Simulation of first-order kinetics

Chemical reaction kinetics: Introduction

Simulation of first-order kinetics

In case of first-order kinetics of the reaction

$\text{A}\longrightarrow \text{B}$ the differential equation that describes the time course of the concentration of substance A can be solved for various parameter values (initial concentrations and reaction rate constant $k$ ) and graphs of solutions can be computed and plotted.

In the following simulation, the time course of the chemical reaction is simulated, and the graphs of the concentrations $[\text{A}]$ and $[\text{B}]$ of substance A and B, respectively, are plotted. You can vary the initial concentrations and rate constants, and see what the effect of this variation. So you can find out for example that a smaller rate constant only means that the conversion goes slower.

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$k$

$C_A$

$C_B$

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We can also determine the time course of the concentration of the produced substance B in the chemical reaction $\text{A}\longrightarrow \text{B}$ , with reaction rate constant $k$ . Let the concentration of substance A and B at time $t$ be denoted as $x(t)$ and $y(t)$ , respectively. In first-order kinetics we have:

$\frac{\dd x}{\dd t}=-k\, x$ This is a differential equation of an exponential decay with the solution

$x(t)=x_0 e^{-kt}\quad\text{with}\quad x_0=x(0)$ Let $y(0)=y_0$ and $x_0+y_0=a$ , for some constant $a$ , then we have at any time

$x(t)+y(t)=a$ and thus also

$x'(t)+y'(t)=0$ It follows that

$y'(t)=-x'(t)=k\, x(t)=k\bigl(a-y(t)\bigr)$ and

$y(t)=a - x_0 e^{-kt}\text.$ The differential equation

$y'(t)=k\bigl(a-y(t)\bigr)$ describes limited exponential growth and its general solution is

$y(t)=a - c\, e^{-kt}$ for some constant $c$ .