Welcome to the first lesson of the Connected Chemistry Ideal Gas Laws unit!

In this unit, we will investigate what causes gas pressure, how it is measured, and how it is affected by the properties of the particles that make the gas and the characteristics of the container they are in. To achieve this goal, we will study gases 

• at the macroscopic level by exploring real world objects such as basketballs, air duster cans, and basketballs.
• at the microscopic level by using computational models.

We will begin our exploration with an air duster can. This everyday object will help us understand:

• How do gases behave in a fixed volume container?
• What are computational models?
  • How do we create them?
  • Why do we need them?
  • How are they related to scientific models?

Let's watch the following video of someone cleverly "hacking" an air duster. This demonstration may give us a better idea on how it works.

Please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If you the discussion has not started yet, please answer the extra credit question below as you are waiting.

In this unit, we are going to work with a few fundamental assumptions about gases. The first of them is: gases are made up of many individual particles (molecules).

However, we cannot see the gas molecules inside the air duster can; they are too small to see with only our eyes 👀. Instead, we will use  NetLogo 🔗 to develop a computational model to visualize the molecules inside the air duster can as the air is being pumped into it. NetLogo is a free programmable environment for building and running computational scientific models that simulate natural and social phenomena.

Scientific models 🔗 are used to explain and predict the behavior of real objects or systems and are used in a variety of scientific disciplines. They can be expressed in many different ways. For example, your written explanations and sketches can be considered as informal models of the gas behavior in the air duster can.

Below is a simple particle sandbox modeling toolkit. We can use this simple modeling toolkit to construct static systems that include basic particles and barriers. The short video above demonstrates how to use this tool.

Now let's re-create an approximation of our sketched models of the air duster can using the simple modeling toolkit below! 

Please do not spend more than 6 minutes on this task.

Again, please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If you the discussion has not started yet, please answer the extra credit question below as you are waiting.

The particle sandbox model allowed us to create simple gas molecules and place them inside containers. However, it did not allow us to see how the particles behaved. We can use computer models to animate the particles, too.

Below is another simple modeling toolkit that allows us to define the behavior of gas particles. For simplicity, this toolkit allows you to work only up to 4 particles. The short video on the right demonstrates how to use the blocks below the model to program the behavior of the particles. Please watch it carefully if you are having difficulty with the blocks.

A few quick points:

• Click  to reset the model and click  to run your model.
• Your code should always start with the  block, or the model will not know what to do.
• Your code will be performed by the particles continuously.
  • For example, if you wrote that "each particle + moves zig-zag", each particle in the model will move one step forward doing a zig-zag movement and then move one more step forward the same way, and so on. This will keep going on as long as the GO button is pressed.

Now let's try to figure out how gas behave inside a container!

Please do not spend more than 6 minutes on this task.

Now we can put our static particle sandbox model and our particle behavior block codes together. Let's use the combined modeling toolkit below to:

1. Load your static model design from Page 4* using the  button.
2. Re-create your blocks code and run it within your sandbox model.
3. If the model does not behave as you anticipated, change your experiment design (e.g., add more particles, walls) or blocks-code to manipulate your model.

Note: If you don't remember the code you wrote in the previous step, you should be able to see a link to your blocks code screenshot below the modeling toolkit.

* If you did not save your design, try to recreate it as best as you can. But this time, don't forget to save it.

One last time, please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If you the discussion has not started yet, please answer the extra credit question below as you are waiting.

In this final section, we will compare all three of our models (written, sketch, and computational).

In this lesson, you will use a scientific model similar to the one you developed in the previous lesson. Using this model, you will observe the effects of pumping air into a bike tire.

The simple bike tire model will help us understand:

• Learning about the Kinetic Molecular Theory and its main assumptions
• Defining pressure (P) in a gas container
• Thinking about the role of gas pressure in real-life objects and events
• Considering the trade-offs of making simplifications when constructing computer-based models

In Lesson 1, we tried to hypothesize about how gas particles behave in a container. We did so by writing our hypotheses verbally, sketching, developing static computational representations, and finally defining the behavior of the gas particles within a block-based programming environment.

The Kinetic Molecular Theory (KMT) is the scientific theory that is used to explain and predict the behavior of different objects that use air pressure to function such as these:

The main assumptions of Kinetic Molecular Theory (KMT) are the following:

1. A gas is composed of a large number of identical molecules moving in random directions, separated by distances that are large compared with their size.
2. Collision between gas particles occur like collisions between billiard balls (i.e. the total energy is conserved; see elastic collision 🔗). Otherwise, they do not interact. There are no attractive or repulsive forces between the particles.
3. Any energy the particles have is because of their motion only (i.e. kinetic energy).
4. These assumptions are simplifications that describe a theoretical "ideal gas". Most real gases behave qualitatively like an ideal gas.

Before we move on to the implications of KMT, let's first reflect on how these assumptions compare to our gas particle models from the Lesson 1.

Study a physical bike tire and a pump (or equivalent objects chosen by your teacher) in the classroom. Try to observe what happens when you pump air into the tire. As you are doing this, keep the assumptions of KMT in your mind.

Then, familiarize yourselves with the simple computational bike-tire model below. To run it, first click the  button, and then click the  button. If you would like to pause the model, you can click the "➤➤ go" button again.

For now, just experiment with the model freely as you experimented with the physical bike tire and pump. Observe what happens when you run the model with different settings. Please do not spend more than 4 minutes on this task.

Pressure 🔗 is defined as force per unit area. It is represented by the ratio of force to area:

(P)ressure = (F)orce / Surface (A)rea

This means that the pressure on a surface depends both on the force that is applied to the surface and the size of the surface. A larger force on a surface leads to larger pressure. A larger surface with the same force against it leads to a smaller pressure.

What does this definition of pressure imply for our Bike Tire model? What constitutes the "surface area"? Where does the force come from? How do we calculate the pressure in a fixed volume gas container like the one in our model?

We will use our simple Bike Tire model to find answers to these questions. Run the model once or twice without changing any parameters. Just observe the "pressure over time" plot for a while (≈30 ticks). Then answer the questions below. 
Please do not spend more than 4 minutes with this task.

Please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If your teacher did not initiate the discussion yet, please start answering the first question below as you are waiting.

In the case of gas particles in a container (e.g., a real bike tire, the air duster can, and the Bike Tire model below) the pressure of the system is created by the particles hitting the walls of the container.

The pressure at a given instant is calculated as the total force applied by the particles hitting the walls at a given time and the surface area is the area of the walls.

That is also why we observe the fluctuations in the "pressure over time" plot: at a given tick, slightly different number of particles may hit the walls, and the total force applied by them might be slightly different. 

Now let's explore what is the difference between the fluctuations in the plot, changes in pressure, and the stability of the system: 

1. Run the model with 50 particles by clicking the  button, and then clicking the  button.
2. Wait until the system stabilizes (≈20 ticks). 
3. Add 150 more particles using the   button three times.
4. Observe the changes in the system until the system stabilizes again.

In the beginning of this lesson, you experimented with a real (physical) bike tire and then you also experimented with a computational bike-tire model. A distinct feature of our model was that it simplified some aspects of the real world object. For example, the walls in our computational model did not expand or shrink. 

Before ending this lesson, let's reflect on the differences between the two and why might we (and scientists) want to use such computational models.

Please set your computer aside briefly (do not close this page) and join the classroom discussion and the physical experiment that your teacher is going to moderate.

In the previous two lessons, we qualitatively explored:

• the behavior of gas particles inside fixed volume containers
• and how the simultaneous behavior of many gas particles at micro-level result in a macro-level property, pressure.

In this lesson, we are going to start developing quantitative relationships between the macro-level properties of gasses. To do so, we will use a slightly different version of the bike tire model from the lesson 2 and a quantitative data analysis application called CODAP.

The goals of this lesson are:

• What is the relationship between the number of particles in a gas container and gas pressure?
• How can we design experiments within computational models to collect quantitative data?
• How can we analyze quantitative data using CODAP to determine a mathematical relationship between our two variables?

Before starting our computational experiments, let us familiarize ourselves with our experimental setup. The first component of our experimental setup is a slightly modified version of the Bike Tire model from the previous lesson.

1. There is no "add particles" button anymore. We will decide the number of particles inside the bike tire before running each experiment. 
2. There is a new   slider. This slider will help us run experiments until the system stabilizes (e.g., the model will stop).

Now let's begin with running three simple experiments:

1. Set the "number-of-particles" parameter to 100 and the "ticks-to-run" parameter to 15.
   1. Click the   button to set-up the experiment.
   2. Then  button to run the experiment.
       
2. Once the experiment is completed, move your mouse over the plot (as shown on the right) and see how the pressure has changed over time. Note down the final pressure of the system.
    
3. Repeat the same procedure with 200 particles and 15 ticks. Note the resulting pressure.
4. Repeat the same procedure with 300 particles and 15 ticks. Note the resulting pressure.
5. Answer the questions below the model.

In the previous lesson, we ran 3 experiments and then tried to plot the data by hand. Even with just three experiments, we  started noticing trends. However, we also noticed two important issues:

1. Reliability issue: The particles' random behavior may lead to fluctuations in our final pressure value.

2. Practicality issue: We might not be able to precisely plot the data by our hands.

Scientists try to overcome such issues by using an approach called "statistical mechanics 🔗". To put it simply, they collect lots of data and then use statistical methods to determine mathematical relationships between variables.

Below, you will see a CODAP workbench. CODAP is a web-based data analysis platform that will make it much easier for us to run many experiments at once. It will also make it much easier to plot our data and find out mathematical relationships between the variables.

Let's begin with familiarizing ourselves with the CODAP environment:

1. Run three experiments for 15 ticks: with 100 particles, with 200 particles, and with 300 particles. (p.s.: you can speed up the model to get the results more quickly)
2. Observe how the results of each experiment is automatically transferred to the "Experiment results" table on the right.
3. Drag the num_particles variable to the x-axis of the plot plot as shown in the GIF on the right.
4. Drag the last_pressure_value variable to the y-axis of the plot plot as shown in the GIF on the right.
5. Observe the resulting plot. Then answer the questions below the CODAP workbench.

Please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If your teacher did not initiate the discussion yet, you can start answering the first four questions below as you are waiting.

Now let's go ahead and conduct our fist computational experiment within CODAP!!!

Note: If you do not remember the exact details, you can see your initial experimental design below the CODAP window.

Also note: If you changed your mind after the classroom discussion, feel free to make changes to your experimental design.

A final note: Do not forget to drag variable names to the axes of your plot widget as you did in the Page 2.

So far, we collected many data points from the Bike Tire model, we plotted them, and saw a consistent relationship. We stated our findings verbally such as "the more particles in the box, the higher is the pressure".

However, verbal statements cannot help us answer questions such as: "What happens if I add 10 more particles to the box?" or "Approximately how many particles are inside the box if the pressure is 440?"

To answer such quantitative questions, we need to develop mathematical equations.

CODAP makes it trivially easy to develop such mathematical equations from data :

1. Load your CODAP savefile by clicking CODAP's hamburger menu () and then clicking "Open".
2. Open your file from the "Local File" section. You should have all your data back in the "Experiment Results" table, as well as your plot.
3. Click anywhere on the plot, and then click the ruler icon ().
4. Click the  option.
5. A line with three anchor points will appear. Move the line as shown on the right to fit the data as best as you can.
6. Note the linear equation 🔗 (with yellow background) that CODAP generates for us.

Congratulations to all of us!!! We developed a mathematical model using computational thinking!

• We analyzed a real world phenomenon using a computational model.
• We designed an experiment to find out the relationship between two variables.
• We collected data and plotted our data points on a graph.
• We used a computational data analysis tool to develop a mathematical equation.

We should now test the validity of our Pressure-Number equation through further experimentation! You can skip directly to the questions below.

In the previous two lessons, we used computational models of a bike tire to:

• describe how "pressure" as a macro-level property emerges from the micro-level interactions between the gas particles and the container's walls
• find a relationship between two variables, the number of particles and pressure, by conducting computational experiments and doing statistical analysis.

In this lesson, we are going to explore the relationship between temperature and pressure. To do so, we are going to use a new NetLogo model and the CODAP data analysis platform.

The goals of this lesson are:

• Defining temperature in a gas container.
• Understanding the relationship between gas particles' movement (kinetic energy) and gas temperature.
• Analyzing the relationship between gas temperature (independent variable) and gas pressure (dependent variable).
• Developing a mathematical model that defines the relationship between gas temperature and pressure quantitatively.

Before moving on to our computational explorations, let us consider the GIF on the right, which shows a the result of an interesting real world experiment:

1. The experimenter places a blown balloon inside liquid nitrogen*.
2. The balloon shrinks while contacting the liquid nitrogen.
3. He takes the shrunk balloon out and holds it in his hands.
4. The balloon automatically expands.

Why does the balloon expand? Let's hypothesize.

*Liquid nitrogen is a very cold substance used in food industry to preserve food by freezing it quickly. It is also used in medicine to treat skin conditions.

Please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If your teacher did not initiate the discussion yet, you can start answering the first four questions below as you are waiting.

Let us begin our exploration of the relationship between gas pressure and gas temperature with another NetLogo model. In this model, we have a fixed-volume gas container similar to our previous Bike Tire model. However, this model allows us to warm up and cool down the walls of the container. 

When a particle hits the wall, two things may happen:

• If it has more kinetic energy than the energy of the wall, it will loose some of its energy.
  • the loss of energy is dependent on the difference between the particle's energy and the wall's energy.
• If it has less energy than the energy of the wall, it will gain some energy.
  • the gain of energy is dependent on the difference between the particle's energy and the wall's energy.
• Particles only have kinetic energy. The higher a particle's kinetic energy, the faster it moves.
  • a particle will speed up if it gains energy from the wall
  • a particle will slow down if it looses energy to the wall

Now begin exploring the model:

1. Run the model by clicking "setup" and then clicking "go".
2. Wait until all three plots stabilize.
3. Warm up the walls (at least 6).
4. Observe the changes in plots.
5. Repeat the same steps, but this time cool down the walls.

Some quick notes: The temperature of a gas is related to the average speed of the particles. When the average particle speed increases, the temperature also increases. However, this is not a linear relationship. Gas temperature is proportional to the square of the average particle speed. For example, if you double the average speed, you will quadruple the temperature.

Even though temperature and average particle speed are not directly proportional, when one increases the other will too. This means we can still use them to make comparisons and develop mathematical models.

Now let's go ahead and conduct our experiment within CODAP!!!

Note: If you do not remember the exact details, you can see your experimental design below the CODAP window. If you are not sure about your experimental design, consult your teacher. Here are a few considerations to keep in mind:

1. Run each experiment long enough so that all three plots stabilize before your model reports the data.
2. Keep non-involved parameters (e.g., number of particles, ticks-to-run) fixed at all experiments.
3. Run at least 3 trials (repetitions) for each combination.
4. Try at least 5 different values for the independent variable (temperature/outside-energy).
5. Speed up the model to conduct the experiments as fast as possible.

Did you also find a mathematical equation using CODAP's "movable line" tool? If you did not, please do it now.

If you do not remember exactly, here is how to do it:

1. Load your CODAP savefile by clicking CODAP's hamburger menu () and then clicking "Open".
2. Open your file from the "Local File" section. You should have all your data back in the "Experiment Results" table, as well as your plot.
3. Click anywhere on the plot, and then click the ruler icon ().
4. Click the  option.
5. A line with three anchor points will appear. Move the line as shown on the right to fit the data as best as you can.
6. Note the linear equation 🔗 (with yellow background) that CODAP generates for us.

In Lesson 3, we explored the relationship between the number of gas particles in a container and pressure. We kept other factors (temperature and volume) constant.

In Lesson 4, we explored the relationship between gas temperature and pressure. We kept other factors (number of particles and volume) constant.

In this lesson, we are going to explore the relationship between volume and pressure.

The goals of this lesson are:

• Analyzing the relationship between container volume (independent variable) and gas pressure (dependent variable).
• Developing a mathematical model to express the relationship between container volume and gas pressure.

Once again, before moving on to our computational explorations, let us consider the experiment on the right:

1. The experimenter places a marshmallow inside an unsealed syringe.
2. He pushes the plunger all the way until the marshmallow.
3. He seals the syringe with his fingers.
4. He pulls the plunger all the way
5. The marshmallow expands.

Why does the marshmallow expand? Let's hypothesize.

Note: If your teacher has syringes (and maybe even marshmallows), you should try to conduct this experiment yourselves.

The models we used in the previous activities did not let us change the volume of the container. However, many gas containers, like balloons and syringes, grow and shrink. In this lesson, we are going to explore the relationship between container volume and pressure. To do so, we will use another slightly different model. This model will allow us to simulate a syringe; it will allow us to chance the position of a wall, so that we can increase and decrease the volume.

Begin experimenting with your model and explore the relationship between your dependent variable (pressure) and your independent variable (volume). Conduct a preliminary experiment as follows:

1. Run the model with the default parameters (initial wall position = 30).
2. Wait for pressure to stabilize. Take note. Also note the volume of the container.
3. Move the wall to the right at least 10 units above 30 (higher volume).
4. Wait for pressure to stabilize. Take note. Also note the volume of the container.
5. Move the wall to the left at least 10 units below 30 (lower volume).
6. Wait for pressure to stabilize. Take note. Also note the volume of the container.

Record the changes in the data table below.

Note: We already know that increasing the number of particles increase pressure. So, do not change the number of particles in your trials. Otherwise, your data may mislead you.

Now let's go ahead and conduct our final computational experiment within CODAP!!!

Note: If you do not remember the exact details, you can see your initial experimental design below the CODAP window. If you are not sure about your experimental design, consult your teacher before starting. 

1. Run each experiment long enough so that pressure stabilizes before your model reports the data.
2. Keep non-involved parameters (e.g., number of particles, ticks-to-run) fixed at all experiments.
3. Run at least 3 trials (repetitions) for each combination.
4. Try at least 5 different values for the independent variable (volume/wall-position).
5. Speed up the model to conduct the experiments as fast as possible.
    

As usual, let's put our mathematical model to a test and if our inverse plotted function approach paid-off!

We did a lot of things in the previous 5 lessons:

• We observed a macro-level gas phenomenon (air duster can) and we tried to reason about the underlying micro-level events such as:
  • What is a gas made up of?
  • How do gas particles move in space?
  • How do gas particles interact with each other and the walls of the container?
• We tried to qualitatively understand how micro-level interactions among gas particles lead to an important macro-level property: pressure.
  • What is kinetic molecular theory?
  • What is an ideal gas?
• We qualitatively and quantitatively explored the relationship between three variables and gas pressure: number of particles, gas temperature, container volume.
  • How can we use computational models to study the behavior of gasses?
  • How can we conduct computational experiments to: collect data, analyze data, and develop mathematical models?
  • How do we validate our mathematical models?
  • In what ways our mathematical models help us reason about real-world phenomena that involves gas pressure?
     

In this final lesson, we will reflect on our hard work and try to put together all three mathematical models, as well as our understanding of the behavior of gas particles, to develop a coherent theory of ideal gases.

We started this unit with an exploration on how the "air duster can" works. Let's revisit the air duster can again, but this time, about another interesting aspect of air duster cans (and all other spray cans), the warnings:

 .         

Can we use the knowledge we constructed through previous 5 lessons to explain why we need these warning texts on spray cans?

What we did in this unit was to re-develop three main gas theories using computational models and quantitative data analysis.

We defined the relationship between number of particles and pressure as:

• Qualitatively: As number of particles increases, pressure increases linearly (given that temperature and container volume are constant)
• Quantitatively: \((P)ressure = m_1 \times (N)umber \) , where \(m_1\) is a constant coefficient.

We defined the relationship between gas temperature and pressure as:

• Qualitatively: As gas temperature increases, pressure increases linearly (given that number of particles and container volume are constant)
• Quantitatively: \(P= m_2 \times (T)emperature \) , where \(m_2\) is a constant coefficient.

And, we defined the relationship between container volume and pressure as:

• Qualitatively: As gas temperature increases, pressure increases linearly (given that number of particles and container volume are constant)
• Quantitatively: \({P} = {m_3 \over (V)olume}\) , where \(m_3\) is a constant coefficient.

Our methodology and findings were analogous to three scientific discoveries made between 17^th and 19^th centuries:

However, as both the warning on the air duster can and the ballon-on-fire experiment showed, often times, we may not be able to explain gas-pressure related phenomena through just one variable. We need be combine our three theories and come up with one ideal gas theory. Let's try to do that!

Please set your computer aside briefly (do not close this page) and join the classroom discussion that your teacher is going to moderate.

Note: If your teacher did not initiate the discussion yet, you can start answering the first four questions below as you are waiting.

Let's review how we would put together our three theories mathematically:

Step 1: We would write them one by one:

\(P = m_1 \times N\)     _{(Eq. 1)}

\(P = m_2 \times T\)     _{(Eq. 2)}

\(P = {m_3 \over V}\)     _{(Eq. 3)}

Step 2: We would combine the right side of the equations:

\(P = {{(m_1 \times N) \times (m_2 \times T) \times (m_3)} \over V}\)     _{(Eq. 4)}

Step 3: We would slightly modify the equation by moving \(V\) to the other side of the equation and we would also get rid of the parentheses:

\(P \times V = {m_1 \times m_2 \times m_3 \times N \times T }\)     _{(Eq. 5)}

Finally, we would combine all three coefficients \((m_1 \times m_2 \times m_3)\)as one coefficient, which we can simply call as \(m_g\) or the ideal gas coefficient:

\(P \times V = {m_g \times N \times T }\)     _{(Eq. 6)}