Learning Scenario - Precipitate Model (AgentSheets)

Basic Model:

Description

This is an agent model of a simple precipitation reaction of a solute in a solvent. The model displays a number of particles that move randomly within a liquid. Whenever two of these particles collide along the horizontal axis, they have a chance of precipitating out. Precipitates will fall to the bottom of the tank and coalesce. Parameters in this model determine the starting density of the particles and the chance that they will precipitate out upon colliding.

Background Information

Many chemical reactions occur between two compounds that are dissolved in water or another suitable solvent. Dissolved molecules have a greater ability to move around and interact with one another than they would have if they remained a solid. Where things get interesting, however, is when both of the reactants in a chemical reaction are water-soluble, but their product is not. This creates a situation where the compounds react in the water, but then their product either sinks to the bottom as a solid or bubbles to the top as gas. This separation of products from reactants is called precipitation. Precipitate reactions have a large number of different applications in chemistry. One of their most useful features is the ability to completely remove a solute from water, something that is normally only possible by boiling and recondensing the water. As a result, precipitate reactions are a very effective way of removing certain types of impurities from water as part of a water treatment process. Precipitate reactions can also be desirable for their quality that any precipitate reaction will go to completion. Since the product is no longer a part of the solution, it cannot react again to become a reactant. If the product of a reaction is the only useful compound, a precipitate reaction will help ensure that all of the reactant is used efficiently.

Science/Math

The fundamental principle behind this model is that the probability of an event occurring is proportional to the number of different ways that it can occur, given that all occurrences have equal probability. Thus, in this model the chance that a particle will precipitate out is proportional to the number of possible pairs of particles that could interact with one another. Each time tick, the following things happen:

1. Each particle that has yet to precipitate moves in a random direction
2. Each particle that has precipitated moves down (if possible)
3. Any two particles that have not yet precipitated but are next to each other on the horizontal axis have a certain chance of precipitating out.

Teaching Strategies

An effective way of introducing this model is to start with "pure" statistics first to ensure that students understand the mathematics behind it. Ask students to enumerate all potential pairs of results that can occur under the following random simulations (order matters):

1. Flip 2 coins
2. Flip 1 coin and roll one 4-sided number cube
3. Roll 2 4-sided number cubes
4. Roll 1 4-sided and 1 6-sided number cube
5. Roll 2 6-sided number cubes
6. Roll 2 8-sided number cubes

When they are done, ask the following questions:

1. How many different pairs of results did you get for each simulation above?
2. Is there a relationship between the number of outcomes of each individual random event and the number of pairs of outcomes? If so, what might that relationship be?
3. If I asked you to tell me the number of possible pairs of results of rolling 2 20-sided dice, what would that be? You shouldn't need to enumerate them one by one.
4. If I increased the sides of the die by 50% to 30, what would happen to the number of possible pairs? How do you know? What is the general relationship between the number of sides of the die and the number of possible combinations of rolls?

Implementation

How to use the Model

There are two different parameters that can be manipulated to change the way this model runs:

1. The "density" parameter determines the proportion of solvent cells that are filled with solute when the pencil tool is used to refresh the container. This determines the initial quantity of solute, and thus the reaction rate
2. The "stickiness" parameter determines the probability that two solute cells that come into contact on the horizontal axis will "stick together" and precipitate out of the solution. This also contributes to the reaction rate

These parameters can be manipulated by clicking on the dropdown arrow in the upper-right hand corner of the worksheet and choosing "Simulation Property Editor". Users are able to input values directly or use the up and down arrows to change each parameter by a preset amount. Changes to stickiness take effect starting on the next tick, without requiring a reset of the simulation. Changes to density affect the initial concentration of solute, so they do not take effect until the simulation is reset. To run the simulation, click the green play button at the bottom of the worksheet. As the simulation runs, the results will automatically be displayed on a graph. Users can also use the "Slow�Fast" slider bar to change the rate at which the simulation runs. To move forward just one tick at a time, click on the gray play/pause button next to the run button instead. For more information about Agentsheets reference the Agentsheets tutorial at: http://shodor.org/tutorials/agentSheets/Introduction.

Learning Objectives

1. Understand the relationship between solute concentration and precipitation rate in a precipitation reaction
2. Understand how reaction probability affects precipitation rate in a precipitation reaction

Objective 1

To accomplish this objective, have students modify the simulation to add a graph of the number of solute molecules that have not yet precipitated at a given time, and then run the simulation several times with different initial solute concentration levels. For each run, record the amount of time it takes for the solute concentration to get (a) below half of the initial concentration, and (b) less than 50. Ask the following questions:

1. Based on your results, approximately how long does it take for a 10% density solution to precipitate half of its solute? How about a 20% density solution? A 50% solution? Which was the fastest? Does that make sense?
2. How long did each of the previous solutions take to get less than 50 solute molecules left? Which was the fastest? Does that make sense?
3. In a simple inverse equation of the form 1/x, it should take exactly the same time to cut the amount of solute in half regardless of the starting point. Is that what you observed? If not, is there any other equation that might model the results more effectively?
4. What is the number of ways that a pair of particles can interact? How does this relate to the equation that seems to model the results here?

Objective2

To accomplish this objective, have students set the "stickiness" of the particles to (a) 1.0, (b) 0.5, (c) 0.25, and (d) 0.1. Then, run the simulation several times at each stickiness level and record the amount of time it takes for the solute to get below half and less than 50 in each case. Ask the following questions:

1. Based on your results, approximately how long does it take for a solution to precipitate half of its solute with each stickiness level? What is the approximate relationship between the two? Does that make sense?
2. How did changing the stickiness of the particles affect the results of the simulation? What equation would best approximate this relationship? Does that make sense?
3. How do the two parameters differ in their effects? Why do you think that the relationship between concentration and reaction rate is 1/x2, while the relationship between stickiness and reaction rate is only 1/x? Hint: think about which events the two parameters affect and how many ways those events can happen.

Extensions

1. Investigate ways of increasing reaction rates without changing the fundamental reaction
2. Extend the model to a new reaction with two types of reactants that must interact in order to precipitate

Extension 1

Have students brainstorm other ways that chemists might try to increase reaction rates without increasing the concentration of solute. Common ideas might include increasing the temperature, stirring/shaking, or removing the precipitate from the reaction as it forms. Have students implement their ideas in the AgentSheets model as effectively as possible. For instance, higher temperatures would increase movement rate, while stirring/shaking might change movement probabilities to be biased in one direction or another. Have students test their new simulations, and then ask the following questions:

1. How did your implemented change affect precipitation rates? Was this what you expected? Why or why not?
2. To what extent does increasing the movement rate of particles increase the precipitation rate?
3. In the real world, increasing the temperature of a reaction only changes the collision rate by a small amount, but it vastly increases the reaction rates nonetheless. How is that possible? What other parameters might temperature affect?

Extension 2

Have students create a new agent within the model and change the agent behavior such that agents can only precipitate if they interact with a different agent, not one of their own. Have students test their new simulations, and then ask the following questions:

1. Compared to the single-reactant model, are precipitation rates in this new model higher, lower, or the same for an equal concentration? Why do you think this is?
2. Change the concentration of both reactants to the same levels that you did before with the single solute, and re-run the simulation. What relationship do you see between concentration and precipitation rates now? Does that make sense given our earlier statistical reasoning? Why or why not?
3. Change the stickiness and re-run the simulation. What relationship do you see between stickiness and precipitation rates now? Does that make sense given our earlier statistical reasoning? Why or why not?

Related Models

Second-Order Chemical Kinetics

This is a closely related model to the Precipitate Model that simulates a chemical reaction wherein two compounds react to form two products. As with the precipitate model, the relationship between the concentration of the compounds and the reaction rate is quadratic rather than linear. The main difference is that second-order chemical kinetics has the potential to be a reversible reaction. In such a situation, the relationship between concentration and reaction rate would still be linear, but it would go to equilibrium rather than completion. This model is best used as an extension of the precipitate model, to introduce multiple reactants and the concept of reversibility.

Beginning Enzyme Kinetics

This model builds on the precipitate model but takes it in a slightly different direction. Here, rather than needing any two particles to collide in order to react, the reactant must bind with an enzyme before it can transform into a product. The enzyme is not consumed or changed in any way by the reaction. In theory, if the number of enzymes is approximately equal to the number of reactants, this reaction will evolve by the same mathematical logic as the precipitate model. However, the concentration of enzymes is usually extremely small compared to the concentration of reactants, so it acts as a rate-limiting step and the actual reaction is approximately linear. This model should be used to help students understand the role of assumptions in chemistry. Depending on whether there is an equal quantity of both types of reactants or a larger amount of one reactant, the mathematical equation for the rate of the chemical reaction can be completely different. Similarly, small changes in the rules of the precipitate reaction can have large effects on the way it unfolds.