9.3 Chain reactions and their characteristics

Chain reactions are multi-step processes where each step generates reactive species that drive the next step forward. They follow a predictable pattern of initiation, propagation, and termination, with free radicals sustaining the cycle. Understanding how these stages work together is essential for deriving rate laws for complex mechanisms and for explaining real-world processes like polymerization and combustion.

Chain Reaction Fundamentals

Characteristics of chain reactions

A chain reaction proceeds through three distinct stages. Each stage plays a specific role in starting, sustaining, or ending the overall process.

• Initiation: A stable molecule breaks apart to produce reactive species (usually free radicals). This bond dissociation is triggered by an external energy source such as heat (thermolysis) or light (photolysis). For example, UV light splits $Cl_2$ into two $Cl \cdot$ radicals.
• Propagation: The reactive species from initiation react with stable molecules, producing product and regenerating a new reactive species. This is what makes the reaction a chain: each propagation step feeds the next. Propagation steps are typically fast and exothermic.
• Termination: Two reactive species combine to form a stable product, or a radical is deactivated by colliding with a vessel wall or an inert molecule. Termination removes radicals from the cycle and stops the chain.

A single initiation event can trigger thousands of propagation cycles before termination occurs. That amplification effect is what makes chain reactions so powerful and, in some cases, explosive.

Role of reactive intermediates

Free radicals are the workhorses of chain reactions. A free radical is a species with one or more unpaired electrons, making it extremely reactive and short-lived.

During initiation, radicals are generated from stable precursors. In propagation, each radical reacts with a stable molecule to form a new radical. The two most common propagation mechanisms are:

• Hydrogen abstraction: A radical pulls an H atom from a stable molecule (e.g., $Cl \cdot + CH_4 \rightarrow HCl + CH_3 \cdot$).
• Addition to a double bond: A radical adds across a $C=C$ bond, creating a new, larger radical. This is the key step in radical polymerization.

During termination, radicals are consumed through recombination (two radicals join to form a single stable molecule) or disproportionation (two radicals exchange an H atom, producing two different stable molecules). Because termination requires two radicals to meet, it becomes more likely at higher radical concentrations.

Kinetics and Applications

Rate equations for chain reactions

Deriving a rate law for a chain reaction involves applying the steady-state approximation to the reactive intermediates. You assume that after a brief induction period, the rate of radical formation (initiation) equals the rate of radical destruction (termination), so the radical concentration stays roughly constant.

For a simple chain reaction with one initiation, one propagation, and one termination step, this approach yields:

$Rate = k_p \sqrt{\frac{k_i}{k_t}}[A]$

where:

• $k_i$ = rate constant for initiation
• $k_p$ = rate constant for propagation
• $k_t$ = rate constant for termination
• $[A]$ = concentration of the reactant consumed in the propagation step

Notice the square-root dependence on $k_i / k_t$. This arises because termination is typically second-order in radical concentration while initiation is zero- or first-order. The ratio inside the square root controls the steady-state radical concentration.

Factors that influence the overall rate:

• Temperature: Increases both $k_i$ and $k_p$, accelerating the reaction. Because initiation often has a high activation energy, temperature has a particularly strong effect on radical production.
• Pressure: For gas-phase reactions, higher pressure increases the frequency of bimolecular collisions in propagation steps.
• Reactant concentration: Higher $[A]$ directly increases the propagation rate.
• Inhibitors: Molecules that react with free radicals to form stable, unreactive products (e.g., antioxidants like BHT) effectively increase the termination rate and slow or halt the chain.

Chain length

Chain length ($\nu$) is the average number of propagation cycles a single radical completes before being terminated:

$\nu = \frac{\text{rate of propagation}}{\text{rate of initiation}} = \frac{k_p [A]}{2 \cdot \text{rate of initiation per radical}}$

A longer chain length means each initiation event produces more product, resulting in a faster overall reaction and higher yield. Chain length depends on the competition between propagation and termination:

• If propagation is much faster than termination, $\nu$ is large (often $10^3$ to $10^5$ in combustion).
• If termination competes effectively with propagation, $\nu$ is small and the reaction is slow.

Chain transfer agents are molecules that react with a growing radical to terminate one chain but simultaneously generate a new radical that starts another. They don't change the overall reaction rate much, but in polymerization they control the average molecular weight of the polymer by limiting how long each individual chain grows.

Examples in polymerization and combustion

Radical polymerization (e.g., production of polyethylene or polystyrene):

1. Initiation: A thermal initiator like benzoyl peroxide decomposes into two radicals upon heating.
2. Propagation: A radical adds across the $C=C$ double bond of a monomer (e.g., ethylene), creating a new radical at the chain end. This repeats thousands of times, building a long polymer chain.
3. Termination: Two growing polymer radicals meet and undergo recombination or disproportionation, producing a dead polymer chain.

The chain length in polymerization directly determines the degree of polymerization and therefore the physical properties (strength, melting point) of the resulting plastic.

Combustion of hydrocarbons (e.g., burning methane in a gas turbine):

1. Initiation: At high temperatures, $O_2$ or fuel molecules dissociate to produce radicals such as $H \cdot$, $O \cdot$, and $OH \cdot$.
2. Propagation: These radicals abstract hydrogen atoms from fuel molecules, generating new radicals and releasing heat. A critical feature of combustion is chain branching, where a single propagation step produces two or more radicals instead of one (e.g., $H \cdot + O_2 \rightarrow OH \cdot + O \cdot$). Branching can cause the radical population to grow exponentially, leading to explosions.
3. Termination: Radicals recombine in the gas phase or are deactivated on surfaces.

Other important chain reactions include stratospheric ozone depletion (where $Cl \cdot$ radicals from CFCs catalytically destroy $O_3$ with chain lengths exceeding $10^4$) and nuclear fission chain reactions (where neutrons, rather than radicals, are the chain carriers).