Unit #1

Activity 7
Feeling Low on Energy? Have Some of Mine

Activity Overview

Continual movement of atoms results in motion that looks random and causes kinetic energy to be distributed evenly among the atoms in a gas.

Part A: Students kinesthetically model a gas, by role playing atoms moving around in a container. By doing this students gain an understanding about the behavior of lots of atoms bouncing around and exchanging kinetic energy.

Part B: Students observe a computer model of a gass (in spatial and thermal equilibrium). They experiment with varying initial conditions (# of atoms, masses of atoms, initial velocity of atoms) to see what kind of predictions the model makes about the eventual state of the total system and each individual atom. They should eventually observe that the kinetic energy [energy of motion] is distributed evenly among atoms and comes to a state of equilibrium.

Learning Objectives

Students will:

• Demonstrate, using a computer model, that all initial states result in a similar equilibrium distribution of kinetic energies and spatial locations of atoms.
• Predict what will happen given some random initial state of a computer model depicting a gas at the atomic level.
• Describe what it means to be in a state of thermal equilibrium.
• Describe what it means to be in a state of spatial equilibrium.
• Define general equilibrium in very simple terms.

Conceptual Prologue

Macro-Micro Connection

The model of many atoms in a box is a good depiction of a gas. Using the models in this activity students will start to get an idea of how the air in the hot air balloon is behaving. All of the air molecules are bumping into each other exchanging kinetic energy with each other and with the fabric of the balloon. Air inside the balloon spreads out and reaches various states of equilibrium both spatially and thermally [see science concepts below]. Understanding how kinetic energy gets distributed among all of the atoms in the balloon is important to the overall picture of how the balloon eventually flies. Later, students will learn that the kinetic energy of atoms is related to their temperature, so understanding the distribution of kinetic energy will give students an understanding of how heat is distributed throughout the balloon.

Other Macro connections:

• Air inside a room reaches a state of equilibrium such that the atoms* are spread out evenly and have about the same kinetic energy.
• Clouds are made of tiny droplets (much bigger than atoms) which are kept aloft by collisions with air molecules all around them.

*Air is actually made of a mixture of molecules (tightly bound groups of atoms) not single atoms.

Science Concepts

When atoms collide with each other, the total amount of kinetic energy [energy of motion] they have is conserved [remains constant]. Typically, when one atom collides with another of the same mass, the faster atom, the one with more energy, will slow down and consequently now have less kinetic energy. The other atom with which it collided will speed up and consequently now have more kinetic energy. The total energy of the atoms before and after the collision is the same.

As the number of atoms being modeled increases, patterns and certain statistical behaviors emerge that are not easily observed in simpler models. Using a computer model with many atoms, we begin to approach the modeling of macro scale phenomena (e.g., the behavior of gasses). With many atoms, we can now study the statistical behavior of molecules in a gas. Specifically, the following can be observed:

• Eventually, if the atoms are allowed to collide randomly, the system reaches a state of dynamic equilibrium. Equilibrium is a state in which measurements of the system or any part of the system remain constant, but not due to reaching some endpoint where everything stops. For example, imagine two sets of people on opposite sides of a bridge. They begin to walk across in such a way that for every person who crosses from side A to side B, one crosses from side B to A. If you were to measure how many people are on either side of the bridge you would always get the same number. However, the specific people are always changing. This is a state of dynamic equilibrium.
• Two kinds of equilibrium will occur:
  • Spatial equilibrium: when the number of atoms in any one part of the box is, on average, the same over time. In other words, the atoms are spread out evenly in the box.
  • Thermal equilibrium: when the kinetic energy of every atom is, on average, the same over time. In other words, when the kinetic energy is spread out evenly among the atoms.
• The starting location or starting kinetic energy will have no effect on the eventual equilibrium of the system.
• Atoms move in straight lines until they collide with other atoms. Because of numerous collisions between atoms, the motion of individual atoms appears random.
• All atoms of a gas have the same average kinetic energy, even if this gas contains a mixture of atoms with different masses. The heavier ones will move more slowly, but they will have the same average kinetic energy as the lighter faster moving particles.

Naive Conceptions

• Equilibrium is an endpoint when everything stops.

Equilibrium is a dynamic state in which measurements on the system or parts of the system remain constant. However, the constant measurements are due to many counterbalancing fluctuations of individual elements that comprise the system.

Activity Design and Execution

Major Science Concepts:	• kinetic energy • equilibrium
Assumed Previous Knowledge:	• Experience with a computer model of two atoms in a container. • That atoms bounce off of each other with 100% elasticity. • That atoms have kinetic energy [energy of motion] which is related to mass and velocity [speed]. • That the total energy in a collision between two atoms is conserved [the same before and after]. • That energy can be transferred between atoms through collisions.
Time:	• Part A: approximately 50 minutes • Part B: approximately 100 minutes (50 min x 2)
Materials:	• Popsicle sticks (a box) • Computers with Workbench Software
Advanced Preparation: (if any)	• None

Investigative Question: How do the atoms of a gas share space and kinetic energy?

Part A :

1. Tell the students that they are going to do some more kinesthetic modeling, except this time many of them will be atoms, instead of just one of them.
2. Ask them to list the rules for how atoms/students should behave in this model. They should have a list of these rules in their notebooks.
3. Teach the students how to play “rock-paper-scissors”.
4. Set aside some part of the room to be a container. The walls can be defined by tables, desks, or floor tiles.
5. Explain the “rules” for this simulation, making connections with students' prior knowledge of atoms and kinetic energy:
   • Each student will walk in a straight line unless they collide with a wall or with another student.
   • The speed each student walks depends upon how many batons they are carrying. Each stick represents some amount of kinetic energy.
   • If they collide with a wall, they should “bounce” off of it just like an atom would.
   • If they collide with another student, then the student/atom with more sticks (kinetic energy) gives one of those batons to the other student/atom and they bounce off each other as would atoms.
   • If two students/atoms collide who have the same kinetic energy, they play “rock-paper-scissors” to determine who gets the energy. The one who wins gets one stick from the other.
6. Start some people off with many sticks and some with very few sticks all in one corner of the container. Write down an initial list of kinetic energies by having each student call out the number of sticks they have.
7. Have students play the collision game for a while, stop them periodically and have them count off out loud how many batons they have. Write this list down so that it is visible to the students. Ideally you would put this data in a spreadsheet, and project this using and LCD projector. Then instant graphs can be made of the data at each stage and at the end.
8. Discuss with them what students noticed about how atoms spread out and how the kinetic energy was distributed after a while.
9. Explain to them that the point when the system seemed to "even out" was when it reached equilibrium. Then distinguish between two kinds of equilibrium - spatial equilibrium when they are spread out evenly, and thermal equilibrium when their energy is spread out evenly.
10. Play the game again with several variations and ask students to predict what will happen before you play. Each time you stop the game to collect data, ask the students if they think they are in equilibrium yet and which kind spatial, thermal, or both. Here are some suggested variations:
    • Start everyone located in one corner with the same kinetic energy.
    • Start everyone distributed evenly in the container, but start some people off with many sticks and some with very few sticks.

 

Part B:

1. The next modeling experience will require that students understand what a "running average" is. You should probably do some exercise to familiarize them with this concept. One possibility would be to provide a fictional list of test grades and ask the students to calculate the average of the first two tests, then the first three tests, and first four tests. The average they calculate is the running average. If you have an LCD projector then you can set up a spreadsheet and graph both the test scores and the running average on the same graph. You should see that the average becomes more and more steady as more and more tests are factored in.
2. Have the students use the Workbench Software to run both the Modeling a Gas: Spatial Equilibruim and Modeling a Gas: Thermal Equilibrium activites.. Explain to them that this is a computer model using rules similar to the ones they used when doing the kinesthetic modeling. However, using the computer allows them to make changes and take measurements much more easily.
3. The software will explain the concept of spatial equilibrium to the students and show them a box full of atoms with various initial states. It will then ask them to start the model and then prompt them to stop the model when they think it has reached spatial equilibrium. After doing a couple of these the software will ask them how they know they have reached spatial equilibrium.
4. The software will then show them a model with the atoms spread out evenly. However, half of the atoms will be of one type and half of another type. They will then be asked to state when this model is in spatial equilibrium.
5. The software will then explain the concept of thermal equilibrium and show the student a box full of atoms with various initial states. The kinetic energy shading will be turned on so students can visually see the kinetic energy of each atom, and a graph of speed distribution will also be displayed. They will then be asked to start and stop the model when they think it has reached thermal equilibrium. After doing a couple of these the software will ask them how they know they have reached thermal equilibrium.
6. The software will display a box of atoms, two drop down menus to bring up predefined initial model states, and a speed distribution graph. The initial states will be:

For the position drop down menu:
- Close to the center.
- Close to a corner.
- Spread evenly.

For the kinetic energy drop down menu:
- Same kinetic energy.
- Random kinetic energy.
- Half high, half low kinetic energy.

7. The students will then be asked to set up various initial states, run the model and indicate, by pushing a button, when they think it has reached spatial and thermal equilibrium. After doing several of these the software will pause to allow class discussion of the activity so far. (You may want to delay the discussion of running average until this point.)
8. The software will resume by showing the students a running model that is in equilibrium and a button which allows them to choose an atom to highlight, as well as a button to show the path of that atom.
9. The students will then be asked to describe the motion of the highlighted atom, to choose several different atoms, and to describe any differences or similarities between the atoms.
10. The software will then show the students a running model that is in equilibrium. It will also allow students to connect two atoms to two graphs which will display current kinetic energy and a running average of the kinetic energy. Another graph of the average of the total KE will be displayed.
11. The students will then be asked to pick an atom and connect it to a graph. When they have done this, they will be asked to start the running average and let it collect data for a few minutes. They will then be asked to compare this to the average of all the atoms.
12. They will be asked to do this for several atoms and asked what they notice about the running average of a single atom when compared with the whole.
13. The software will then display a model with some heavy and some light atoms. They will then be asked to compare, a heavy atom, to a light atom, and both with the total average.
14. The software will then ask the students what was the same and different between the heavy and light atom.
15. Discuss with students what they observed and ask students to come up with a general definition for equilibrium.

Assessment

Have students write several things in their notebooks:

1. Imagine you are at a party where someone dared you to pop one of the helium balloons in a room that has all the windows and doors closed. Describe what would happen to those Helium atoms once they have been released from the balloon. Specifically talk about their eventual position inside the room and what their kinetic energies would be like if you could actually measure them.
2. Define spatial equilibrium.
3. Define thermal equilibrium.
4. The term equilibrium can apply to many different things. Think about what is similar between spatial and thermal equilibrium and give a general definition of equilibrium.
5. When the burner of the hot air balloon is turned on, the kinetic energy of atoms near the burner is greatly increased. This makes the air inside the balloon go out of thermal equilibrium. Explain what will happen to all the atoms in the balloon as they eventually come back to thermal equilibrium.