Learning Scenario - Reaction Data (Excel)

Basic Model

Description

This is a set of experimental data for a series of chemical reactions. It tracks the concentrations of reactants and products for four different reactions over the course of 100 time steps. Since reactions are not guaranteed to occur at precisely the same rate every time, this model includes data for four repetitions of the experiment for each of the reactions. The data in this model is already generated, so there are no parameters to manipulate; instead, users attempt to determine what the parameters used to generate the data were.

Background Information

In chemistry class, the most common way that students learn about a new reaction is to be given the chemical equation for the reaction first, and then experiment with how it looks in the real world. For a chemist, however, these steps occur in reverse. Rather than knowing a reaction's formula ahead of time, chemists often have to run the reaction without much foreknowledge, and then use the results to deduce what the reactants and products are, and how they interact with one another. The first step to doing this is to measure exactly how the concentrations of reactant and product change over time as the reaction takes place. Some reactions go to completion, some go to equilibrium, and some don't happen at all under normal conditions.

Once the reaction data is acquired, the chemist can then try changing the relative concentrations of the reactants and products to get an idea of their placement in the equation. Recording how changing the concentration of the reactant affects reaction rates can help to illuminate what type of reaction is taking place and whether there is another limiting factor. The rate of change of the concentrations of reactants and products is also an important factor - if the concentration of product is rising at twice the rate that the concentration of the reactants is falling, then the reaction likely produces two product molecules for every reactant molecule.

Science/Math

The fundamental principle behind this model is HAVE = HAD + CHANGE. Each time step, the following things happen:

1. A certain proportion of the reactants react to form a product
2. 2) A certain proportion of the products react to form either a reactant or another product

The relative rates of these two changes determine what happens to the overall solution. If the proportions for both (1) and (2) are positive, then the reaction will asymptotically approach an equilibrium where both reactants and products exist proportional to their reaction rate. If the proportion for (2) is zero, then the reaction will go to completion; that is, in the long run all of the reactant will react to form product, but none of the product will turn back into reactant. Completion reactions are most commonly found when the reaction occurs in a solution but the product is either a solid or a gas that precipitates out of the reaction environment.

Teaching Strategies

An effective way of introducing this model is to have students work through an M&M reaction. For this activity, give each student 20 M&Ms and ask them to place the candies such that they are all "m"-side down. Then, model a reaction by the following procedure:

1. Flip a coin for each M&M that is "m"-side down
2. If the coin lands heads, flip the M&M to "m"-side up
3. Record the number of "m"s you see, and repeat until all M&Ms are "m"-side up

This procedure models a simple reaction going to completion and gives students a good idea of how reaction rates change over time. Ask the following questions:

1. How many M&Ms did you flip on the first time step? The second? The fifth?
2. Why do you think that the rate of M&M's flipping decreased over time?
3. Does the number of M&Ms over time look like any function you're familiar with? If so, which one? Why do you think this function makes sense

Now, repeat the above experiment, but this time flip a coin for all M&Ms, and if it lands on tails, flip the M&M back to "m"-side down. Ask the following questions:

1. How does the number of "m"-side up M&Ms change over time now?
2. Do you think it will ever be the case that all M&Ms are "m"-side up? Why or why not?

Implementation

How to use the model

This model doesn't have any specific parameters to modify, but there are plenty of things that can be done to extract information from the dataset:

1. To make the chemical reaction easier to visualize, create a scatter plot with time step as the independent variable and the concentrations of reactants and products as the dependent variable
2. To determine the "half-life" of a reaction that goes to completion, count the number of time steps required for the concentration of the reactant to be half of its starting value. Then, count the number required for it to half again, and repeat until the minimum concentration is reached. The average number of time steps required to cut the concentration in half each time is the half-life.
3. To determine the equilibrium condition of the reaction for reactions that do not go to completion, find the ratios between the concentrations of each substance at the final time step, and then average over the four repetitions of the experiment. This proportion provides a clue as to the chemical formula of the reaction.
4. Add an additional column to the right of the data set whose cells are defined as the variance of the concentration of one of the particles over the four repetitions of the experiment. This variance defines the degree to which the reaction rate differs from experiment to experiment. The main cause of such a difference is experimental error - different conditions for the reaction, different rates of mixing, or inaccurate measurement techniques.

Learning Objectives

1. Understand what it means to say that a reaction goes to completion or goes to equilibrium
2. Understand how to model the concentrations of reactants and products mathematically

Objective 1

To accomplish this objective, have students create scatter plots for each of the four reactions and examine the concentrations of each particle over time. Ask the following questions:

1. How does the concentration of particle A change in the first and second reactions? Which of these reactions do you think goes to completion, and which to equilibrium? How do you know?
2. If a reaction goes to equilibrium, does that mean that the concentrations of reactants and products will be the same in the long run? If not, which of these reactions provides a counterexample? What might the reaction formula look like for the third reaction?
3. Reaction four consists of 3 particles rather than two. Which of these particles do you think are reactants, and which are products? Does the data here suggest a single reaction with two products, or a two-step reaction? How do you know?
4. What might the reaction formula(s) look like for the fourth reaction?

Objective 2

To accomplish this objective, have students calculate the half-life of the reactant in the first reaction. Ask the following questions:

1. What sort of relationship do the reactant and product in the first reaction appear to have as they change over time (linear, quadratic, exponential, etc)? How do you know?
2. Assuming that the equation is exponential with the half-life that you calculated, what would the formula for the concentration of the reactant be?
3. Graph the concentrations of the reactant in the first reaction and add an exponential trend line to the graph. How does the trend line compare to the equation you found earlier?

Extensions:

1. Understand how to model more complex chemical reactions mathematically
2. Calculate the experimental error in the data set

Extension 1

Have students graph and calculate the half-life of the reactant in the second, third, and fourth reactions. Ask the following questions:

1. What are the half-lives of the reactants in each of the reactions?
2. Based on these half-lives, what are the exponential equations for the reaction in each of these reactions? It may help to normalize the concentrations first by subtracting the smallest reaction concentration from all of the values.
3. Is an exponential equation a good fit for the concentration of reactants in chemical reactions that go to equilibrium as opposed to completion? Why or why not?

Extension 2

Have students calculate the variance in the concentrations of each particle at each time step for the various reactions. Next, have students calculate the standard deviation by taking the square root of the variance. Then, divide the standard deviation by the mean to get a percentage. Finally, multiple that percentage by 1.96 to get the 95% confidence interval. Ask the following questions:

1. What is the average variance for the concentrations of particles in each reaction? How do you know?
2. What is the 95% confidence interval for the concentration of the different particles? What does that mean, intuitively?
3. How could we make the confidence interval smaller?
4. 4) What are some potential sources of the difference in reaction rates between different trials? How could we fix or alleviate these problems?

Related Models

Generic Chemical Dynamics

This is a simpler reaction model with just a single two-stage chemical reaction. Unlike in Reaction Data, this model assumes no experimental error and just models the concentrations of the reactions as a population problem. As a result, there is no need for multiple trials because all trials will give the same results. This makes it easier to understand functions for the rate of change of concentrations over time, but it has the disadvantage of being less immediately applicable to the real world. This model can be effectively used to drive home the mathematical foundations for chemical reactions without the distraction of imperfect data.

Epidemic

This epidemic model is surprisingly similar to a chemical reaction model in that a starting population of reactants "healthy people" is eventually converted into a product "sick people". Like the reaction model, this is nondeterministic; so multiple trials may be needed to ensure accurate results despite random chance. The main difference between the models is that, in the epidemic model, the "reaction rate" depends on the number of sick people instead of being fixed. This creates a slightly different result, approaching a Cumulative Distribution Function rather than an exponential function. This model is a useful way to compare other types of reactions and understand how the same principles that were used to determine the mathematical formula for chemical reactions can be applied in a variety of different situations.