kinetics_generalities.tex

A chemical reaction is ``a bunch of molecules
turning into another bunch of molecules'', kinetics\footnote{%
from the greek $\kappa\iota\nu\eta\sigma\iota\varsigma$, ``kinesis'', movement, to move}
is about answering the question ``how fast?''.
Thus chemical kinetics is the mathematical model
to calculate the rate at which the molecules disappear and
appear.

\subsubsection{Going forward}

\Antioch's kinetics is based on the elementary step
hypothesis. It means that, as far as the kinetics is
concerned, every reaction is an elementary step:
the reactants get together and produce the products
immediatly. Mathematically, it means
the partial orders are the absolute
value of the stoichiometric coefficients (see next).

Let's consider a chemical reaction:
\begin{chemicalEquation}
\ce{\scoefabs[A] A + \scoefabs[B] B ->[\rcons] \scoefabs[C] C + \scoefabs[D] D}
\label{genericX}
\end{chemicalEquation}
with \rcons\ the rate constant.
We want to model the evolution of the system, that is we want to
characterize 
$\doverdt{\conc[A]}$,
$\doverdt{\conc[B]}$,
$\doverdt{\conc[C]}$,
$\doverdt{\conc[D]}$.
Using the kinetics theory, we have:
\begin{equation}
\frac{1}{\scoef[A]}\doverdt{[A]} = 
\frac{1}{\scoef[B]}\doverdt{[B]} = 
\frac{1}{\scoef[C]}\doverdt{[C]} = 
\frac{1}{\scoef[D]}\doverdt{[D]} = 
\rcons\conc[A]^{\scoefabs[A]}\conc[B]^{\scoefabs[B]}
\end{equation}
with \scoef[A]\ being the stoichiometric coefficient, which is defined by:\\[5pt]
$\left\{\begin{array}{ll}
\scoef[S] = \scoefabs[S] & \text{if \ce{S} is a product} \\
\scoef[S] = -\scoefabs[S] & \text{if \ce{S} is a reactant} \\
\end{array}\right.$\\[5pt]
So the game is to define the rate constant. 

A rate constant is characterized by two things:
\begin{itemize}
\item the kinetics model,
\item the chemical process.
\end{itemize}
The kinetics model will mathematically describe the rate constant's dependence with
the temperature, it is noted \kinMod\ in this manual, the chemical process will
possibly add a pressure dependency, it is noted \chemProc, with \conc[M]\
denoting the pressure dependence.
\Antioch\ propose six different kinetics models and five chemical processes.
The rate constant is characterized thus, for a choice of a chemical process and
a kinetics model:
\begin{equation}
\rateCons = \chemProc
\end{equation}

\subsubsection{Going backward}

Usually, a reaction will be reversible, which means, if we consider
that reaction~\ref{genericX} is reversible, we should note it:
\begin{chemicalEquation}
\ce{\scoefabs[A] A + \scoefabs[B] B <=>[\fwdratecons][\bkwdratecons] \scoefabs[C] C + \scoefabs[D] D}
\label{genericXrev}
\end{chemicalEquation}
with \fwdratecons\ the forward rate constant and \bkwdratecons\ the backward rate constant.
In a given physico-chemical environment, this reaction will eventually reach
steady state, characterized by an equilibrium constant \Eqconst. 
This equilibrium constant is given by
\begin{equation}
\Eqconst[r] = \frac{\fwdratecons[r]}{\bkwdratecons[r]}
\label{therm:K_kin}
\end{equation}
for a reaction $r$.
It is possible to estimate it from thermodynamics considerations, 
using the relation
\begin{equation}
\Eqconst[r] = \left(\frac{\pz}{\Rg \Temp}\right)^\gamma \exp\left(-\frac{\DGibbsZ[r](\Temp)}{\Rg \Temp}\right)
\label{therm:K_therm}
\end{equation}
The demonstrations are given in appendix~\ref{demo-eq_kin} and \ref{demo-eq_therm}. 
Thus the backward rate constant is therefore known given:
\begin{itemize}
\item the forward rate constant;
\item the thermodynamics of the molecules.
\end{itemize}

\subsubsection{Going nowhere: steady state, a.k.a equilibrium}
\label{phys:equilibrium}
\input{kinetics_equilibrium}