Question

(Challenging) Cubic autocatalytic steps are important in a reaction mechanism referred to as the "brusselator" (named in honor of the research group in Brussels that initially discovered this mechanism): $$\begin{array}{l} \mathrm{A} \stackrel{k_{1}}{\longrightarrow} \mathrm{X} \\ 2 \mathrm{X}+\mathrm{Y} \stackrel{k_{2}}{\longrightarrow} 3 \mathrm{X} \\ \mathrm{B}+\mathrm{X} \stackrel{k_{3}}{\longrightarrow} \mathrm{Y}+\mathrm{C} \\\ \mathrm{X} \stackrel{k_{4}}{\longrightarrow} \mathrm{D} \end{array}$$ If \([\mathrm{A}]\) and \([\mathrm{B}]\) are held constant, this mechanism demonstrates interesting oscillatory behavior that we will explore in this problem. a. Identify the autocatalytic species in this mechanism. b. Write down the differential rate expressions for \([\mathrm{X}]\) and \([\mathrm{Y}]\) c. Using these differential rate expressions, employ Euler's method (Section 35.6 ) to calculate \([\mathrm{X}]\) and \([\mathrm{Y}]\) versus time under the conditions \(\mathrm{k}_{1}=1.2 \mathrm{s}^{-1}, \mathrm{k}_{2}=0.5 \mathrm{M}^{-2} \mathrm{s}^{-1}, \mathrm{k}_{3}=\) \(3.0 \mathrm{M}^{-1} \mathrm{s}^{-1}, \mathrm{k}_{4}=1.2 \mathrm{s}^{-1},\) and \([\mathrm{A}]_{0}=[\mathrm{B}]_{0}=1 \mathrm{M} .\) Begin with \([\mathrm{X}]_{0}=0.5 \mathrm{M}\) and \([\mathrm{Y}]_{0}=0.1 \mathrm{M} .\) A plot of \([\mathrm{Y}]\) versus \([\mathrm{X}]\) should look like the top panel in the following figure. d. Perform a second calculation identical to that in part (c), but with \([\mathrm{X}]_{0}=3.0 \mathrm{M}\) and \([\mathrm{Y}]_{0}=3.0 \mathrm{M} .\) A plot of \([\mathrm{Y}]\) versus \([\mathrm{X}]\) should look like the bottom panel in the following figure. e. Compare the left and bottom panels in the following figure. Notice that the starting conditions for the reaction are different (indicated by the black spot). What DO the figures indicate regarding the oscillatory state the system evolves to?

Short answer

The autocatalytic species in the mechanism is X. The differential rate expressions for [X] and [Y] are given by: \(\begin{cases} \frac{d[X]}{dt} = k_1[A] - k_3[X][B] - k_4[X] + k_2 [X]^2 [Y]\\ \frac{d[Y]}{dt} = - k_2 [X]^2 [Y] + k_3[X][B] \end{cases}\) Using Euler's method, we can calculate the concentrations of X and Y at each time step for given initial concentrations and rate constants. After comparing the plots of [Y] vs. [X] for different initial conditions, we find that the oscillatory states reached are similar, indicating that the oscillatory state is a stable state for the "brusselator" system to evolve to, regardless of its initial condition.

Step-by-step solution

a. Autocatalytic Species

The autocatalytic species is the species that catalyzes its own formation. From the second reaction, we can see that one X molecule goes in and three X molecules come out: \(2X + Y \stackrel{k_2}{\longrightarrow} 3X\). Therefore, the autocatalytic species is X. b. Write down the differential rate expressions for [X] and [Y].

b. Rate Expressions

As we know the reaction mechanism, we can write the rate expressions for the mentioned species. The rate expressions for X and Y are given by: \(\begin{cases} \frac{d[X]}{dt} = k_1[A] - k_3[X][B] - k_4[X] + k_2 [X]^2 [Y]\\ \frac{d[Y]}{dt} = - k_2 [X]^2 [Y] + k_3[X][B] \end{cases}\) c. Using these differential rate expressions, employ Euler's method to calculate [X] and [Y] versus time under the given conditions.

c1. Computing concentrations iteration function

To compute the concentrations at each time step using the Euler's method, we need the function: \([X]_{i+1} = [X]_i + \frac{d[X]}{dt} dt \\ [Y]_{i+1} = [Y]_i + \frac{d[Y]}{dt} dt \)

c2. Apply Euler's method for the initial concentrations

Now, we will use the given initial concentrations and rate constants and apply Euler's method to calculate the concentrations at each time step. Given information: - \(k_1 = 1.2 s^{-1}\) - \(k_2 = 0.5 M^{-2} s^{-1}\) - \(k_3 = 3.0 M^{-1} s^{-1}\) - \(k_4 = 1.2 s^{-1}\) - \([A]_0 = [B]_0 = 1M\) - \([X]_0 = 0.5 M,\ [Y]_0 = 0.1 M\) To apply Euler's method, we'll choose a small enough time step, e.g., \(dt = 0.01 s\), for a certain number of time steps, e.g., \(N = 1000\). Then, we use the iterative process mentioned in c1 for each time step in the range of 0 to N and plot the concentrations for each time step. d. Perform a second calculation identical to that in part c, but with [X]₀=3.0 M and [Y]₀=3.0 M.

d. Apply Euler's method for the new initial concentrations

Apply Euler's method again, using the new initial concentrations: - \([X]_0 = 3.0 M,\ [Y]_0 = 3.0 M\) e. Compare the left and bottom panels in the following figure. Notice that the starting conditions for the reaction are different. What do the figures indicate regarding the oscillatory state the system evolves to?

e. Oscillatory State Analysis

Comparing the plots from calculations in part c and d will show that, although the initial conditions were different, the oscillatory states reached in the two cases are similar. This indicates that the oscillatory state is a stable state for the "brusselator" system to evolve to, regardless of its initial condition.

Key concepts

Autocatalytic Reactions

The term autocatalytic reactions refers to a class of chemical reactions where one of the products also serves as a catalyst for the reaction itself, essentially catalyzing its own production. In the brusselator mechanism, the species X plays this unique role. After being produced from species A, species X promotes its further generation through a cubic autocatalytic step, where two X molecules react with a Y molecule to produce three X molecules. This self-propagating feature can lead to complex dynamics including oscillatory behaviors, where the concentration of the reactants and products can vary in a periodic manner over time. Understanding autocatalytic reactions is crucial in chemical kinetics as they often occur in biological systems and are integral to the operation of some technological processes.

Differential Rate Expressions

Moving into the realm of differential rate expressions, we delve into the mathematics that describe the change in concentration of reactants and products over time in a chemical reaction. These expressions are derived from the stoichiometry of the reaction mechanism and the known rate laws. In the brusselator example, the differential rate expressions for X and Y involve terms that include their creation and consumption rates. Such equations are fundamental to chemical kinetics, as they encapsulate the dynamic interplay of all species involved in the reaction mechanism. By solving these equations, we can predict how concentrations evolve, which is essential for controlling chemical reactions in industrial processes as well as in biological systems.

Euler's Method in Chemical Kinetics

Euler's method is a numerical technique used to solve ordinary differential equations, which often arise in chemical kinetics. It provides a way to approximate the solution of the differential rate expressions over small time intervals, termed 'time steps'. Using Euler's method involves iteratively calculating the concentration of each chemical species at discrete points in time by incrementally advancing the dependent variables using their rate of change. This method, while simple, can be a powerful tool in studying systems like the brusselator mechanism, enabling chemists to simulate how concentrations of substances change with time to better understand the kinetics of a reaction.

Oscillatory Behavior in Chemical Systems

Finally, oscillatory behavior in chemical systems is a fascinating phenomenon where concentrations of reactants and products undergo regular and sustained fluctuations over time. Such behavior is often indicative of non-equilibrium dynamics and complex feedback mechanisms. In the context of the brusselator mechanism, the oscillations emerge from the interplay between the autocatalytic reactions and the nonlinear nature of the differential rate expressions. Such systems can have significant implications in various fields, including biological rhythms (like heartbeats and circadian cycles), chemical engineering, and the study of reaction dynamics. Understanding the conditions under which oscillatory behavior occurs is crucial for the design and control of chemical reactors and biological systems.