Editing Chain reaction (section)

==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 [[photochemistry|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 10<sup>6</sup> molecules of the product [[HCl]]. Nernst suggested that the photon dissociates a Cl<sub>2</sub> molecule into two Cl atoms which each initiate a long chain of reaction steps forming HCl.<ref name=Laidler288>Laidler K.J., ''Chemical Kinetics'' (3rd ed., Harper & Row 1987) p.288-290  {{ISBN|0-06-043862-2}}</ref>

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 [[Jacobus Henricus van 't Hoff|van' t Hoff]].<ref name=Nobel1956/>

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.<ref>{{cite web |url=http://www.marka-art.ru/catalogs/StampSeries.jsp?%26id%3D29264%26lang%3Den |title=Postal stamps series |access-date=2012-04-17 |url-status=dead |archive-url=https://web.archive.org/web/20090116032126/http://www.marka-art.ru/catalogs/StampSeries.jsp?&id=29264&lang=en |archive-date=2009-01-16 }}</ref> Semyonov shared the Nobel Prize in 1956 with Sir [[Cyril Norman Hinshelwood]], who independently developed many of the same quantitative concepts.<ref name=Nobel1956>http://nobelprize.org/nobel_prizes/chemistry/laureates/1956/press.html History of the chemical chain reaction from 1913 to the Nobel work recognized in 1956</ref>

===Typical steps===
The main types of steps in chain reaction are of the following types.<ref name=Laidler288/>
* [[Chain initiation|Initiation]] (formation of [[Active center (polymer science)|active particles]] or chain carriers, often [[free radical]]s, in either a thermal or a photochemical step)
* [[Chain propagation|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).
* [[Chain termination|Termination]] (elementary step in which the active particle loses its activity; e. g. by [[Recombination (chemistry)|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.<ref name=Laidler288/>

Some chain reactions have complex [[rate equation]]s with [[Order of reaction#Fractional order|fractional order]] or [[Order of reaction#Mixed order|mixed order]] kinetics.

=== Detailed example: the hydrogen-bromine reaction ===
The reaction H<sub>2</sub> + Br<sub>2</sub> → 2 HBr proceeds by the following mechanism:<ref name=Laidler291>[[Keith J. Laidler|Laidler K.J.]], ''Chemical Kinetics'' (3rd ed., Harper & Row 1987) p.291-4  {{ISBN|0-06-043862-2}}</ref><ref name=Atkins>P. Atkins and J. de Paula ''Physical Chemistry'' (8th ed., W.H. Freeman 2006), p.830-1  {{ISBN|0-7167-8759-8}}</ref>
* Initiation 
: Br<sub>2</sub> → 2 Br• (thermal) or Br<sub>2</sub> + 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• + H<sub>2</sub> → HBr + H•
: H• + Br<sub>2</sub> → HBr + Br•
: the sum of these two steps corresponds to the overall reaction H<sub>2</sub> + Br<sub>2</sub> → 2 HBr, with [[catalysis]] by Br• which participates in the first step and is regenerated in the second step.

* Retardation (inhibition)
: H• + HBr → H<sub>2</sub> + Br•
: this step is specific to this example, and corresponds to the first propagation step in reverse.

* Termination 2 Br• → Br<sub>2</sub>
: recombination of two radicals, corresponding in this example to initiation in reverse.

As can be explained using the [[Steady state (chemistry)|steady-state approximation]], the thermal  reaction has an initial rate of [[Order of reaction#Fractional order|fractional order]] (3/2), and a complete rate equation with a two-term denominator ([[Order of reaction#Mixed order|mixed-order kinetics]]).<ref name=Laidler291/><ref name=Atkins/>

===Further chemical examples===
* The reaction 2 H<sub>2</sub> + O<sub>2</sub> → 2 H<sub>2</sub>O 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.<ref>Laidler K.J., ''Chemical Kinetics'' (3rd ed., Harper & Row 1987) p. 323-8 {{ISBN|0-06-043862-2}}</ref>
** H• + O<sub>2</sub> → •OH + •O•
** •O• + H<sub>2</sub> → •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]]'' [[enzyme|enzymatic]] [[DNA replication|replication]] using a [[DNA polymerase]].

===Acetaldehyde pyrolysis and rate equation===

The [[pyrolysis]] (thermal decomposition) of [[acetaldehyde]], CH<sub>3</sub>CHO (g) → CH<sub>4</sub> (g) + CO (g), proceeds via the Rice-Herzfeld mechanism:<ref name=LM>{{cite book |last1=Laidler |first1=Keith J. |last2=Meiser |first2=John H. |title=Physical Chemistry |date=1982 |publisher=Benjamin/Cummings |isbn=0-8053-5682-7 |page=417}}</ref><ref>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Julio |title=Atkins' Physical Chemistry |date=2006 |publisher=W. H. Freeman |isbn=0-7167-8759-8 |pages=830–1 |edition=8th}}</ref>

*Initiation (formation of [[Radical (chemistry)|free radicals]]):

: CH<sub>3</sub>CHO (g) → •CH<sub>3</sub> (g) + •CHO (g)	              k<sub>1</sub>

The methyl and CHO groups are [[Radical (chemistry)|free radicals]].

*Propagation (two steps):

: •CH<sub>3</sub> (g) + CH<sub>3</sub>CHO (g) →  CH<sub>4</sub> (g) + •CH<sub>3</sub>CO (g)            	k<sub>2</sub>

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

: •CH<sub>3</sub>CO (g) → CO (g) + •CH<sub>3</sub> (g)		        k<sub>3</sub>

The product  •CH<sub>3</sub>CO (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 CH<sub>3</sub>CHO (g) → CH<sub>4</sub> (g) + CO (g), [[Catalysis|catalyzed]] by a methyl radical •CH<sub>3</sub>.

*Termination:

: •CH<sub>3</sub> (g) + •CH<sub>3</sub> (g) → C<sub>2</sub>H<sub>6</sub> (g)              	k<sub>4</sub>

	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]] (CH<sub>3</sub>COCH<sub>3</sub>) and [[Propionaldehyde|propanal]] (CH<sub>3</sub>CH<sub>2</sub>CHO).

Applying the [[Steady state (chemistry)|Steady State Approximation]] for the intermediate species CH<sub>3</sub>(g) and CH<sub>3</sub>CO(g), the rate law for the formation of methane and the order of reaction are found:<ref name=LM/><ref name=Atkins/>

The rate of formation of the product methane is

<math chem>(1)... \frac{d\ce{[CH4]}}{dt} = k_2\ce{[CH3]} \ce{[CH3CHO]}</math>

For the intermediates

<math chem>(2)... \frac{d\ce{[CH_3]}}{dt} = k_1 \ce{[CH3CHO]} - k_2 \ce{[CH3]} \ce{[CH3CHO]} + k_3 \ce{[CH3CO]} - 2 k_4 \ce{[CH3]}^2 = 0</math> and

<math chem>(3)... \frac{d\ce{[CH3CO]}}{dt} = k_2 \ce{[CH3]} \ce{[CH3CHO]} - k_3 \ce{[CH3CO]} = 0</math>

Adding (2) and (3), we obtain <math chem>k_1 \ce{[CH3CHO]} - 2 k_4 \ce{[CH3]}^2 = 0</math>

so that <math chem>(4)...\ce{[CH3]} = \frac{k_1}{2k_4}\ce{[CH3CHO]}^{1/2}</math>

Using (4) in (1) gives the rate law <math chem>(5) \frac{d\ce{[CH4]}}{dt} = \frac{k_1}{2k_4} k_2 \ce{[CH3CHO]}^{3/2}</math>, which is order 3/2 in the reactant CH<sub>3</sub>CHO.