Chemical Kinetics

kinetics.py module

So far, it only contains the KORD class

Kinetic Order of Reaction Determination, KORD

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:

\[\begin{split}\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}\end{split}\]

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:

\[\begin{split}a(t)=\begin{cases} a_{0}-\alpha kt & t\leq\frac{a_{0}}{\alpha k}\\ 0 & t>\frac{a_{0}}{\alpha k} \end{cases}\end{split}\]

  • 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:

\[\begin{split}b(t)=\begin{cases} \beta kt & t\leq\frac{b_{\infty}}{\beta k}\\ 0 & t>\frac{b_{\infty}}{\beta k} \end{cases}\end{split}\]

  • 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:

\[\begin{split}\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}\end{split}\]

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:

\[\begin{split}\hat{G}^{[0]}(t)=\begin{cases} G_{0}+\beta\dfrac{kt}{b_{\infty}}\left(G_{\infty}-G_{0}\right) & t\leq\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k}\\ G_{\infty} & t>\dfrac{a_{0}}{\alpha k}=\dfrac{b_{\infty}}{\beta k} \end{cases}\end{split}\]