Teachers Guide
This classroom activity introduces students to the idea of reaction rates, timesteps, timecourse, equilibrium and randomness. Each student is given a party hat and a die. For each “reaction” below, students act as a chemical reactant. They follow the rules for the reaction using the dice to determine their chemical state, which is indicated by whether or not they are wearing their hat. The instructor or a designated data recorder should keep track of the numbers of students with or without hats on each timestep.
Each of the simulated reactions described below is characterized by a reaction order (how many reacting species the rate depends on) and directionality (irreversible = oneway, reversible = bothways). For reaction order we have:

Zeroorder – The rate is constant and therefore doesn’t depend on the amount of any reacting species.

Firstorder – The rate depends on the amount of a single reacting species.

Secondorder – The rate depends on the amount of two reacting species. This order fits what we think of as the typical chemical reaction, A + B > C, where two reactants have to interact in order for a reaction to occur. Second order reactions aren’t introduced in this activity, but are left for another activity. While this may seem disconcerting, it turns out that in many of the biological reactions that require two species to interact one of the species is an enzyme that isn’t transformed by the reaction, so it stays at the same concentration as the reaction proceeds.
The simulated reactions described in this activity include:

Reaction #1  A firstorder irreversible reaction (rate depends on amount of one species) in which students have a 1/3 chance of putting on their hat (undergoing chemical transformation) on each turn (timestep). The reaction reaches exhaustion after a few timesteps because there is no one left with their hat off to roll their die. Students are asked to predict what will happen on steps that haven’t happened. By plotting numbers of students with or without hats versus time, students see that the rate changes over the course of the reaction. From their predictions and the graph they should see that the rate depends on amount for this reaction.

Reaction #2  Another firstorder irreversible reaction, but this time students have a 2/3 chance of putting on their hat on each timestep. Questions direct students to think about how the difference in the chance of an individual putting on their hat leads to a change in the overall rate for the reaction, and vice versa.

Reaction #3  A zeroorder irreversible reaction in which a “reaction master” rolls a die at the start to see how many students put on their hats on each time step. This example is used to demonstrate the idea of constant change, and can be used to distinguish a deterministic model, where we can predict exactly how many students will put on hats on each time step, from a probabilistic model, where the random roll of each students die means that we don’t now exactly how many will change on each step. Systems that can show zeroorder kinetics include enzymes when high concentrations of substrate are present. In this case, while the rate depends on the concentration of substrate, but the substrate concentration changes only slightly on each timestep.

Reaction #4  A firstorder reversible reaction in which students have a ½ chance of putting on their hat, and students with their hats on now have a 1/6 chance of removing their hat on each time step. This reaction introduces the idea of equilibrium. Because the system is closed (no students are being added or removed during the reaction) the system will find equilibrium, but it is a dynamic equilibrium.

Reaction #5  Another firstorder reversible reaction with different chances for the haton and hatoff transformations. In this case, the differences in behavior from reaction #4 are not only in the rates at which the overall reaction happens, but where the reaction finds its equilibrium. Questions are directed at trying to get students to think about the relationship between the rates for the conversions and the numbers of hatless and hatwearing students the system tends to stay close to.

Reaction #6  A zeroorder reversible reaction. In this case the forward and reverse rates are constant. If the rates are unequal the system will use exhaust the type of student that is consumed faster than it is formed (leaving just what is formed on each step to be consumed on the next). If the rates are equal then the system will be stationary with no change in total amounts after the first timestep.

Reaction #7  A firstorder sequential reaction in which hatless students have 1/3 chance of putting on their hats, then students with hats have 1/3 chance of removing their hat, or 1/3 chance of raising their hand. This reaction introduces the idea that the products of one reaction are often the reactants for another reaction. This reaction also resembles the MichaelisMenten model for enzyme kinetics.
Remove this description and print the rules below to distribute to the students, along with their hat and die.

Party Hat Chemistry Rule Set
Reaction #1 Rules
Everyone starts with his or her hat off.

If you are not wearing a hat, once each timestep roll your die.

If you roll a 1 or a 2, put on a hat.

If you roll a 3, 4, 5, or 6 don’t do anything (except smile).

Make sure that the data recorder has tallied how many people have hats on.

If you still don’t have your hat on, go back to step 1.
Questions:

Before you start rolling dice, how many (or what proportion of) people do you think will put their hats on after the first roll? _____

How many people do you expect will put their hats on after the second roll? _____

Are the answers to the previous two questions the same or different? Why?

How many rolls should it take until everyone has their hat on? ____ Can there be less than 1 but more than 0 people (are there any fractional people in the room)?

How many timesteps did it take for everyone to put on their hats? ______

How many (what proportion of) people put on their hat on each timestep? _____, _____, _____, _____, _____, _____, _____, _____, _____, _____, _____

If we start over and do it again, will everything happen just the same? Why?
Reaction #2 Rules
Everyone starts with his or her hat off.

If you are not wearing a hat, once each timestep roll your die.

If you roll a 1, 2, 3 or 4 put on a hat.

If you roll a 5 or a 6 don’t do anything (except smile).

Make sure that the data recorder has tallied how many people have hats on.

If you still don’t have your hat on, go back to step 1.
Questions:

Before you start rolling dice, do you expect everyone to have their hats on faster (fewer rolls) or slower (more rolls) than in Reaction #1? Why?

How many timesteps did it take for everyone to put on their hats? ______

How many (what proportion) people put on their hat on each timestep?

What does a graph of the number of people without hats on versus timestep look like?

How does the graph for Reaction #1 differ from the graph for Reaction #2?

How would you explain the difference between the two graphs? What feature of the two reaction accounts for this difference?
Reaction #3 Rules
Everyone starts with his or her hat off.

First, a reaction master is chosen.

Before starting, the reaction master rolls a die.

On each timestep, the reaction master picks the number of people they rolled in step 2 to put their hat on.
Questions:

Before you start taking timesteps, but after the reaction master has rolled a die, how many timesteps will it take before everyone has his or her hat on?

How many timesteps did it take for everyone to put on his or her hat? ______

How many (what proportion of) people put on their hat on each timestep?

How does the plot of the hat reaction differ this time from the first two reactions?
Reaction #4 Rules:
Everyone starts with his or her hat off.

Once each timestep roll your die.

If you are not wearing a hat:

If you roll a 1, 2 or 3 put on a hat.

If you roll a 4, 5 or 6 don’t do anything (except smile).

If you are wearing a hat:

If you roll a 1 take off your hat.

If you roll a 2, 3, 4, 5 or 6 don’t do anything (except smile).
Questions:

Before you start rolling:

How many timesteps do you think it will take before everyone has his or her hat on?

How do you think the reaction timecourse will differ from the previous reactions?

How many timesteps did it take for everyone to put on his or her hat?

How many (what proportion) people put on their hat on each timestep?

How many (what proportion) people took off their hat on each timestep?

How many people were wearing hats after 1 timestep? ______

How many people were wearing hats after 5 timesteps? ______

How many people were wearing hats after 10 timesteps? ______

How many people were wearing hats after 20 timesteps? ______

Were there any timesteps where nobody changed their hat status? ______

Were there any timesteps where the number of people wearing hats didn’t change? How do you explain this?

What proportion of people had their hats on during the last 5 timesteps? How could you find a good estimate for this answer? (so what proportion had their hats off?)
Reaction #5 Rules:
Everyone starts with his or her hat off.

Once each timestep roll your dice.

If you are not wearing a hat:

If you roll a 1 or a 2 put on a hat.

If you roll a 3, 4, 5 or 6 don’t do anything (except smile).

If you are wearing a hat:

If you roll a 1, 2, 3 or 4 take off your hat.

If you roll a 5 or a 6 don’t do anything (except smile).
Questions:

Before you start rolling, what proportion of people do you think will have their hat on after 10 timesteps? (Off?)

What proportion of people had their hat on after 10 timesteps?

How does the proportion of the students with or without hats at equilibrium (people with hats : people without hats) compare to the proportion of the chance of putting your hat on or taking your hat off (haton chance : hatoff chance)?
Reaction #6 Rules:
Everyone starts with his or her hat off.

First, two reaction masters are chosen. One for putting hats on, one for taking hats off.

Each reaction master rolls a die.

On each timestep, the reaction master picks the number of people they rolled in step 2 to put on, or take off their hat. If there aren’t that many people available to do what the reaction master says, then all of the people available are chosen.

Before starting this reaction, can you tell the answer to the next question?
Questions:

Before you start, how do you think the behavior of this reaction might differ from the last reaction?

What proportion of people ended up with a hat on? ______

How many timesteps did it take for everyone to put on their hats? ______

How many (what proportion) people put on their hat on each timestep? ______

How many (what proportion) people took off their hat on each timestep? ______

How many people were wearing hats after 1 timestep? ______

After 5 timesteps? ______

After 10 timesteps? ______

After 20 timesteps? ______

Were there any timesteps where nobody changed their hat status? ______

Were there any timesteps where the number of people wearing hats didn’t change? ______
Reaction #7 Rules:
Everyone starts with his or her hat off.

Once each timestep roll your dice.

If you are not wearing a hat:

If you roll a 1 or a 2 put on a hat.

If you roll a 3, 4, 5 or 6 don’t do anything (except smile).

If you are wearing a hat:

If you roll a 1 or a 2 take off your hat.

If you roll a 3 or a 4 don’t do anything (except smile).

If you roll a 5 or a 6 raise your hand.

If your hand is up:

Be patient.
Questions:

Before starting this reaction, what proportion of people do you think will end up wearing a hat? ______

Was equilibrium reached? Can you explain this?

What rules might you add or change that would cause this reaction to change from finding equilibrium to exhausting resources, or vice versa?
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