Chain reaction

Chain reactions are one way that systems which are not in thermodynamic equilibrium can release energy or increase entropy in order to reach a state of higher entropy. For example, a system may not be able to reach a lower energy state by releasing energy into the environment, because it is hindered or prevented in some way from taking the path that will result in the energy release. If a reaction results in a small energy release making way for more energy releases in an expanding chain, then the system will typically collapse explosively until much or all of the stored energy has been released.

A macroscopic metaphor for chain reactions is thus a snowball causing a larger snowball until finally an avalanche results ("snowball effect"). This is a result of stored gravitational potential energy seeking a path of release over friction. Chemically, the equivalent to a snow avalanche is a spark causing a forest fire. In nuclear physics, a single stray neutron can result in a prompt critical event, which may finally be energetic enough for a nuclear reactor meltdown or (in a bomb) a nuclear explosion.

Another metaphor for a chain reaction is the domino effect, named after the act of domino toppling, where the simple action of toppling one domino leads to all dominoes eventually toppling, even if they are significantly larger.

Numerous chain reactions can be represented by a mathematical model based on Markov chains.

Chemical chain reactions

History

In 1913, the German chemist Max Bodenstein first put forth the idea of chemical chain reactions. If two molecules react, not only molecules of the final reaction products are formed, but also some unstable molecules which can further react with the parent molecules with a far larger probability than the initial reactants. (In the new reaction, further unstable molecules are formed besides the stable products, and so on.)

In 1918, Walther Nernst proposed that the photochemical reaction between hydrogen and chlorine is a chain reaction in order to explain what is known as the quantum yield phenomena. This means that one photon of light is responsible for the formation of as many as 10⁶ molecules of the product HCl. Nernst suggested that the photon dissociates a Cl₂ molecule into two Cl atoms which each initiate a long chain of reaction steps forming HCl.

In 1923, Danish and Dutch scientists J. A. Christiansen and Hendrik Anthony Kramers, in an analysis of the formation of polymers, pointed out that such a chain reaction need not start with a molecule excited by light, but could also start with two molecules colliding violently due to thermal energy as previously proposed for initiation of chemical reactions by van' t Hoff.

Christiansen and Kramers also noted that if, in one link of the reaction chain, two or more unstable molecules are produced, the reaction chain would branch and grow. The result is in fact an exponential growth, thus giving rise to explosive increases in reaction rates, and indeed to chemical explosions themselves. This was the first proposal for the mechanism of chemical explosions.

A quantitative chain chemical reaction theory was created later on by Soviet physicist Nikolay Semyonov in 1934. Semyonov shared the Nobel Prize in 1956 with Sir Cyril Norman Hinshelwood, who independently developed many of the same quantitative concepts.

Typical steps

The main types of steps in chain reaction are of the following types.

• Initiation (formation of active particles or chain carriers, often free radicals, in either a thermal or a photochemical step)
• Propagation (may comprise several elementary steps in a cycle, where the active particle through reaction forms another active particle which continues the reaction chain by entering the next elementary step). In effect the active particle serves as a catalyst for the overall reaction of the propagation cycle. Particular cases are:
  • chain branching (a propagation step where one active particle enters the step and two or more are formed);
  • chain transfer (a propagation step in which the active particle is a growing polymer chain which reacts to form an inactive polymer whose growth is terminated and an active small particle (such as a radical), which may then react to form a new polymer chain).
• Termination (elementary step in which the active particle loses its activity; e. g. by recombination of two free radicals).

The chain length is defined as the average number of times the propagation cycle is repeated, and equals the overall reaction rate divided by the initiation rate.

Some chain reactions have complex rate equations with fractional order or mixed order kinetics.

Detailed example: the hydrogen-bromine reaction

The reaction H₂ + Br₂ → 2 HBr proceeds by the following mechanism:

• Initiation

Br₂ → 2 Br• (thermal) or Br₂ + hν → 2 Br• (photochemical)

each Br atom is a free radical, indicated by the symbol "•" representing an unpaired electron.

• Propagation (here a cycle of two steps)

Br• + H₂ → HBr + H•

H• + Br₂ → HBr + Br•

the sum of these two steps corresponds to the overall reaction H₂ + Br₂ → 2 HBr, with catalysis by Br• which participates in the first step and is regenerated in the second step.

• Retardation (inhibition)

H• + HBr → H₂ + Br•

this step is specific to this example, and corresponds to the first propagation step in reverse.

• Termination 2 Br• → Br₂

recombination of two radicals, corresponding in this example to initiation in reverse.

As can be explained using the steady-state approximation, the thermal reaction has an initial rate of fractional order (3/2), and a complete rate equation with a two-term denominator (mixed-order kinetics).

Further chemical examples

• The reaction 2 H₂ + O₂ → 2 H₂O provides an example of chain branching. The propagation is a sequence of two steps whose net effect is to replace an H atom by another H atom plus two OH radicals. This leads to an explosion under certain conditions of temperature and pressure.
  • H• + O₂ → •OH + •O•
  • •O• + H₂ → •OH + H•
• In chain-growth polymerization, the propagation step corresponds to the elongation of the growing polymer chain. Chain transfer corresponds to transfer of the activity from this growing chain, whose growth is terminated, to another molecule which may be a second growing polymer chain. For polymerization, the kinetic chain length defined above may differ from the degree of polymerization of the product macromolecule.
• Polymerase chain reaction, a technique used in molecular biology to amplify (make many copies of) a piece of DNA by in vitro enzymatic replication using a DNA polymerase.

Acetaldehyde pyrolysis and rate equation

The pyrolysis (thermal decomposition) of acetaldehyde, CH₃CHO (g) → CH₄ (g) + CO (g), proceeds via the Rice-Herzfeld mechanism:

• Initiation (formation of free radicals):

CH₃CHO (g) → •CH₃ (g) + •CHO (g) k₁

The methyl and CHO groups are free radicals.

• Propagation (two steps):

•CH₃ (g) + CH₃CHO (g) → CH₄ (g) + •CH₃CO (g) k₂

This reaction step provides methane, which is one of the two main products.

•CH₃CO (g) → CO (g) + •CH₃ (g) k₃

The product •CH₃CO (g) of the previous step gives rise to carbon monoxide (CO), which is the second main product.

The sum of the two propagation steps corresponds to the overall reaction CH₃CHO (g) → CH₄ (g) + CO (g), catalyzed by a methyl radical •CH₃.

• Termination:

•CH₃ (g) + •CH₃ (g) → C₂H₆ (g) k₄

This reaction is the only source of ethane (minor product) and it is concluded to be the main chain ending step.

Although this mechanism explains the principal products, there are others that are formed in a minor degree, such as acetone (CH₃COCH₃) and propanal (CH₃CH₂CHO).

Applying the Steady State Approximation for the intermediate species CH₃(g) and CH₃CO(g), the rate law for the formation of methane and the order of reaction are found:

The rate of formation of the product methane is

${\displaystyle (1)...\frac{d[{\text{CH}}_4^{}]}{dt}=k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]}$

For the intermediates

${\displaystyle (2)...\frac{d[{\text{CH}}_3^{}]}{dt}=k_1[{\text{CH}}_3^{}\text{CHO}]-k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]+k_3[{\text{CH}}_3^{}\text{CO}]-2k_4{[{\text{CH}}_3^{}]}^2=0}$ and

${\displaystyle (3)...\frac{d[{\text{CH}}_3^{}\text{CO}]}{dt}=k_2[{\text{CH}}_3^{}][{\text{CH}}_3^{}\text{CHO}]-k_3[{\text{CH}}_3^{}\text{CO}]=0}$

Adding (2) and (3), we obtain ${\displaystyle k_1[{\text{CH}}_3^{}\text{CHO}]-2k_4{[{\text{CH}}_3^{}]}^2=0}$

so that ${\displaystyle (4)...[{\text{CH}}_3^{}]=\frac{k_1}{2k_4}{[{\text{CH}}_3^{}\text{CHO}]}^{1/2}}$

Using (4) in (1) gives the rate law ${\displaystyle (5)\frac{d[{\text{CH}}_4^{}]}{dt}=\frac{k_1}{2k_4}{k}_2{[{\text{CH}}_3^{}\text{CHO}]}^{3/2}}$, which is order 3/2 in the reactant CH₃CHO.