_3N_2CH_3}] \ll k_2\):

\[\frac{\mathrm{d}[\mathrm{C_2H_6}]}{\mathrm{d}t} = k_1[\mathrm{CH_3N_2CH_3}]^2\]

The reaction shows second-order kinetics with \(k = k_1\). This prediction was experimentally verified in 1927, providing strong support for the Lindemann mechanism.

7.2 Third-Order Reactions

Some reactions exhibit third-order kinetics, such as the oxidation of nitric oxide:

\[2\mathrm{NO} + \mathrm{O_2} \rightarrow 2\mathrm{NO_2}\]

with rate law:

\[\frac{\mathrm{d}[\mathrm{NO_2}]}{\mathrm{d}t} = k[\mathrm{NO}]^2[\mathrm{O_2}]\]

7.2.1 Mechanistic Implications

A simple termolecular collision mechanism seems implausible - simultaneous three-body collisions are extremely rare. Moreover, the reaction rate decreases with increasing temperature, suggesting a more complex mechanism.

An alternative mechanism involving a pre-equilibrium better explains these observations:

\[\mathrm{NO} + \mathrm{NO} \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} (\mathrm{NO})_2\] \[(\mathrm{NO})_2 + \mathrm{O_2} \xrightarrow{k_2} 2\mathrm{NO_2}\]

At equilibrium:

\[K = \frac{k_1}{k_{-1}} = \frac{[(\mathrm{NO})_2]}{[\mathrm{NO}]^2}\]

The overall rate:

\[\frac{\mathrm{d}[\mathrm{NO_2}]}{\mathrm{d}t} = 2k_2[(\mathrm{NO})_2][\mathrm{O_2}] = 2k_2K[\mathrm{NO}]^2[\mathrm{O_2}]\]

This mechanism explains both the observed third-order kinetics and the negative temperature dependence (as the pre-equilibrium favours dissociation at higher temperatures).

7.3 Formation of Ozone

Another important third-order reaction is ozone formation:

\[\mathrm{O} + \mathrm{O_2} \rightarrow \mathrm{O_3}\]

This reaction exhibits third-order kinetics with the rate law:

\[\frac{\mathrm{d}[\mathrm{O_3}]}{\mathrm{d}t} = k[\mathrm{O}][\mathrm{O_2}][\mathrm{M}]\]

where M represents a third body (typically another gas molecule like N₂ or O₂).

7.3.1 Why a Third Body is Necessary

The requirement for a third body arises from two critical factors:

1. High Exothermicity: The formation of ozone is highly exothermic, releasing approximately 107 kJ/mol when the new O-O bond forms. This substantial energy release creates a vibrationally excited ozone molecule (O₃*).

2. Momentum Conservation Constraints: When an O atom collides with an O₂ molecule, the laws of momentum conservation severely restrict how the released energy can be distributed. Consider a head-on collision:

For a simple bimolecular collision, both momentum and energy must be conserved:

\[m_\mathrm{O}v_\mathrm{O} + m_\mathrm{O_2}v_\mathrm{O_2} = m_\mathrm{O_3}v_\mathrm{O_3}\]

The conservation of momentum forces much of the released energy into the internal vibrational modes of the O₃ molecule rather than translational motion. This leads to a highly excited O₃* that rapidly dissociates back to O + O₂ unless the excess energy can be removed.

This is where the third body (M) becomes essential. When the vibrationally excited O₃* collides with M, the excess energy can be transferred away, allowing the ozone molecule to stabilize:

# Insert diagram showing energy transfer to third body

The reaction therefore proceeds through two steps:

1. Formation of excited ozone: \[\mathrm{O} + \mathrm{O_2} \rightleftharpoons \mathrm{O_3^*}\]

2. Collisional stabilisation: \[\mathrm{O_3^*} + \mathrm{M} \rightarrow \mathrm{O_3} + \mathrm{M}\]

Applying a pre-equilibrium approximation to this mechanism yields the observed third-order kinetics:

\[\frac{\mathrm{d}[\mathrm{O_3}]}{\mathrm{d}t} = k[\mathrm{O}][\mathrm{O_2}][\mathrm{M}]\]

7.3.2 Fundamental Molecular Constraints

The necessity for a third body in ozone formation illustrates an important principle in reaction dynamics: highly exothermic reactions involving simple molecules (with few internal degrees of freedom) often cannot effectively distribute the released energy internally. The product molecule becomes vibrationally excited in specific modes that can break newly formed bonds.

Third-body collisions provide a mechanism to remove this excess energy, allowing the product to stabilize. This principle applies broadly to association reactions between atoms and small molecules, explaining the prevalence of third-order kinetics in many atmospheric and combustion processes.