kinetics

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 \(G_\mathrm{THEO}\) 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 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:

\[v = -\frac{1}{\alpha} \frac{dA}{dt} = k A^{n}\]

  • Order 0:

\[A(t)=A_{0}-\alpha kt\]

  • 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:

\[B(t)=\beta kt\]

  • Order 1:

\[B(t) = B_{\infty} \{1 - \exp(-\alpha kt)\}\]

  • Order 2:

\[B(t)=B_{\infty}\left(\frac{1}{1+\frac{\alpha}{A_{0}\alpha^{2}kt}}\right)\]

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:

\[G_{\mathrm{THEO}}(t)=G_{0}+\frac{\alpha kt}{A_{0}}(G_{\infty}-G_{0})\]

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 \(t_{end} = \frac{A_{0}}{\alpha k}\). These values are mathematical artifacts and should be ignored.


Standard Formula for Order 1:

\[G_\mathrm{THEO}(t) = G_{\infty} + \exp(-\alpha kt)(G_{0} - G_{\infty})\]

Standard Formula for Order 2:

\[G_{\mathrm{THEO}}(t)=G_{\infty}-\frac{1}{1+A_{0}\alpha kt}(G_{\infty}-G_{0})\]