added biblio

paper.md

@@ -53,29 +53,25 @@ In the first section of this work, we define the general problem, in the second
 
 # Problem Definition
 Let’s start with a simple example of interacting species:
-\begin{equation}\label{eq:1}
-\frac{[AB_2]}{[A][B]^2}=K_1
-\end{equation}
-\begin{equation}\label{eq:2}
-\frac{[AB_2C]}{[AB_2][C]}=K_2
-\end{equation}
+$$A + 2B ↔ AB_2$$
+$$AB_2 + C ↔ AB_2C$$
 
 And their associated equilibrium constants, defined as the ratio between forward and backward reaction rates:
-\begin{equation}\label{eq:3}
-\frac{[AB_2]}{([A][B]^2)} = K_1
+\begin{equation}\label{eq:1}
+\frac{[AB_2]}{[A][B]^2} = K_1
 \end{equation}
-\begin{equation}\label{eq:4}
-\frac{[AB_2C]}{([AB_2][C])} = K_2
+\begin{equation}\label{eq:2}
+\frac{[AB_2C]}{[AB_2][C]} = K_2
 \end{equation}
 
 And the associated mass conservations:
-\begin{equation}\label{eq:5}
+\begin{equation}\label{eq:3}
 [A]_{tot} = [A] + [AB_2] + [AB_2C]
 \end{equation}
-\begin{equation}\label{eq:6}
+\begin{equation}\label{eq:4}
 [B]_{tot} = [B] + 2[AB_2] + 2[AB_2C]
 \end{equation}
-\begin{equation}\label{eq:7}
+\begin{equation}\label{eq:5}
 [C]_{tot} = [C] + [AB_2C]
 \end{equation}
 
@@ -88,21 +84,41 @@ The algorithm presented here the advantage of operating on a user-friendly set o
 
 # Mathematical Treatment
 In a system with n different species $X_{1…n}$, the mass conservation relationship for the $i^{-th}$ species can be stated as the sum over all the species contributions with their relative stoichiometries (a). We can define the conservation of mass for species $X_i$ as:
-\begin{equation}\label{eq:8}
+\begin{equation}\label{eq:6}
 a_1[X_1] + a_2[X_2] + ... + a_n[X_n] = [X_i]_{tot}
 \end{equation}
 
 Or equivalently:
-\begin{equation}\label{eq:9}
+\begin{equation}\label{eq:7}
 \sum_{j=1}^n a_j[X_j] = [X_i]_{tot}
 \end{equation}
 
 In order to express such conservation of mass as a linear function of the logarithm of concentrations of the reactants, following the approach by Wall we must first transform the summations to products using the theory of the arithmetic-geometric mean inequality from Passy 3 as applied by Baker 4. We reorganize Eq. 7 so that the summation over all strictly positive terms *a* and *X* is rewritten as the following:
 
-\begin{equation}\label{eq:9}
+\begin{equation}\label{eq:8}
 \frac{[X_i]_{tot}}{\sum_{j=1}^n a_j[X_j]} = 1
 \end{equation}
 
+Then we “condense” the sum in the denominator of Eq. 8 into a product:
+\begin{equation}\label{eq:9}
+\sum_{j=1}^n a_j[X_j] = \prod_{j=1}^n \bigg(\frac{a_j[X_j]}{W_j}\bigg)^{W_j}
+\end{equation}
+
+With *W* for a given species *j* part of a mass conservation relationship being equal to:
+\begin{equation}\label{eq:10}
+W_j = \frac{a_j[X_j]}{\sum_{p=1}^n a_p[X_p]}
+\end{equation}
+
+So that Eq. 8 becomes:
+\begin{equation}\label{eq:11}
+\frac{[X_i]_{tot}}{\bigg(\frac{a_1[X_1]}{W_1}\bigg)^{W_1} * \bigg(\frac{a_2[X_2]}{W_2}\bigg)^{W_2} * ... * \bigg(\frac{a_n[X_n]}{W_n}\bigg)^{W_n}} = 1
+\end{equation}
+
+We can then reorganize the fraction:
+\begin{equation}\label{eq:12}
+\left\{ [X_1]^{-1}\*[X_2]^{-2}\*...\*[X_n]^{-n} \right\}\*[X_i]_{tot}
+\end{equation}
+
 # Old Stuff
 
 Single dollars ($) are required for inline mathematics e.g. $f(x) = e^{\pi/x}$