demonstrations.tex

\section{\texorpdfstring{$\frac{1}{\Mm[\text{mix}]} = \sum_s \frac{\massfrac[s]}{\Mm[s]}$}{Mixture molar mass}}
\label{demo-Mm}
\[
\begin{split}
\Mm[\text{mix}] & = \frac{\mass[\text{mix}]}{\conc_\text{mix}}
                  = \frac{\mass[\text{mix}]}{\sum_s \conc_s}
                  = \frac{\mass[\text{mix}]}{\sum_s \frac{\mass[s]}{\Mm[s]}}\\
\Rightarrow
\frac{1}{\Mm[\text{mix}]}
               & = \frac{\sum_s\frac{\mass[s]}{\Mm[s]}}{\mass[\text{mix}]}
                 = \sum_s\frac{\frac{\mass[s]}{\Mm[s]}}{\mass[\text{mix}]}
                 = \sum_s\frac{\frac{\mass[s]}{\mass[\text{mix}]}}{\Mm[s]}
                 = \sum_s\frac{\massfrac[s]}{\Mm[s]}
\end{split}
\]

\section{Steady state--inverse rate constant}

Let's consider the reaction:
\begin{chemicalEquation}
\ce{{\scoefabs[A]} A + {\scoefabs[B]} B <=>[\fwdratecons(T,p)][\bkwdratecons(T,p)] {\scoefabs[C]} C + {\scoefabs[D]} D}
\label{eq:equat}
\end{chemicalEquation}


We consider this reaction being composed of elemental steps
(\ce{{\scoefabs[A]} A + {\scoefabs[B]} B -> {\scoefabs[C]} C + {\scoefabs[D]} D} and
 \ce{{\scoefabs[C]} C + {\scoefabs[D]} D -> {\scoefabs[A]} A + {\scoefabs[B]} B}
are elementary processes). 

\subsection{Kinetics}
\label{demo-eq_kin}

By definition: $\frac{\dd \conc}{\dd t} = 0$ for any species concerned by the steady state.

Thus, for any species \ce{S}: 
$\frac{1}{\scoef[S]}\frac{\dd\conc[S]}{\dd t} =  0$, which can be rewritten as
\begin{equation}
\fwdratecons \prodReac = \bkwdratecons \prodProd
\label{eq-conccond}
\end{equation}
The equilibrium constant\footnote{which is representative of a \emph{steady state}, not
an equilibrium. I know, life is hard, quit whining and get ovet it.} is, by definition, the
ratio between the concentrations of the products and the concentrations of the reactants,
which is
\begin{equation}
\Eqconst 
         = \frac{\displaystyle\prodProd}
                {\displaystyle\prodReac}
         = \prod_{s}\conc[S]^{\scoef[S]} 
\label{eq:Kdef}
\end{equation}
We deduce, from \ref{eq-conccond} and \ref{eq:Kdef}
\begin{equation}
\Eqconst = \frac{\fwdratecons}{\bkwdratecons}
\end{equation}

\subsection{Thermodynamics}
\label{demo-eq_therm}

Starting from the definition \eqref{eq:Kdef}, knowing that
for ideal gas we have the relation
\begin{equation}
\press = \conc\Rg \Temp \Rightarrow \conc = \frac{\press}{\Rg \Temp}
\end{equation}
with \Rg\ the ideal gas constant and \conc\ the concentration, we
deduce:
\begin{equation}
\begin{split}
\Eqconst 
  & = \prod_s \left(\frac{\press[s]^{(\eq)}}{\Rg \Temp}\right)^{\scoef[s]} \\
  & = \prod_s \left(\frac{\press[s]^{(\eq)}}{\pz}\frac{\pz}{\Rg \Temp}\right)^{\scoef[s]} \\
  & = \left(\frac{\pz}{\Rg \Temp}\right)^{\sum_s\scoef[s]} \prod_s \left(\frac{\press[s]^{(\eq)}}{\pz}\right)^{\scoef[s]}
\label{eq:K}
\end{split}
\end{equation}
Also, for ideal gas:
\begin{equation}
\gibbs(\Temp) = \gibbsZ(\Temp) + \Rg \Temp \ln\left(\frac{\press}{\pz}\right)
\end{equation}
wich gives
\begin{equation}
\frac{\press}{\pz} = \exp\left(\frac{\gibbs(\Temp) - \gibbsZ(\Temp)}{\Rg \Temp}\right)
\end{equation}
replacing in \ref{eq:K}
\begin{equation}
\begin{split}
\Eqconst
  & = \left(\frac{\pz}{\Rg \Temp}\right)^{\sum_s\scoef[s]}
           \prod_s \exp\left(\scoef[s] \frac{\gibbs[r]^{(\eq)}(\Temp) - \gibbsZ(\Temp)}{\Rg \Temp}\right) \\
  & = \left(\frac{\pz}{\Rg \Temp}\right)^{\sum_s\scoef[s]} 
           \exp\left(\frac{\DGibbs[r]^{(eq)}(\Temp)}{\Rg \Temp} - \frac{\DGibbsZ[r](\Temp)}{\Rg \Temp}\right) \\
\end{split}
\end{equation}
yet, by definition, at steady state
\begin{equation}
\DGibbs[r]^{(eq)}(\Temp) = 0
\end{equation}
we obtain, noting $\gamma = \sum_s \scoef[s]$,
\begin{equation}
\Eqconst = \left(\frac{\pz}{\Rg \Temp}\right)^\gamma \exp\left(-\frac{\DGibbsZ[r](\Temp)}{\Rg \Temp}\right)
\end{equation}