Rate equation

Rate equations describe the concentration curves of different species over time, for example with coupled chemical reactions or with excitation and de-excitation processes of different levels in atoms or molecules . Therefore they describe the reaction kinetics .

The rate of change in the concentration of a species is the sum of the rates of change in the concentrations which are caused by different reactions: ${\displaystyle c_{i}}$ ${\displaystyle i}$

${\displaystyle {\frac {\mathrm {d} c_{i}}{\mathrm {d} t}}=\sum _{j=1}^{N_{\mathrm {R} }}{\frac {\mathrm {d} c_{i,r}}{\mathrm {d} t}}=\sum _{j=1}^{N_{\mathrm {R} }}\nu _{i,j}r_{j},}$,

where is the species under consideration, its concentration and an index that runs over all reactions that occur (i.e. also over the back and forth reaction). is the reaction rate of the reaction . The reaction rate is proportional to the product of the reactant activities of the reaction , with the rate constant of the reaction as the proportionality constant: ${\displaystyle i}$ ${\displaystyle c_{i}}$ ${\displaystyle j}$ ${\displaystyle r_{j}}$ ${\displaystyle j}$ ${\displaystyle r_{j}}$ ${\displaystyle a_{j,l}^{|\nu _{j,l}|}}$ ${\displaystyle j}$ ${\displaystyle k_{j}}$

${\displaystyle r_{j}=k_{j}\prod _{l}a_{j,l}^{|\nu _{j,l}|}}$.

This gives the rate equation as:

${\displaystyle {\frac {\mathrm {d} c_{i}}{\mathrm {d} t}}=\sum _{j=1}^{N_{\mathrm {R} }}\nu _{ij}k_{j}\prod _{k=1}^{N_{j}}a_{k}^{b_{kj}},}$

in which

• ${\displaystyle a_{i}}$the activity of the species - the concentration is often used for simplicity -, ${\displaystyle i\,}$ ${\displaystyle c_{i}}$
• ${\displaystyle \nu _{ij}}$the stoichiometric coefficients of the species in the reaction , ${\displaystyle i}$ ${\displaystyle j}$
• ${\displaystyle b_{ij}}$the amounts of the stoichiometric coefficients, if the rate equation is set up with activities, or the partial reaction orders of the species in the reaction (generally not equal to the stoichiometric coefficients), if the rate equation is set up with concentrations, ${\displaystyle i}$ ${\displaystyle j}$
• ${\displaystyle k_{j}}$the rate coefficients (generally rate constants ),
• ${\displaystyle N_{\mathrm {R} }}$ the number of reactions and
• ${\displaystyle N_{j}}$is the number of substances involved in the reaction . ${\displaystyle j}$

The rate equations are generally a system of coupled, rigid , nonlinear first-order differential equations for which component conservation must apply. In the stationary case the law of mass action results . Rate equations can be represented compactly using the stoichiometric matrix . ${\displaystyle (\mathrm {d} c_{i}/\mathrm {d} t=0)}$

Derivation

The rate equations can be derived for all species involved by setting up the continuity equation with source or sink term (or a balance equation ) for the particle concentrations :

${\displaystyle {\frac {\partial c_{i}}{\partial t}}+\operatorname {div} {\vec {j}}_{i}=f_{i}(\{a_{i}\})}$,

where is the source term which depends on the activities . Again, these activities are generally nontrivially dependent on all concentrations. ${\displaystyle f_{i}(\{a_{i}\})}$ ${\displaystyle a_{i}(\{c_{i}\})}$

Since an equilibrium reaction always has a forward reaction and a reverse reaction, the forward reaction rate and the reverse reaction rate exist . The source term is given by the sum of all reactions: ${\displaystyle k_{\text{+,r}}}$ ${\displaystyle k_{\text{-,r}}}$

${\displaystyle f_{i}(\{a_{i}\})=\sum _{r}|\nu _{i,r}|\left[-k_{\text{-,r}}\prod _{{\text{Reaktanden j}},r}a_{r,j}^{|\nu _{r,j}|}+k_{\text{+,r}}\prod _{{\text{Produkte j}},r}a_{r,j}^{|\nu _{r,j}|}\right].}$

Note that the partial reaction order (the exponent with which the concentrations enter) corresponds to the magnitude of the stoichiometric coefficients only when activities are used. If concentrations in the source term are also used instead of activities and if particle interactions are present, the amounts of the stoichiometric coefficients must be replaced with the partial reaction orders. The partial reaction order can assume any value (e.g. 0) and is determined experimentally.

Different cases

• In equilibrium there are no particle flows ( ) and the particle concentrations no longer change over time. Therefore, in equilibrium: ${\displaystyle {\vec {j}}_{i}={\vec {0}}\,\forall i}$

${\displaystyle 0=f_{i}(\{a_{i}\}).}$

Assuming that every reaction (as a pair of back and forth reactions) is individually balanced, the law of mass action is obtained for every reaction by transformation:

${\displaystyle K_{r}={\frac {k_{\text{-,r}}}{k_{\text{+,r}}}}={\frac {\prod _{{\text{Produkte j}},r}a_{r,j}^{|\nu _{r,j}|}}{\prod _{{\text{Reaktanden j}},r}a_{r,j}^{|\nu _{r,j}|}}}}$

• If the system is not in equilibrium but is homogeneous, no particle flows occur, but the concentrations change over time until equilibrium is reached:

${\displaystyle {\frac {\partial c_{i}}{\partial t}}=f_{i}(\{a_{i}\})}$

• In the event that an inhomogeneous system in non-equilibrium is considered, the particle flow is and can be determined by Fick's first law ${\displaystyle {\vec {j}}_{i}\neq 0}$

${\displaystyle {\vec {j}}_{i}=-{\vec {\nabla }}(D_{i}c_{i}({\vec {r}}))+{\vec {j}}_{i}^{\text{excess}}}$

(whereby the non-ideal excess term occurs only for non-ideal systems). A reaction diffusion equation is then obtained .

• In the event that there is additional flow in the system, convection in the particle flow must be taken into account and the convection-diffusion equation is obtained .

Rate coefficients

→ Main article : rate constant

The reaction occurring in the rate equations advise coefficient can generally regarded as arbitrary functions of the respective, possibly time-dependent temperature (see also Plasma Physics : Thermal equilibrium) are considered. In general, rate coefficients for chemical processes of the heavy particles must be taken from the literature ( 'rate constant' of a chemical reaction ), the rate coefficients for the electron impact-induced processes can be obtained with the help of electron kinetics .

The Boltzmann equation for the electron energy distribution forms the basis for the kinetic treatment of the electrons, both for calculating such rate coefficients and for electronic transport processes ( electrical conductivity ) .

example

Hydrogen oxidation

Hydrogen oxidation is used for clarification:

${\displaystyle \displaystyle \mathrm {2H_{2}+O_{2}\longrightarrow 2H_{2}O} }$

(Rate coefficient ) ${\displaystyle \displaystyle k_{1}}$

a part dissociates

${\displaystyle \displaystyle \mathrm {2H_{2}O\longrightarrow H_{3}O^{+}+OH^{-}} }$

(Rate coefficient ) ${\displaystyle \displaystyle k_{2}}$

The rate equations (Eq. 1) for the five species are:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{2}]} =-2k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[O_{2}]} =-k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{2}O]} =+2k_{1}\mathrm {[H_{2}]^{2}[O_{2}]} -2k_{2}\mathrm {[H_{2}O]^{2}} }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[H_{3}O^{+}]} =+k_{2}\mathrm {[H_{2}O]^{2}} }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathrm {[OH^{-}]} =+k_{2}\mathrm {[H_{2}O]^{2}} }$

The concentrations of the species:

${\displaystyle c_{1}\equiv \mathrm {[H_{2}]} ,\;\;c_{2}\equiv \mathrm {[O_{2}]} ,\;\;c_{3}\equiv \mathrm {[H_{2}O]} ,\;\;c_{4}\equiv \mathrm {[H_{3}O^{+}]} ,\;\;c_{5}\equiv \mathrm {[OH^{-}]} }$

Numerical solution methods

Since the rate equations are a system of rigid differential equations, one is forced to choose a method with the largest possible stability area so that the integration steps do not become too small. A-stable processes are most favorable .

For the rate equations, 'stiff' means that the time constants of the various species differ greatly: Compared to others, some concentrations change only very slowly. Two examples of absolutely rigid-stable integration methods are the Implicit Trapezoidal Method and the Implicit Euler Method , and some BDF methods (backward differentiation formula) are also suitable.

Block preservation

The principle of block preservation provides a way of checking the quality of the numerical solutions, because the following applies at all times:

${\displaystyle \sum _{i=1}^{N_{\mathrm {Sp} }}c_{i}\beta _{ik}=\gamma _{k}={\text{const.}},\quad \forall k=1,\dots ,N_{\mathrm {B} }}$

in which

${\displaystyle N_{\mathrm {B} }\,}$    Minimum number of modules

${\displaystyle N_{\mathrm {Sp} }\,}$   Number of species involved in the reactions.

Derivation

A species i , written here as , is composed of the following building blocks : ${\displaystyle A_{i}}$ ${\displaystyle B_{k}}$

${\displaystyle A_{i}=\sum _{k=1}^{N_{\mathrm {B} }}\beta _{ik}B_{k}\quad \Rightarrow \quad \sum _{i=1}^{N_{\mathrm {Sp} }}\nu _{ij}\beta _{ik}={\underline {\underline {0}}}}$  .

Inserted into the rate equation (Eq. 1) and added up over all species, yields the above-mentioned building block maintenance . ${\displaystyle \sum _{i=1}^{N_{\mathrm {Sp} }}{\dot {c}}_{i}\beta _{ik}=0}$

Example for the matrix β _ik

${\displaystyle {\begin{pmatrix}\mathrm {H_{2}} \\\mathrm {O_{2}} \\\mathrm {H_{2}O} \\\mathrm {H_{3}O^{+}} \\\mathrm {OH^{-}} \end{pmatrix}}=\underbrace {\begin{pmatrix}2&0&0\\0&2&2\\2&1&1\\3&1&0\\1&1&2\end{pmatrix}} _{\beta _{ik}}\cdot {\begin{pmatrix}\mathrm {H} \\\mathrm {O^{+}} \\e^{-}\end{pmatrix}}}$

literature

• W. Frie: Calculation of the gas compositions and the material functions of ${\displaystyle SF_{6}}$ . In: Zeitschrift für Physik , 201, 269, 1967; Springer-Verlag, Berlin / Heidelberg / New York
• C. Schwab: Contributions to the kinetic modeling of partially ionized non-equilibrium plasmas . Dissertation at the Faculty of Physics at the Eberhard Karls University of Tübingen, 1989
• HR Schwarz: Numerical Mathematics . BG Teubner, Stuttgart 1986, ISBN 3-519-02960-X
• G. Wedler: Textbook of Physical Chemistry , Wiley-VCH, 2004, ISBN 3-527-31066-5
• DA McQuarrie, JD Simon, J. Choi: Physical Chemistry: A Molecular Approach . University Science Books, 1997, ISBN 0-935702-99-7