# Chemical Kinetics **`kinetics.py` module** So far, it only contains the `KORD` class ## Kinetic Order of Reaction Determination, KORD * [1. Principle](#principle) * [2. Theoretical Model](#theoretical-model) --- ### 1. Principle #### 1.1 Experimental Measurements ($G_\mathrm{EXP}$) In chemical kinetics, we track the evolution of molar concentrations over time: $C_{i}(t)$. Experimentally, we measure a physical quantity proportional to these concentrations: $G_\mathrm{EXP}(t)$. * **Spectrophotometry**: $G_\mathrm{EXP} = A = \sum_{i} \epsilon_{i} \cdot l \cdot C_{i}$, where $A$ is the absorbance * **Polarimetry**: $G_\mathrm{EXP} = \alpha = \sum_{i} [\alpha]_{i} \cdot l \cdot C_{i}^{w}$ * **Conductivity**: $G_\mathrm{EXP} = \sigma = \sum_{i} \lambda_{i} \cdot C_{i}$ **General Form:** $G_\mathrm{EXP}(t) = \sum_{i} \eta_{i} \cdot C_{i}(t)$ #### 1.2 Theoretical Model ($G_\mathrm{THEO}$) The model has three different expressions (see Theoretical model section), depending on the order of the reaction: **Standard Formula for Order 0**: $$\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}$$ **Standard Formula for Order 1**: $$\hat{G}^{[1]}(t)=G_{\infty}+\exp\left(-\alpha kt\right)\left(G_{0}-G_{\infty}\right)$$ **Standard Formula for Order 2**: $$\hat{G}^{[2]}(t)= G_{0}+\dfrac{\left(G_{\infty}-G_{0}\right)}{1+\dfrac{\beta}{b_{\infty}\alpha^{2}kt}}=G_{0}+\frac{\left(G_{\infty}-G_{0}\right)b_{\infty}\alpha^{2}kt}{b_{\infty}\alpha^{2}kt+\beta}$$ $G_\mathrm{THEO}$ is defined by two types of values: **fixed parameters** (input by the user) and **adjustable variables** (optimized by the algorithm). **Fixed Parameters:** * **Reaction Order**: $n \in \{0, 1, 2\}$ (The user selects the order to test). * **Stoichiometry**: $\alpha$ and $\beta$ are known constants, provided by the user. $\alpha$ and $\beta$ must be the smallest possible positive integers * **Initial Concentration**: $a_{0}$ (Note: For Order 1, $G_{THEO}$ is independent of $a_{0}$). The concentration must also be provided by the user The final concentration of B, $b_{\infty}$ is related to $a_0$ by the relation: $$\frac{a_0}{\alpha} = \frac{b_{\infty}}{\beta}$$ **Adjustable Variables:** The model fits the experimental data by adjusting the following: * **Rate Constant**: $k$ * **Final Value**: $G_{\infty}$ * **Initial Value**: $G_{0}$ (While $G_{0}$ is measured, the algorithm also adjusts it to ensure the best fit starting point). The optimization is performed for a specific reaction order at a time to determine which model best describes the experimental data.
By default, KORD chooses the first and last $G_\mathrm{EXP}$ values as $G_{0}$ and $G_{\infty}$. And a default $k$ value is also setup by KORD. If you need to change that because of a convergence issue, ensure your starting values for $k$, $G_{0}$ and $G_{\infty}$ are realistic to help the algorithm converge.
#### 1.3 Optimization (RMSD) The algorithm minimizes the **Root-Mean-Square Deviation** to fit the theoretical curve to the experimental data: $$RMSD = \sqrt{\frac{1}{n} \sum_{k=1}^{n} \{G_\mathrm{EXP}(t_{k}) - G_\mathrm{THEO}(t_{k})\}^{2}}$$ #### 1.4 Input Data input is performed through a structured Excel file. Users simply provide the kinetic parameters ($\alpha$, $\beta$), the initial concentration $[A]_0$, and the experimental data series (time $t$ and property $G_{\mathrm{exp}}$). --- ### 2. Theoretical Model The reaction model used in KORD is designed to be as simple as possible based on the following criteria: - Single-component reaction: A **unique reactant** $A$ transforms into a **unique product** $B$ ($\alpha A \rightarrow \beta B$) - **Total reaction**: The reaction goes to completion (the extent of reaction is 100%) - **Closed system**: No exchange of matter occurs between the system and its environment; only energy exchanges are possible - **Homogeneous system**: The concentration of any compound $C_i$ is uniform throughout the entire system - **Isochoric system**: The volume of the system remains constant throughout the reaction. #### 2.1 Reactant Expression $a(t)$ The rate law is defined as: $$v = -\frac{1}{\alpha} \frac{d[A]}{dt} = k [A]^{n}$$ * **Order 0**: $$a(t)=\begin{cases} a_{0}-\alpha kt & t\leq\frac{a_{0}}{\alpha k}\\ 0 & t>\frac{a_{0}}{\alpha k} \end{cases}$$ * **Order 1**: $$a(t) = a_{0} \exp(-\alpha kt)$$ * **Order 2**: $$a(t) = \frac{1}{\frac{1}{a_{0}} + \alpha kt}$$ #### 2.2 Product Expression $B(t)$ Derived from mass balance ($M_{A} a(t) + M_{B} b(t) = M_{A} a_{0} = M_{B} b_{\infty}$): * **Order 0**: $$b(t)=\begin{cases} \beta kt & t\leq\frac{b_{\infty}}{\beta k}\\ 0 & t>\frac{b_{\infty}}{\beta k} \end{cases}$$ * **Order 1**: $$b(t) = b_{\infty} \{1 - \exp(-\alpha kt)\}$$ * **Order 2**: $$b(t)=\frac{b_{\infty}}{1+\frac{\beta}{b_{\infty}\alpha^{2}kt}}$$ #### 2.3 Global Expression $G_\mathrm{THEO}(t)$ The theoretical quantity is a linear combination of $a(t)$ and $b(t)$: $$G_\mathrm{THEO}(t) = \frac{G_{0}}{a_{0}} \cdot a(t) + \frac{G_{\infty}}{b_{\infty}} \cdot b(t)$$ **Standard Formula for Order 0**: $$\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}$$
Warning: This mathematical model for Order 0 is a linear equation. Unlike Order 1 or 2, this linear model does not naturally plateau. Depending on the values of $k$ and $t$, the model may predict non-physical values (e.g., negative absorbance or negative concentration) if the time $t$ exceeds the theoretical completion time $t_{\mathrm{end}} = \frac{a_{0}}{\alpha k}=\frac{b_{\infty}}{\beta k}$. These values are mathematical artifacts and should be ignored.

**Standard Formula for Order 1**: $$\hat{G}^{[1]}(t)=G_{\infty}+\exp\left(-\alpha kt\right)\left(G_{0}-G_{\infty}\right)$$ **Standard Formula for Order 2**: $$\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}$$