Unit 5. Empirical and Theoretical Rate Expressions

This website provides learning and teaching tools for a first course on kinetics and reaction engineering. The course is divided into four parts (I through IV). Here, in Part II of the course, the focus is on chemical reaction kinetics, and more specifically, on rate expressions, which are mathematical models of reaction rates. As you progress through Part II, you will learn how rate expressions are generated from experimental kinetics data.

This first section of Part II of the course focuses upon the selection of an equation to be tested as a rate expression. The equation to be tested can be chosen simply for its mathematical convenience. Alternatively, theory can be used to select the mathematical form of the equation to be tested. For some reactions, theory can be applied directly. In other cases the reaction must be described in terms of a group of reactions that comprise what is known as a reaction mechanism. In the latter case theory can be applied to the reactions in the mechanism which are then combined to get the mathematical form of the equation to be tested.

Unit 5 takes a closer look at rate expressions, starting with a common empirical form for a rate expression known as a power-law rate expression. It then defines an elementary reaction and presents two theories that allow the prediction of the mathematical form of the rate expression for an elementary reaction. Those theories are called the Collision Theory and the Transition State Theory.

Learning Resources

• Archive (.zip) - Contains all learning resources listed below for this unit
• Documents to Read:
  • Unit 05 Informational Reading (.pdf)
  • Example 5.1 (.pdf)
  • Example 5.2 (.pdf)
  • Example 5.3 (.pdf)
  • Example 5.4 (.pdf)
  • Example 5.5 (.pdf)
• Videos to Watch (please right-click and save, then play back locally on your computer):
  • Unit 05 Informational Video (.mov)
  • Example 5.1 Video (.mov)
  • Example 5.2 Video (.mov)
  • Example 5.3 Video (.mov)
  • Example 5.4 Video (.mov)
  • Example 5.5 Video (.mov)
• Reference Files:
  • Unit 5 Definitions, Nomenclature and Equations (.pdf)
• Other Files:
  • 05_Example_1.xlsx - Excel file used in the solution of Example 5.1

Teaching Resources

• Archive (.zip) - Contains all teaching resources listed below for this unit
• Sample Class
  • Online Pre-class Quiz (.pdf file)
  • Redacted slides to make available to students before class
    • Keynote 09
    • PowerPoint
  • Spreadsheet (05_Activity_1_Handout.xlsx) that will be used by the students during Activity 5.1
  • Class Lesson Plan (.pdf)
  • Complete slides to make available to students after class
    • Keynote 09
    • PowerPoint
  • Learning activity files to make available to students after class
    • Note, slides 7 through 9 of the complete slides above represent one possible solution to Activity 5.1
    • 05_Activity_1.xlsx - Excel spreadsheet used in the solution of Activity 5.1
• Alternative Questions (.pdf) that could be used in a pre-class quiz
• Alternative In-Class Learning Activities
  • Alternative Activity 5.1 (.zip) - an activity where students explore the shape of plots of rate versus conversion for various empirical rate expressions.
  • Alternative Activity 5.2 (.zip) - an activity where students analyze different chemical reactions and determine whether they could be elementary reactions.

Practice Problems

1. Suppose that for a quick preliminary calculation you need an approximate value for the rate of reaction (1a) below for a mixture containing 46% H₂, 31% CO₂, 22% CO, and 1% CH₃OH at a total pressure of 49.3 atm and a temperature of 327 °C. Suppose further that you have obtained an old company report which says that the rate expression given in equation (1b) below was shown to fit experimental data from reaction (1a) at similar compositions and pressures, but at the temperatures given in the table below. Using the data in that table, what is your best estimate for the rate of reaction (1a) at the conditions of interest to you. (Note: the rate expression used in this example is made up and should not be used for any purpose other than answering this question.)

(Problem Statement as .pdf file)

2. Collision theory can't be applied directly to a unimolecular reaction like that given in equation (2a) below. One approach to developing a theory for unimolecular reactions is to assume that the reactant molecule must first undergo a collision that results in it gaining internal energy. Collision theory can be used to estimate the rate of this preliminary step. Assuming a system contains pure ethane (collision diameter equal to 0.53 nm) at atmospheric pressure and 300 °C, estimate the corresponding pre-exponential factor.

(Problem Statement as .pdf file)

3. Consider the reaction between a diatomic molecule and an atom where the activated complex is non-linear. Use transition state theory to write out an expression that explicitly shows all the places that temperature appears in the rate coefficient. You may leave your answer in terms of masses, moments of inertia, vibrational frequencies, etc. of relevant species, but you must expand all summations and continuous products. Determine how many times the temperature appears.

(Problem Statement as .pdf file)

4. According to simple collision theory the rate coefficient for reaction (4a) will depend upon temperature according to equation (4b). Using the rate coefficient data in the table below (and this Excel file), determine whether this is true, and, if it is, find the best values of the pre-exponential factor and the activation energy.

(Problem Statement as .pdf file)