Question

For the sequential reaction \(\mathrm{A} \stackrel{k_{A}}{\rightarrow} \mathrm{B} \stackrel{k_{B}}{\rightarrow} \mathrm{C}$$k_{A}=1.00 \times 10^{-3} \mathrm{s}^{-1} .\) Using a computer spreadsheet program such as Excel, plot the concentration of each species for cases where \(k_{B}=10 k_{A}, k_{B}=1.5 k_{A},\) and \(k_{B}=0.1 k_{A} .\) Assume that only the reactant is present when the reaction is initiated.

Short answer

To create the plots for each case, use the Excel spreadsheet and input the following formulas for each case: Case 1: \(k_b = 10 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\] Case 2: \(k_b = 1.5 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\] Case 3: \(k_b = 0.1 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\] Set up a column for time (t), and the concentration of A, B, and C in each case. Use Excel's scatter plot function to create three different plots for the three cases with the x-axis representing time (t) and the y-axis representing concentration, using different colors or markers to differentiate between A, B, and C in each plot.

Step-by-step solution

Defining the reactions

Reactant A is converted to intermediate B with a rate constant \(k_A = 1.00 \times 10^{-3}\mathrm{s}^{-1}\), and intermediate B is converted to product C with a rate constant \(k_B\). The reaction scheme is given as: \[A\stackrel{k_{A}}{\rightarrow}B\stackrel{k_{B}}{\rightarrow}C\]

Calculating the concentration of species for the three different cases

Since the reaction only starts with the reactant A, we can assume the initial concentration of B and C is zero. We will use numerical methods through the Excel spreadsheet to calculate the concentration of A, B, and C at different time points. For each case, we need to set up a column in Excel for time (t), and the concentration of A, B, and C. We can use the following formulas for the three cases, where \(t\) is the time interval: Case 1: \(k_b = 10 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\] Case 2: \(k_b = 1.5 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\] Case 3: \(k_b = 0.1 k_a\) \[A(t) = A_0 e^{-k_A t}\] \[B(t) = \frac{k_A A_0}{k_B-k_A}(e^{-k_A t}-e^{-k_B t})\] \[C(t) = A_0 - A(t) - B(t)\]

Creating a spreadsheet with the formulas and plotting the concentrations

Now that we have formulas for the concentration of each species at different time intervals, input these equations into an Excel spreadsheet to obtain the concentrations for each time interval. Each case will have a separate table with columns for time, and concentrations of A, B, and C. Use the scatter plot function in Excel to create three plots for the three cases, with the x-axis representing time (t) and the y-axis representing concentration, using different colors or markers to differentiate between A, B, and C in each plot.

Key concepts

Numerical Methods in Chemistry

Numerical methods in chemistry are essential tools for solving complex equations related to chemical reactions that cannot be easily solved analytically. In the context of the exercise, we use these methods to determine the concentrations of chemical species involved in a sequential reaction over time. When the behavior of reactions under different conditions is modeled, such as various reaction rate constants, finding solutions requires calculations at numerous time points. These problems are often solved using iterative methods to approximate the changes in concentrations as reactions proceed. Here, numerical methods help simulate the sequential reactions: when reactant A converts to intermediate B, and subsequently, B converts to product C. By calculating these conversions at small time intervals and adjusting the concentrations step-by-step, chemists can follow the progression and kinetics of the reaction in a more detailed manner. Without such methods, understanding these dynamical systems would be more challenging, especially when dealing with multiple time-dependent components.

Spreadsheet Modeling

Spreadsheet modeling allows chemists to simulate and visualize chemical reactions effectively. Programs like Excel are conducive to creating models because they handle calculations and visualize data efficiently through graphs and charts. The exercise involves using Excel to plot the concentration of each species (A, B, and C) over time. By setting up a spreadsheet with columns for different time points and concentrations, we can apply the formulas derived from the numerical methods. This setup allows you to calculate the concentrations of each species at specified intervals easily by dragging down the formula across cells.

Visualizing Data

Using Excel's plotting features, such as scatter plots, you can effectively visualize the changes in concentrations for each scenario. Here, the scatter plots show how concentrations of A, B, and C evolve as the reaction progresses. Differentiating data with colors or markers for various scenarios (variations in rate constant, \(k_B\)), helps to discern patterns and predict reaction behavior under different conditions.

Chemical Concentration Calculations

Chemical concentration calculations are vital for understanding and predicting the outcomes of reactions. For the sequential reaction in the exercise, we ensured accurate concentration values over time by applying the correct mathematical equations and adjusting for different rate constants (\(k_B\)).Using the initial concentration of reactant A and assuming that intermediates B and product C start at zero, we calculate the concentration at various points in time using the provided equations:- For A: \( A(t) = A_0 e^{-k_A t} \)- For B: the concentration is calculated based on the rate constants,- For C: it is the remaining concentration from both A and B based on mass balance.These calculations help determine the speed and extent to which reactants convert into intermediates and products, critical for designing and controlling chemical processes. The concentration equations applied in the exercise also illustrate how varying the rate constant \(k_B\) affects the formation of intermediates and final products, showcasing the delicate balance and interdependence of reaction rates.