derivations_equations.tex

What we want is 
$\mdot[S] = \doverdt{\mass[S]}$, and its derivatives:
$\doverdT{\mdot[S]}$, $\doverdm[j]{\mdot[S]}$ with respect to species $j$.
%
The kinetics model's value is noted \kinMod, the rate constant is noted \rateCons, the
rate of the reaction is noted \rate.
%
We use the equations
\begin{equation}
\mass[S] = \Mm[S] \conc[S]
\end{equation}
and
\begin{equation}
\dd\mass[S] = \Mm[S]\,\dd\conc[S]
\end{equation}
as everything in kinetics theory is expressed in concentrations and 
not in terms of mass.
We have thus:
\begin{equation}
\mdot[S] = \doverdt{\mass[S]} = \Mm[S] \doverdt{\conc[S]}
\end{equation}
\begin{equation}
\doverdm[E]{\mdot[S]} = \frac{\Mm[S]}{\Mm[E]}\doverdc[E]{\doverdt{\conc[S]}}
\label{derive:m_T}
\end{equation}
\begin{equation}
\doverdT{\mdot[S]} = \Mm[S]\doverdT{\doverdt{\conc[S]}}
\label{derive:m_c}
\end{equation}
but for Paul's sake, we will do the full differentiation with mass as well
as the differentiation with respect to concentration.

Kinetics theory gives us:
\begin{equation}
\begin{split}
\doverdt{\conc[S]} & = \sum_{\text{reactions}} \scoef[r,S] \rate[r] \\
                   & = \sum_{\text{reactions}} \scoef[r,S] \left[\fwdrate[r] - \bkwdrate[r]\right]\\
                   & = \sum_{\text{reactions}} \scoef[r,S]
                        \left[ \fwdratecons[r] \prodReac - \bkwdratecons[r] \prodProd \right]
\end{split}
\end{equation}

The game now is thus to differentiate the forward and backward components.
\subsection{Forward rate}

The forward rate \fwdrate[r] for a reaction $r$ is given by:
\begin{equation}
\fwdrate[r] = \fwdratecons[r] \prodReac
\label{ratefDef}
\end{equation}
with
\begin{equation}
\fwdratecons = \chemProc
\end{equation}
The differentiates are straightforward, just look at Tab.~\ref{antioch::kinMod}
and \ref{antioch::chemProc}

\subsubsection{To derive with respect to \Temp}
For any species \ce{S} participating in the reaction.
\begin{equation}
\begin{split}
\doverdT{\fwdrate} & = \doverdT{\fwdratecons} \prodReac \\
                   & = \doverdT{\left(\chemProc\right)} \prodReac \\
\end{split}
\end{equation}

\subsubsection{To derive with respect to \mass}
Knowing that $\conc = \frac{\mass}{\Mm}$, we substitute in~\ref{ratefDef}
\begin{equation}
\fwdrate[r] = \fwdratecons[r] \prod_\text{reactants}  \left(\frac{\mass[\reac]}{\Mm[\reac]}\right)^{\scoefabs[\reac]}
\end{equation}
The differenciation with respect to concentration is:
\begin{equation}
\doverdc[E]{\fwdrate[r]} = \left\{\begin{array}{ll}
                              \fwdrate[r]\left[ \frac{1}{\fwdratecons[r]} \doverdc[E]{\left(\chemProc[r]\right)} + \frac{\scoefabs[E]}{\conc[E]}\right] & \text{if \ce{E} is a reactant} \\[5pt]
                              \fwdrate[r]\left[ \frac{1}{\fwdratecons[r]} \doverdc[E]{\left(\chemProc[r]\right)}\right] & \text{else} \\
                              \end{array}\right.
\end{equation}
We obtain for a derivation with respect to \mass[E]\ of species \ce{E}:
\begin{equation}
\doverdm[E]{\fwdrate[r]} = \left\{\begin{array}{ll}
                             \fwdrate[r] \left[\doverdm[E]{\fwdratecons[r]} \frac{1}{\fwdratecons[r]} + \frac{\scoefabs[E]}{\mass[E]}\right] & \text{if \ce{E} is a reactant} \\[5pt]
                             \fwdrate[r] \left[\doverdm[E]{\fwdratecons[r]} \frac{1}{\fwdratecons[r]}\right] & \text{else} 
                           \end{array}\right.
\end{equation}


\subsection{Backward}

The backward rate is
\begin{equation}
\bkwdrate[r] = \bkwdratecons[r]\prodProd
\end{equation}
The backward rate constant is given by
\begin{equation}
\bkwdratecons[r] = \frac{\fwdratecons[r]}{\Eqconst}
\end{equation}

\subsubsection{Derive with respect to \Temp}
It makes:
\begin{equation}
\doverdT{\bkwdrate[r]} = \doverdT{\bkwdratecons[r]}\prodProd
\end{equation}
Decomposition gives
\begin{equation}
\doverdT{\bkwdratecons[r]} = \doverdT{\fwdratecons[r]}\Eqconst^{-1} - \doverdT{\Eqconst}\frac{\bkwdratecons[r]}{\Eqconst^{2}}
\label{rateb-decomp}
\end{equation}
For \Eqconst we have \eqref{therm:K_therm}
\begin{equation}
\begin{split}
\doverdT{\Eqconst} & = \doverdT{\left[\left(\frac{\pz}{\Rg \Temp}\right)^{\sum_s \scoef[s]} \exp\left(-\frac{\DGibbsZ(\Temp)}{\Rg \Temp}\right)\right]} \\
                   & = \Eqconst\left[-\frac{\sum_s\scoef[s]}{\Temp} - \doverdT{\left[\frac{\DGibbsZ(\Temp)}{\Rg \Temp}\right]}\right]
\end{split}
\end{equation}
The derived value of \DGibbsZ(\Temp) is easily given, 
see Eq.~\ref{therm:DH} and \ref{therm:DS}:
\begin{equation}
\begin{split}
\doverdT{\left[\frac{\DGibbsZ(\Temp)}{\Rg \Temp}\right]} 
        & = \doverdT{\left[\frac{\DenthZ(T)}{\Rg \Temp}\right]} - \doverdT{\left[\frac{\DentrZ(\Temp)}{\Rg}\right]} \\
        & = 2\tc{0}\Temp^{-3} + \tc{1}\Temp^{-2}\left(1+\ln(\RedTemp)\right) + \frac{\tc{3}}{2} + 
           \frac{2}{3}\tc{4}\Temp + \frac{3}{4}\tc{5}\Temp^{2} + \frac{4}{5}\tc{6}\Temp^3 \\ 
        &  \spacehack - \tc{8}\Temp^{-2} + \tc{0}\Temp^{-3} + \tc{1}\Temp^{-2} + \tc{2}\Temp^{-1} + \tc{3} + \tc{4}\Temp + \tc{5} \Temp^{2} + \tc{6} \Temp^{3}
\end{split}
\end{equation}
with \RedTemp\ the reduced temperature: $\RedTemp = \frac{\Temp}{\Tref}$, here $\Tref = 1~\unit{K}$.
Finally,
\begin{equation}
\doverdT{\left[\frac{\DGibbsZ(\Temp)}{\Rg \Temp}\right]} =
        3\tc{0}\Temp^{-3} + \tc{1} \Temp^{-2} \left(2 + \ln(\RedTemp)\right) + \frac{3}{2} \tc{3} + \frac{5}{3} \tc{4} \Temp
        + \frac{7}{4}\tc{5} \Temp^2 + \frac{9}{5}\tc{6} \Temp^3 - \tc{8} \Temp^{-2}
\end{equation}

\subsubsection{Derivation with respect to \mass}
Using~\ref{rateb-decomp}
\begin{equation}
\begin{split}
\doverdm[E]{\bkwdratecons[r]} & = \doverdm[E]{\fwdratecons[r]}\Eqconst^{-1} - \doverdm[E]{\Eqconst}\frac{\bkwdratecons[r]}{\Eqconst^{2}} \\
                              & = \doverdm[E]{\fwdratecons[r]}\Eqconst^{-1}
\end{split}
\label{rateb-decomp-m}
\end{equation}

\subsection{Net rate}

The net rate is
\begin{equation}
\rate[r] = \fwdrate[r] - \bkwdrate[r]
\end{equation}
thus,
\begin{equation}
\doverdT{\rate[r]} = \doverdT{\fwdrate[r]} - \doverdT{\bkwdrate[r]}
\end{equation}
and
\begin{equation}
\doverdm{\rate[r]} = \doverdm{\fwdrate[r]} - \doverdm{\bkwdrate[r]}
\end{equation}

\subsection{Back to \texorpdfstring{\mdot}{omega dot}}
\begin{equation}
\begin{split}
\mdot[S] &= \Mm[S] \doverdt{\ce{S}} \\
         &= \Mm[S] \sum_{\text{reactions}}\scoef[r,S]\rate[r]
\end{split}
\end{equation}

\begin{equation}
\doverdT{\mdot[S]} = \Mm[S] \sum_{\text{reactions}}\scoef[r,S]\doverdT{\rate[r]}
\end{equation}
and
\begin{equation}
\doverdm[E]{\mdot[S]} = \Mm[S] \sum_{\text{reactions}}\scoef[r,S]\doverdm[E]{\rate[r]}
\end{equation}