edited paper

README.md

@@ -42,4 +42,4 @@ $$[C]_{tot} = [C] \+ [AB_2C] \label{Eq. 5}$$
 
 We can define a system comprising these equations to be simultaneously solved.
 In this system, we can see that chemical equilibria consist of nonlinear functions, meaning that they cannot be expressed as a sum of their variables each raised to the power of one.
-'equpy' solves this problem by linearizing these equations to make them suitable to be solved employing linear algebra and an iterative numerical method.
+`equpy` solves this problem by linearizing these equations to make them suitable to be solved employing linear algebra and an iterative numerical method equivalent to the Newton search of the logarithmic equations over the logarithm of the variables.

paper.md

@@ -80,14 +80,11 @@ And the associated mass conservations:
 \end{equation}
 
 We can define a system comprising these equations to be simultaneously solved.
-In this system, we can see that chemical equilibria consist of nonlinear functions, meaning that they cannot be expressed as a sum of their variables each raised to the power of one. A common approach to solve this problem is to linearize these equations first to make them suitable to be solved employing numerical methods.
-
-Common solutions in literature revolve around the so-called kinetic approach and thermodynamic approach, which involves finding the extent of reaction at the energy minimum of the system.
-
-The algorithm presented here the advantage of operating on a user-friendly set of equations that is intuitively employed by any user with a basic chemistry knowledge. These equations are solved with an approach equivalent to the Newton search of the logarithmic equations over the logarithm of the variables.
+In this system, we can see that chemical equilibria consist of nonlinear functions, meaning that they cannot be expressed as a sum of their variables each raised to the power of one. equpy approach to solve this problem is to linearize these equations first to make them suitable to be solved employing fast numerical methods.
+The algorithm presented here has the advantage of operating on a user-friendly set of equations that is intuitively employed by any user with a basic chemistry knowledge. These equations are solved with an approach equivalent to the Newton search of the logarithmic equations over the logarithm of the variables.
 
 # 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:
+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:6}
 a_1[X_1] + a_2[X_2] + ... + a_n[X_n] = [X_i]_{tot}
 \end{equation}
@@ -97,6 +94,7 @@ Or equivalently:
 \sum_{j=1}^n a_j[X_j] = [X_i]_{tot}
 \end{equation}
 
+With *a* for a given species that does not take part of a mass conservation relationship being equal to zero.
 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 [@Passy:1972] as applied by Baker [@Baker:1980]. We reorganize \autoref{eq:7} so that the summation over all strictly positive terms *a* and *X* is rewritten as the following:
 
 \begin{equation}\label{eq:8}