# Petersen matrix

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The '''Petersen matrix''' is a comprehensive description of systems of [biochemical reactions](/source/biochemistry) used to model [reactors](/source/chemical_reactor) for [pollution control](/source/Biodegradability_prediction) (engineered [decomposition](/source/decomposition)) as well as in [environmental systems](/source/environmental_systems). It has as many columns as the number of relevant involved components ([chemicals](/source/chemicals), [pollutants](/source/pollutants), [biomass](/source/biomass)es, [gases](/source/gases)) and as many rows as the number of involved [processes](/source/Chemical_process) (biochemical reactions and physical degradation). One further column is added to host the description of the [kinetics](/source/kinetics_(chemistry)) of each transformation ([rate equation](/source/rate_equation)).<ref name=Russell>{{cite book|last=Russell|first=David L.|title=Practical wastewater treatment|year=2006|publisher=Wiley|location=Hoboken, NJ|isbn=978-0-471-78044-1|pages=288|url=http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471780448.html}}</ref><ref name=Fang>{{cite book|last=Fang|first=editor, Herbert H.P.|title=Environmental anaerobic technology : applications and new developments|year=2010|publisher=Imperial College Press|location=London|isbn=9781848165427}}</ref>

== Matrix structure ==
The mass conservation principle for each process is expressed in the rows of the matrix. If all components are included (none omitted) then the mass conservation principle states that, for each process:

: <math>
\text{for all process } i:\sum_{j=1}^n a_{ij} \dot{\rho_j} = 0 \;,
</math>

where <math>\dot{\rho_j}</math> is the density rate of each component. This can also be seen as the process [stoichiometric relation](/source/Stoichiometry).

Moreover, the rate of variation of each component for all processes simultaneous effect can be easily assessed by summing the columns:

: <math>
\text{for all component } j: \frac{\partial C_j}{\partial t} = \sum_{i=1}^m a_{ij} r_i \; ,
</math>

where <math>r_i</math> are the reaction rates of each process.

== Example ==
A system of a third [order reaction](/source/Order_of_reaction) followed by a [Michaelis–Menten](/source/Michaelis%E2%80%93Menten) enzyme reaction.

:<chem>
{A} + 2B -> S
</chem>
:<chem>
{E} + S <=>[k_f][k_r] ES ->[k_\mathrm{cat}] {E} + P
</chem>

where the reagents A and B combine forming the substrate S (S = AB<sub>2</sub>), which with the help of enzyme E is transformed into the product P.
Production rates for each substance is:

:<math chem>\begin{align}
\frac{d [\ce A]}{d t} &= -k_1[\ce A][\ce B]^2
\\[6pt]
\frac{d [\ce B]}{d t} &= -2 k_1[\ce A][\ce B]^2
\\[6pt]
\frac{d [\ce S]}{d t} &= k_1[\ce A][\ce B]^2 - k_f[\ce E][\ce S] + k_r[\ce{ES}]
\\[6pt]
\frac{d [\ce E]}{d t} &= - k_f[\ce E][\ce S] + k_r[\ce{ES}] + k_\ce{cat}[\ce ES]
\\[6pt]
\frac{d [\ce ES]}{d t} &= k_f[\ce E][\ce S] - k_r[\ce{ES}] - k_\ce{cat}[\ce{ES}]
\\[6pt]
\frac{d [\ce P]}{d t} &= k_\ce{cat}[\ce ES]
\end{align}</math>

Therefore, the Petersen matrix reads as
{| class="wikitable"
|-
! {{Diagonal split header|Process | Components<br/>(kmol/m³)}} !! A !! B !! S !! E !! ES !! P !! Reaction rate  
|-
| P1: 2nd order formation of S from A and B || −1 || −2 || +1 || 0 || 0 || 0 || <math chem> k_1[\ce A][\ce B]^2</math>
|-
| P2: Formation of ES from E and S|| 0 || 0 || −1 || −1 || +1 || 0 || <math chem> k_f[\ce E][\ce S]</math>
|-
| P3: Back decomposition of ES into E and S|| 0 || 0 || +1 || +1 || −1 || 0 || <math chem> k_r[\ce{ES}]</math>
|-
| P4: Forward decomposition of ES into E and P || 0 || 0 || 0 || +1 || −1 || +1 || <math chem> k_\ce{cat}[\ce{ES}]</math>
|}

The Petersen matrix can be used to write the system's rate equation

:<math chem>
  \begin{pmatrix} 
     \frac{d}{d t}[\ce A] \\
     \frac{d}{d t}[\ce B] \\
     \frac{d}{d t}[\ce S] \\
     \frac{d}{d t}[\ce E] \\
     \frac{d}{d t}[\ce{ES}] \\
     \frac{d}{d t}[\ce P]
  \end{pmatrix}
  =
  \begin{bmatrix}
    -1 & 0 & 0 & 0  \\
    -2 & 0 & 0 & 0  \\
    +1 & -1 & +1 & 0  \\
    0 & -1 & +1 & +1  \\
    0 & +1 & -1 & -1  \\
    0 & 0 & 0 & +1
  \end{bmatrix}
  \begin{pmatrix} 
     k_1[\ce A][\ce B]^2 \\
     k_f[\ce E][\ce S] \\
     k_r[\ce{ES}] \\
     k_\ce{cat}[\ce{ES}]
  \end{pmatrix}
</math>

== References ==
{{Reflist}}

Category:Biodegradation
Category:Biodegradable waste management
Category:Chemical processes

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Adapted from the Wikipedia article [Petersen matrix](https://en.wikipedia.org/wiki/Petersen_matrix) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Petersen_matrix?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.