Physical Chemistry

CHEM 441/442

College of Charleston

Chemical Kinetics Studies

Studying Chemical Kinetics

without the Student

Actually Solving the

Differential Equations

Paper 1021

Presented at the 219th ACS National Meeting,

San Francisco, CA 30 March 2000

Abstract

Solutions of the differential equations describing the kinetics of simple reaction mechanisms are presented in several undergraduate courses. However, discussions of more complex mechanisms are often limited to physical chemistry courses where students should have the mathematical maturity required to understand these more complicated systems. Often, physical chemistry students lack the skills to solve or to locate the general solutions of these more complicated differential equations.

An interactive study of both simple and complex reaction mechanisms by the student is possible using graphically based modeling software such as STELLA which is independent of solving the differential equations. The reaction mechanism is easily modeled using a visual interface and the rate constant-concentration changes are entered in appropriate dialog boxes. The program generates the respective differential equations and produces a real-time solution expressed in graphical or other formats. Student assignments using the free, downloadable run-
time version of the software include generating models of simple mechanisms and analyzing models representing more complex mechanisms.

Software Introduction

STELLA (High Performance Systems, Inc., https://www.hps-inc.com) is a dynamic modeling program in which relational diagrams are created using a graphical interface to which numerical values and mathematical functions may be assigned. A free run-time version of the software (excluding the "Save" and "Help" capabilities) is available for downloading at the HPS website.

Once the program is opened, there are three different work areas known as "layers" that are available for use. To change from one layer to another, click on the "up" or "down" navigation arrowhead icons at the far left of the screen. The functionality of each layer includes:

1) The upper "Mapping" layer is used for placing text; showing output of calculations in various forms; imputing data for calculations using dials, sliders, etc.; and showing photos and QuickTime movies. This level is usually used by those who want to simply run the model once the model has been highly developed and by those not concerned with either the modeling itself or the mathematics involved.

2) The center "Modeling" construction layer is used to construct the model using the various "Building Blocks", "Tools", and "Objects" available. Our models will be constructed and run from this layer.

3) The lower "Equation" layer lists a summary of the initial values of the various "Stocks", the equations representing the time dependence of the "Stocks", and the data within the "Converters". We will use this layer to confirm the accuracy of our model construction by comparing the entries to the known differential equations describing the chemical processes being considered.

The construction of a model is done in the "Modeling" layer. Click the "Globe" icon at the far left of the screen to change it to the "x2" icon which is used for modeling entry. The icons at the top of this layer fall into several groups. The ones that we will use include:

1) Building Blocks

A "Stock" represents a value undergoing a change. We will use a "Stock" to represent the number of molecules of reactant or product present during a chemical reaction at a given time. Normally, we will set the initial number of reactant molecules at 1000 and the initial number of product molecules at 0.

A "Flow" represents a change with respect to time of a "Stock". During the process a "Stock" attached at the left decreases as time passes and one at the right increases. For out study of chemical kinetics, a "Stock" at the left represents a chemical reactant, a "Stock" at the right represents a chemical product, and the "Flow" represents the chemical reaction arrow.

A "Converter" contains constants and transformation equations. We will use "Converters" to enter values of the rate constants for the chemical reactions being studied. Normally, we set the values of the rate constants in the range of 0 to 100.

A "Connector" links the various "Building Blocks" within the model.

2) Tools

The "Hand" is the general purpose editing tool used for selecting, moving, opening, and editing various objects in the work area.

The "Dynamite" stick is the delete tool. Not that STELLA does not have an "undo" feature.

3) Objects

This icon creates a "Graph Pad". Use the "Hand" to double click in the "Graph Pad" to open a dialog box for setting up the graphing parameters. An icon for printing the "Graph Pad" is available at the bottom.

This icon creates a "Table Pad". Use the "Hand" to double click in the "Table Pad" to open a dialog box for setting up the table parameters. The output is very similar to that from a spreadsheet. An icon for printing the "Table Pad" is available at the top.

4) Run Controller

Opens a dialog box for running the model and obtaining results.

Listed below are some simple directions on using the 'Building Blocks" described above:

1) Drawing a "Stock".

A) Click the "Stock" icon.

B) Click in the work area at the desired location. A "Noname Stock" appears.

C) The "Stock" can be named by moving the cursor over the name and typing.

D) Double clicking the "Stock" opens the dialog box for data entry.

2) Drawing a "Flow" between "Stocks".

A) Click the "Flow" icon.

B) Move the cursor to the center of the "Outflow Stock".

C) Click and drag to center of the "Inflow Stock".

D) Release. A "Noname Flow" appears.

E) The "Flow" can be named by moving the cursor over the name and typing.

F) Double clicking the "Flow" opens the dialog box for data entry.

3) Replacing a "Cloud" on a "Flow" with a "Stock" already in the work area.

A) Click the "Hand" icon.

B) Click and move the "Stock" to the "Cloud" until the "Cloud" is highlighted.

C) Release.

4) Drawing a "Controller".

A) Click the "Controller" icon.

B) Click in the work area at the desired location. A "Noname Controller" appears.

C) The "Controller" can be named by moving the cursor over the name and typing.

D) Double clicking the "Controller" opens the dialog box for data entry.

5) Drawing a "Connector".

A) Click the "Connector" icon.

B) Place the cursor inside the "from Building Block".

C) Click and drag to the "target Building Block" until the "target" is highlighted.

D) Release.

E) The locations of attachments can be adjusted, if desired.

During the execution of a completed model, any differential equations describing the time dependence of the "Stock" y are solved using one of three numerical methods based on a Taylor series

where ym represents the value of the "Stock" at time t = tm and represents the value of the "Stock" at time . Unless changed, Euler's method is the default method used by STELLA

which is based on the first two terms of the series. STELLA also offers "Runge-Kutta 2"

which is based on the first three terms of the series and "Runge-Kutta 4"

which is based on the first five terms of the series. Normally, we will use the "RK4" method because this method produces the most accurate results for our type of calculations.

Some of the default settings in STELLA are a little unusual. The following changes are recommended for the following study of chemical kinetics:

1) Click on "File".

2) Click on "Default Settings".

3) Click on "Model" and change the "Variable Name Font Size" to "12 pt".

4) Click on "Time Specs" and change "To" to "10" and "DT" to "0.1". Also change "Integration Method" to "Runge-Kutta 4".

5) Click on "OK".

Chemical Kinetics Studies

In this series of exercises we will consider several different mechanisms involving first order reactions. If you have difficulties in preparing the model or would simply like to see the final model, the appropriate file can be downloaded.

Simple First Order Reaction

Consider the reaction

which is first order in A. The differential rate equations are

and the integrated rate equations are

The STELLA model (simple.stm) is

and is created using the following steps:

1) Create "Stock" A in the work area.

A) Place the "I" text cursor on the "Stock" name and replace "Noname" by "A".

B) Double click the "Stock" and enter "1000" for "Initial (A)".

C) Click "OK".

2) Create "Stock" B in the work area. Name it "B" and enter "0" for "Initial(B)".

3) Draw the "Flow" connecting A to B and name it "Chemical Reaction". Do not worry about the question mark that appears on the "Flow" symbol–it will be eliminated later.

4) Place the "Controller" in the work area.

A) Name it "k1".

B) Double click the "Controller" and enter "1" for "k1 = ".

C) Click "OK".

5) Connect "Stock" A to the "Flow" using a "Connector".

6) Connect "Controller" k1 to the "Flow" using a "Connector".

7) Enter the kinetics of the reaction.

A) Double click the "Flow".

B) Click on the icon for k1.

C) Type "*" for multiplication.

D) Click on the icon for A.

E) The "Chemical_Reaction =" entry should read "k1*A".

F) Click "OK".

8) Create a "Graph Pad" in the work area.

A) Double click the "Graph Pad".

B) To add A to the plot, click on the A entry at the left and the ">>" icon.

C) To set the scale for A, click on the A entry at the right and the double headed arrow to the far right. Enter "0" and "1000" for the "Min" and "Max" entries for "Scale" and click "Set".

D) Likewise add B with the same scaling.

E) Click "OK".

F) Move the "Graph Pad" to a convenient location in the work area by clicking and dragging by the title bar.

G) Click the "Stick Pin" icon at the upper left of the "Graph Pad" to return to the modeling layer.

Change to the "Equation" layer by clicking the "down" navigation arrowhead at the far left of the screen. Check the entries for the "Stocks" to assure that they agree with the differential rate equations. For example, the "Outflow" for A is given by "k1*A" which agrees with the differential equation (dA/dt) = -k1 A and that the value of A at a given time is the old value less the "Outflow" times dt.

Return to the "Modeling" layer by clicking the "up" navigation arrowhead and run the model by

1) Click the "Run Controller" icon at the bottom left of the screen.

2) Click the "Run" (forward arrow) icon.

3) Plots of A and B will be generated on the "Graph Pad".

4) Print the graph and record appropriate notes.

Answer the following questions:

1) What is the sum of A and B at any given time?

2) At what approximate time does A = B (50 % completion)?

3) How would the graph change if k1 = 2? You can check your answer by double clicking the k1 "Controller", entering the value of "2", and rerunning the model.

Parallel First Order Reactions Producing a Common Product

Consider the reactions in which C is formed as the product of two independent first-order chemical reactions

Such reactions are common in radioactive decay series when two different branches terminate with a common product. The differential rate equations are

and the integrated rate equations are

Using the techniques described above, construct the model (parsprod.stm) shown at the right. Note that the direction of a "Flow" can be changed by hitting the "Shift" key while dragging. Set k1 = k2 = 1 and the initial values of A = 1000, B = 1000, and C = 0. Check the entries in the "Equation" layer to assure that they agree with the differential rate equations. Note that the upper limit on the "Graph Pad" will need to be 2000.

Run the model and answer the following questions:

1) What is the sum of A, B, and C at any given time?

2) How will the graph change if k1 = 2? You can check this answer.

3) How will the graph change if k1 = 2 and the initial value of A is 1000 and k2 = 1 and the initial value of B is 500? You can check this answer.

Parallel First Order Reactions Involving the Same Reactant

Consider the reactions in which A reacts to form two different products

Such reactions are common in radioactive decay series when a parent can undergo decay by two different processes. The differential rate equations are

and the integrated rate equations are

Construct the model (parsreac.stm) as shown. Set k1 = k2 = 1 and the initial values of A = 1000 and B = C = 0. Check the entries in the "Equation" layer to assure that they agree

with the differential rate equations.

Run the model and answer the following questions:

1) What is the sum of A, B, and C at any given time?

2) How will the graph change if k1 = 2? You can check this answer.

3) After reaction completion (at long time), is the ratio of C/B the same as k2/k1?

Reversible First Order Reactions

Consider the reversible reaction system

The differential rate equations are

and the integrated rate equations are

Construct the model (reverse.stm) as shown. Set k1 = k2 = 1 and the initial values of A = 1000 and B = 0. Check the entries in the "Equation" layer to assure that they agree with the differential rate equations.

Run the model and answer the following questions:

1) What is the sum of A and B at any given time?

2) How will the graph change if k1 = 2 and k2 = 1? You can check this answer.

3) Once equilibrium is reached, is the ratio of B/A the same as k1/k2?

Consecutive First Order Reactions

Consider the series of two first-order reactions

The differential rate equations are

and the integrated rate equations are

Construct the model (consec.stm) as shown. Set k1 = k2 = 1 and the initial values of A = 1000 and B = C = 0. Check the entries in the "Equation" layer to assure that they agree with the differential rate equations.

Run the model and answer the following questions:

1) What is the sum of A, B, and C at any given time?

2) How will the graph change if k1 > k2 such as k1 = 5 and k2 = 1? You can check this answer. Under these conditions, the series of reactions are known to be in "disequilibrium".

3) How will the graph change if k1 < k2 such as k1 = 1 and k2 = 5? You can check this answer. Under these conditions, the series of reactions are known to be in "transient equilibrium".

4) How will the graph change if k1 << k2 such as k1 = 0.5 and k2 = 10? You can check this answer. Under these conditions, the series of reactions are known to be in "secular equilibrium".

Consider the series of first order reactions

Write the differential rate equations for dA/dt, dB/dt, dC/dt, dD/dt, and dE/dt. Construct the model (sevconsc.stm). Set the initial values of A =1000 and B = C = D = E = 0 and use k1 = k2 = k3 = k4 = 1. Check the entries in the "Equation" layer to assure that they agree with the differential rate equations. Run the program. What is the trend for the maximum of each of the peaks?

Reversible Consecutive First-Order Reactions

Consider the reversible reaction system

Write the differential rate equations for dA/dt, dB/dt, and dC/dt. Construct the model (revsonsc.stm) for this system of reactions. Set the initial value of A =1000 and B = C = 0 and use k1 = k2 = k3 = k4 = 1. Check the entries in the "Equation" layer to assure that they agree with the differential rate equations. Run the program.

This model can be used to illustrate the "steady state" and "equilibrium" approximations often used in stepwise (complex) mechanisms by setting k4 = 0. The steady state approximation assumes that the intermediate B is very reactive so that at any given time B is very small and dB/dt = 0. Under these conditions (k2 + k3) >> k1. Run the model with k2 = k3 = 10 and k1 = 0.1 to observe the results for the steady state approximation. The equilibrium approximation assumes that B/A is constant. Under these conditions (k2 + k3) >> k1 and k2 >> k3. Run the model with k2 =10, k3 = 1, and k1 = 0.5 to observe the results for the equilibrium approximation.

Placing Pertinent Equations on Mapping Layer

For each STELLA .stm file there is a .pct with the same name which contains the differential and integrated rate equations. These files can be placed on the "Mapping" layer using the following procedure:

1) Click on the "Graphics Form" icon and click on desired space in "Mapping" layer.

2) Resize the box to full screen by clicking and dragging the 'Resizing Handles".

3) Double click one of the "Resizing Handles" to open the dialog box.

4) Click "From File" and "Size to Form".

5) Click "Import Picture" and choose .pct file with corresponding name as the STELLA .stm file.

6) Click "OK".

7) Resize if necessary.

Acknowledgments

1999 Shodor Computational Science Institute

Shodor Education Foundation

Supported in part by the National Science Foundation DUE-9752815

and the National Computational Science Alliance (NSF-funded activity

through the Partnerships for Advanced Computational Infrastructure)

Department of Chemistry and Biochemistry

College of Charleston

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