Simple Reaction

• Problem Description

  • Reactants \(A\) and \(B\) react to give product \(C\), and \(C\) is eliminated to \(E\)

  • Reaction rate constant is \(k\) and elimination rate constant is \(k_e\)

\[A \: + \: B \: {\stackrel{k}{\rightarrow}} \: C \: {\stackrel{k_e}{\rightarrow}} \: E\]

• Model Descriptions

  • The differential equations model is given by the following nonlinear system

\[\begin{array}{rcl} {d A \over d t} & = & -k A B \\ {d B \over d t} & = & -k A B \\ {d C \over d t} & = & k A B \: - \: k_e C \\ {d E \over d t} & = & k_e C \end{array}\]

• • A nonlinear response function is built to model the creation of product \((C)\) within a single system, where the inputs to the system are: initial concentrations \(C_0 =\) 0 mM, \(k\) = 0.75, \(k_e\) = 1.0, and three different initial \(A\),\(B\) concentrations \(A_0\) = \(B_0\) = 10, 20, and 50 mM.

• Results

  • Resulting \(C(t)\) curves are generated for the three \(A_0,B_0\) concentrations as well as one predicted \(C\) curve using \(A_0\) = \(B_0\) = 30 mM.

  • The \(C(t)\) curves are compared to “data,” where “data” represents the solution to the nonlinear system of differential equations

  • All of the resulting plots are shown here. Model curves are labeled M1, M2, M3, and prediction curve is labeled P1.

• Conclusions

  • The nonlinear response function approach accurately models simple reaction kinetics, which represent solutions to a nonlinear system of differential equations.

  • The nonlinear response function model requires no additional fitting of parameters beyond those which are contained in the response function.

  • The nonlinear response function model is also able to indicate peak concentrations that are not contained in the data used to build the model.

  • The nonlinear response function model represents a more universal formulation for use in modeling simple reactions.