# Autocatalysis

Autocatalysis ( gr. Αυτοκατάλυση, aftokatálissi : "the self-disintegration") refers to a special type of catalytic chemical reaction in which a final product as a catalyst is effective for the reaction. The continuous formation of this catalyst accelerates the reaction as long as there are still enough starting materials available. It is a positive feedback .

## example

${\ displaystyle \ mathrm {5 \ HOOC-COOH + 2 \ MnO_ {4} ^ {-} + 6 \ H ^ {+} \ rightarrow 10 \ CO_ {2} +2 \ Mn ^ {2 +} + 8 \ H_ {2} O}}$

The resulting manganese (II) ions act as a catalyst for this reaction, so that the initially hesitant discoloration of the permanganate proceeds faster and faster. If manganese (II) ions are made available at the start of the reaction, the reaction proceeds quickly from the start.

## Time law

Sigmoid variation of the product concentration in autocatalytic reactions

Auto catalysis are second order reactions :

${\ displaystyle A + B \ rightarrow \; 2 \ B}$

${\ displaystyle \ v = - {\ frac {d [A]} {dt}} = k [A] [B]}$

This differential equation (DGL) is solved by deriving an expression for from the conservation of mass and expressing it as a variable : ${\ displaystyle [B (t)]}$${\ displaystyle [A (t)]}$${\ displaystyle x}$

${\ displaystyle [B (t)] = [A] _ {0} + [B] _ {0} - [A (t)] = [A] _ {0} + [B] _ {0} -x }$

This then gives the DGL:

${\ displaystyle {\ frac {dx} {dt}} = - kx ([A] _ {0} + [B] _ {0} -x)}$

Which solves as follows:

${\ displaystyle \ int {\ frac {dx} {([A] _ {0} + [B] _ {0}) xx ^ {2}}} = \ int -kdt}$

${\ displaystyle {\ frac {1} {[A] _ {0} + [B] _ {0}}} \ cdot ln \ left ({\ frac {[A] _ {0} [B (t)] } {[B] _ {0} [A (t)]}} \ right) = kt}$

${\ displaystyle [B (t)] = {\ frac {[A] _ {0} + [B] _ {0}} {1 + {\ frac {[A] _ {0}} {[B] _ {0}}} \ exp {\ left (- ([A] _ {0} + [B] _ {0}) kt \ right)}}}}$

The following results analogously for : ${\ displaystyle [A (t)]}$

${\ displaystyle [A (t)] = {\ frac {[A] _ {0} + [B] _ {0}} {1 + {\ frac {[B] _ {0}} {[A] _ {0}}} \ exp {\ left (([A] _ {0} + [B] _ {0}) kt \ right)}}}}$

The graph for this equation follows the sigmoid function typical for autocatalytic reactions : The chemical reaction begins slowly because initially only a low concentration of catalyst is present. The speed of the reaction increases steadily because the concentration of the catalyst increases as the reaction proceeds. Then the speed drops again because the concentrations of the starting materials decrease.

## Chemical oscillation

The presence of one or more autocatalytic steps can lead to chemical oscillation .