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kinetics_equilibrium.tex
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kinetics_equilibrium.tex
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\paragraph{With kinetics.}
A steady state is defined by
%
\begin{equation}
\forall\:s,\quad \doverdt{\conc[s]} = 0
\label{equilibrium:def}
\end{equation}
%
So basically it means that for each species $S$:
\begin{equation}
\sum_{\text{reactions}}\scoef[r,S] \rate[r] = 0
\end{equation}
%
\paragraph{With thermodynamics.}
A thermodynamic phase is at steady state
for a minimized Gibbs energy (at \Temp, \Press\ constant), with
the relation
\begin{equation}
\dd\Gibbs = \Vol\dd\Press - \Entr\dd\Temp + \sum_s\chempot[s]\dd\Mol[s]
\end{equation}
and we have Euler's identity
\begin{equation}
\Gibbs = \sum_s \Mol[s]\chempot[s]
\label{Euler_id}
\end{equation}
with also,
\begin{equation}
\left(\doverdext[r]{\Gibbs[s]}\right)_{\Temp,\Press} = \scoef[s,r]\chempot[s]
\end{equation}
We note
\begin{equation}
\DGibbs_r = \sum_s \scoef[s,r]\chempot[s] \left[= \sum_s \left(\doverdext[r]{\Gibbs[s]}\right)_{\Temp,\Press}\right]
\end{equation}
Considering
\begin{equation}
\chempot_s = \doverdn[s]{\Gibbs[s]}
\end{equation}
We have, deduced from~\ref{Euler_id}
\begin{equation}
\chempot[s] = \gibbs[s]
\end{equation}
Thus,
\begin{equation}
\chempot[s] = \chempotZ[s] + \Rg\Temp\ln\left(\frac{\press[s]}{\pz}\right)
\left[ = \chempotZ[s] + \Rg\Temp\ln\left(\frac{\Press}{\pz}\right) + \Rg\Temp\ln\left(\molarfrac[s]\right) \right]
\end{equation}
The story behind chemical extent is:
\begin{equation}
\Mol[s] = \Mol[s](t=0) + \sum_r \scoef[s,r] \ext[r]
\end{equation}
Using the ideal gas equation:
\begin{equation}
\Press = \conc \Rg \Temp
\end{equation}
thus
\begin{equation}
\begin{split}
\chempot[s] & = \chempotZ[s] + \Rg\Temp\ln\left(\frac{\Rg \Temp \molar[s]}{\pz}\right) \\
& = \chempotZ[s] + \Rg\Temp\ln\left(\frac{\Rg \Temp}{\pz}\left(\molar[s](t=0) + \sum_r \scoef[s,r]\frac{\ext[r]}{\Vol}\right)\right) \\
\end{split}
\end{equation}
and therefore
\begin{equation}
\begin{split}
\doverdext[r]{\chempot[s]} & = \frac{\pz}{\Vol}\frac{\scoef[s,r]}{\molar[s](t=0) + \sum_{r'} \scoef[s,r']\frac{\ext[r']}{\Vol}}\\
\ddoverddext{\chempot[s]}{r}{l} & = -\frac{\pz}{\Vol^2}\frac{\scoef[s,l]\scoef[s,r]}{\left(\molar[s](t=0) + \sum_{r'} \scoef[s,r']\frac{\ext[r']}{\Vol}\right)^2}\\
\end{split}
\end{equation}
Usually, it's better to consider the system per unit of volume,
using an extent of reaction per volume, thus the equations
become:
\begin{equation}
\begin{split}
\chempot[s] & = \chempotZ[s] + \Rg\Temp\ln\left(\frac{\Rg \Temp}{\pz}\left(\molar[s](t=0) + \sum_r \scoef[s,r]\ext[r]\right)\right) \\
\doverdext[r]{\chempot[s]} & = \pz\frac{\scoef[s,r]}{\molar[s](t=0) + \sum_{r'} \scoef[s,r']\ext[r']}\\
\ddoverddext{\chempot[s]}{r}{l} & = -\pz\frac{\scoef[s,l]\scoef[s,r]}{\left(\molar[s](t=0) + \sum_{r'} \scoef[s,r']\ext[r']\right)^2}\\
\end{split}
\end{equation}
Steady state is defined by
\begin{equation}
\forall\; r,\; \DGibbs_r = 0
\end{equation}
or
\begin{equation}
\min \Gibbs(\{\chempot[s]\}_s)
\end{equation}
\subsubsection{\texorpdfstring{$\forall\;r,\;\DGibbs_r = 0$}{Reaction enthalpy}}
\begin{equation}
\begin{split}
\DGibbs_r & = \sum_s \scoef[s,r] \chempot[s]\\
\doverdext[i]{\DGibbs_r} & = \sum_s\scoef[s,r]\doverdext[i]{\chempot[s]} \\
\ddoverddext{\DGibbs_r}{i}{j} & = \sum_s\scoef[s,r]\ddoverddext{\chempot[s]}{i}{j} \\
\end{split}
\end{equation}
For $R$ reactions:
\begin{equation}
\left[\begin{array}{cccc}
\sum_s\scoef[s,1]\doverdext[1]{\chempot[s]} & \sum_s\scoef[s,1]\doverdext[2]{\chempot[s]} & \cdots & \sum_s\scoef[s,1]\doverdext[R]{\chempot[s]}\\
\sum_s\scoef[s,2]\doverdext[1]{\chempot[s]} & \sum_s\scoef[s,2]\doverdext[2]{\chempot[s]} & \cdots & \sum_s\scoef[s,2]\doverdext[R]{\chempot[s]}\\
\vdots &\vdots &\vdots &\vdots \\
\sum_s\scoef[s,R]\doverdext[1]{\chempot[s]} & \sum_s\scoef[s,R]\doverdext[2]{\chempot[s]} & \cdots & \sum_s\scoef[s,R]\doverdext[R]{\chempot[s]}\\
\end{array}\right]
\times
\left[\begin{array}{c}
\Delta\ext[1] \\
\Delta\ext[2] \\
\vdots \\
\Delta\ext[R]
\end{array}\right]
=
\left[\begin{array}{c}
\sum_s \scoef[s,1] \chempot[s] \\
\sum_s \scoef[s,2] \chempot[s] \\
\vdots \\
\sum_s \scoef[s,R] \chempot[s]
\end{array}\right]
\end{equation}
\subsubsection{\texorpdfstring{$\min\Gibbs(\{\chempot[s]\}_s)$}{Phase enthalpy minimization}}
\begin{equation}
\begin{split}
\Gibbs & = \sum_s\Mol[s]\chempot[s]
= \sum_s\left(\Mol[s]^0 + \sum_r\scoef[s,r]\ext[r]\right)\chempot[s] \\
\doverdext[r]{\Gibbs} & = \sum_s \left[\left(\Mol[s]^0 + \sum_{r'}\scoef[s,r']\ext[r']\right)\doverdext[r]{\chempot[s]}
+ \scoef[s,r]\chempot[s]\right] \\
\ddoverddext{\Gibbs}{r}{l} & = \sum_s \left[\left(\Mol[s]^0 + \sum_{r'}\scoef[s,r']\ext[r']\right)\ddoverddext{\chempot[s]}{r}{l}
+ \scoef[s,l]\doverdext[r]{\chempot[s]}
+ \scoef[s,r]\doverdext[l]{\chempot[s]}\right] \\
\end{split}
\end{equation}
Considering these equations per unit of volume, one obtains:
\begin{equation}
\begin{split}
\Gibbs & = \sum_s\left(\molar[s]^0 + \sum_r\scoef[s,r]\ext[r]\right)\chempot[s] \\
\doverdext[r]{\Gibbs} & = \sum_s \left[\left(\molar[s]^0 + \sum_{r'}\scoef[s,r']\ext[r']\right)\doverdext[r]{\chempot[s]}
+ \scoef[s,r]\chempot[s]\right] \\
\ddoverddext{\Gibbs}{r}{l} & = \sum_s \left[\left(\molar[s]^0 + \sum_{r'}\scoef[s,r']\ext[r']\right)\ddoverddext{\chempot[s]}{r}{l}
+ \scoef[s,l]\doverdext[r]{\chempot[s]}
+ \scoef[s,r]\doverdext[l]{\chempot[s]}\right] \\
\end{split}
\end{equation}