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# 3.2. Modelling a Gas

Now we have looked at a few key ideas to do with temperature, let’s apply this to study the behaviour of gases. Intuitively you are probably happy with the ideas that if you compress a gas, it’s pressure goes up; or the reasons why you should never heat a sealed can. I always like to start this topic with some classroom demonstrations to illustrate some of these behaviours. One of my favourites is the ‘crushing can experiment’. The Mythbusters did a scaled up version of this using a train carriage – take a look.

Overall, we start by building upon some of the Gas Laws ideas we started upon in GCSE, before exploring further these ideas using some kinetic theory. There are a few tricky bits related to this that the IB likes to ask questions on, so make sure you familiarise these.

The section has been divided up as follows:

The Gas Laws : Combining the 3 gas laws to obtain the ideal gas equation

Moles and Avogadro’s Number : A chemist’s best friend, what is a mole?

Kinetic Model of an Ideal Gas : How do gases behave on a molecular level?

The 3 Gas Laws

#### The 3 Gas Laws

We looked at the 3 gas laws at GCSE. These look at the effects of pressure, volume and temperature on a gas. There are three gas laws in total:

• Boyle's Law

• Charles' Law

• Pressure Law (or Avogadro's Law)

TedEd summarises these laws (the ABC's of gas) below:

Boyle's Law states that the pressure of a gas is inversely proportional to its volume, assuming a fixed mass of gas and constant temperature.

p ∝ 1 / V          (or pV = constant)

Note - this inverse proportion relationship holds true no matter what units we use, whether that be volume in m³, ml or pints; or pressure in pascals, mmHg or psi.

Plotting P against V gives our inverse proportion relationship. We can linearise this by plotting P against 1/V (or V against 1/P).

Charles' Law states that the volume of a gas is directly proportional to its absolute temperature, assuming a fixed mass of gas and constant pressure

V ∝ T          (or V/ T = constant)

Note - we only observe a proportional relationship when temperature is given in Kelvin. If other temperature scales are used, the graph still will be linear, but will intercept the x-axis at absolute zero.

Lastly, the Pressure Law (or Avogadro's Law) states that the pressure of a gas is directly proportional to its absolute temperature, assuming a fixed mass of gas and constant volume

p ∝ T          (or p/ T = constant)

Note - as with Charles' Law, we only observe our proportional relationship when temperature is given in Kelvin.

Notice each of these looks at the effect of one variable on another, while the third is kept constant. We also assume a fixed mass of gas, i.e. the number of molecules in the container is fixed (no leaks of gas).

## Required Practical - Investigate one gas law experimentally

The IB require you to investigate one of these three gas laws in class. That could be through actually performing the investigation, or you could alternatively investigate the gas laws using a simulation. There are many different variations on these practicals, but the video below talks you through how to investigate Boyle's Law experimentally. You may also be interested to watch some videos for Charles' Law and the Pressure Law.

Alternatively you can investigate these gas laws with a simulation. This one by PHET is one of my favourites on this topic. You can change the amount of gas present in the vessel, and look at the effect of changing the different variables (with one variable being kept constant). If you want to do the experiment properly, try plotting one variable against the other in Excel, then linearising to obtain a straight line relationship.

1. Plot width of vessel against pressure in atmospheres to investigate Boyle's Law (holding temperature constant) - the width is proportional to the volume here.

2. Plot temperature in Kelvin/ Celsius against vessel width (again, proportional to volume) to investigate Charles Law (holding pressure constant)

3. Plot pressure in atmospheres against temperature in Kelvin/ Celsius to investigate the pressure law (holding volume constant here).

4. You can continue investigating these with different amounts of gas, or different types of gas to investigate the effect of changing these ideas.

The Ideal Gas Law

#### The Ideal Gas Law

At IB we now can use the proportionality relationships of these three laws to derive something called the Ideal Gas Law. Combining our three laws we get:

pV/T = constant

In each of our 3 gas laws, we assumed that the mass of gas (or the number of moles of gas in the vessel) is constant. In our proportionality relationship above we need to include the number of moles of gas. Our ideal gas law therefore becomes:

We introduce a constant of proportionality here called the Gas Constant, R = 8.31 JKˉ¹molˉ¹. It is worth pointing out that with our proportionality relationships previously, the units did not matter (other than absolute temperature). As we now have an equation here, we now must use our standard base units.

Another Gas Law simulation here, this time by Geogebra. This one is quite nice as it allows you to see the mathematical relationship of pV = nRT a little more clearly and visualise the graphical relationship.

## Worked Example - Gases in two connected vessels

Q. Gas in two vessels, A and B, are connected with a thin tube. The volume of vessel B is three times that of vessel A. The vessels A and B are held at 100K and 200K respectively. What is the ratio of the mass of gas in A to the mass of gas in B?

So this sort of question is a favourite in the IB Paper 1 - using ratios and the ideal gas law for a multiple choice. Here, your clue to solving this is that these two vessels are connected - i.e. they are at the same pressure.

If they are at the same pressure, we can equate our gas laws for pressure for the two vessels.

Now, let's use some of the info given in the question to work out our ratio of nA: nB (the ratios of the masses of gas).

We know that TB = 2 TA. We also know that VB = 3VA. Simplifying and rearranging we get:

## Video Lessons

 Khan Academy Molecular Theory of Gases Ideal Gas Law Kelvin Scale and Ideal Gases Study Nova Ideal Gas Law https://www.youtube.com/watch?v=o0lgches1io&list=PL6LH0ngwf3HtX-zu3skDM2MKCUiWjVu_H&index=24 Chris Doner Gas Laws and Ideal Gases IB Specific Crash Course Ideal Gas Law Science Shorts The Gas Laws

## Resources

 IB Physics Topic 3 Notes IB-Physics.net Chapter 3 Summary IB Revision Notes Isaac Physics The Ideal Gas Law Partial Pressure not relevant Mr. G 3.2 Teaching Notes 3.2 Student Notes Physics and Maths Tutor Thermal Definitions Thermal Detailed Notes Thermal Key Points PDF Thermal Flashcards A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 3: Add Qs Topic 3: Add Qs MS Topic 3: MCQs CUP Website Link Freely available online Grade Gorilla 3.2 (Ideal Gases) MCQ Topic 3 (Thermal) End Quiz Quick IB Specific Mixed MCQs Isaac Physics The Gas Laws Mr. G 3.2 Formative Assessment Topic 3 Summary Qs IB Specific Questions Physics and Maths Tutor Ideal Gases (AQA 2) Ideal Gases MS (AQA 2) MCQ Thermal (AQA 2) MCQ Thermal MS (AQA 2) A-Level Qs: overlapping content Physics and Maths Tutor Thermal Physics (AQA 1) Thermal Physics MS (AQA 1) A-Level Qs: overlapping content
The Mole

If we look back to Chapter 1, you may remember that the SI unit for amount of something is the mole. Well, what exactly is a mole? If you are a chemist, you should already be familiar with this idea. If not, read on!

For most practical applications, it does not make sense to count the exact number of atoms – clearly even for small samples this would result in ridiculously big numbers (one glass of water contains about 10 million billion billion molecules). A mole is essentially a very big number, which we use to quantify the number of atoms. The mole is 602 214 076 000 000 000 000 000, or 6.02 x 10²³. This is commonly called Avogadro’s number, NA (and can be found in your formula book). TedEd have a nice little video here discussing the mole.

The mole is a very useful number in calculating the number of molecules in a macroscopic quantity of matter. It is defined as:

One mole (i.e. Avogadro's Number, NA) is the number of atoms in 12 grams of Carbon-12

If you are not a chemist, it's worth just making sure you are happy with how to convert between molar mass and moles. Although it's not strictly necessary for IB Physics, it is still a helpful idea when looking at ideas such as radioactive decay.

The most important equation to know here is this:

n = N / NA

Where: N = the total number of molecules (big N, big number)

n = the number of moles of molecules (small n, smaller number)

NA = Avogadro's number (6.02 × 10²³)

As an aside, the webcomic XKCD has a series called What If? which involves them answering a series of absurd hypothetical questions from readers using proper physical analysis along with quite a lot of tongue in cheek humour (it's more entertaining than it sounds, I promise!). One of my favourites is their answer to the question - What would happen if you had a mole of moles?

## Resources

 IB Physics Topic 3 Notes IB-Physics.net Chapter 3 Summary IB Revision Notes Mr. G 3.2 Teaching Notes 3.2 Student Notes Physics and Maths Tutor Thermal Definitions Thermal Detailed Notes Thermal Key Points PDF Thermal Flashcards A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 3: Add Qs Topic 3: Add Qs MS Topic 3: MCQs CUP Website Link Freely available online Grade Gorilla 3.2 (Ideal Gases) MCQ Topic 3 (Thermal) End Quiz Quick IB Specific Mixed MCQs Mr. G 3.2 Formative Assessment Topic 3 Summary Qs IB Specific Questions
Kinetic Theory

#### Kinetic Model of an Ideal Gas

Kinetic theory describes how we can explain macroscopic properties (e.g. pressure, temperature, volume) using ideas and assumptions about the movement of molecules. When you first came across the idea of pressure, you learnt that this was due to the collisions of millions of molecules with the vessel wall, each exerting a tiny force, but if these individual forces are summed over a certain area we know this as pressure.

We can model gas behaviour as the motion of many tiny discrete particles. We make a number of assumptions to make the maths easier. This gives us a theoretical model of a gas that follows these assumptions that we call an ideal gas. Some of these assumptions are:

1. The collision between particles is elastic (no kinetic energy lost)

2. The volume of the vessel is much larger than the total volume occupied by gas molecules

3. There are no intermolecular forces between gas molecules (or in other words, the molecules kinetic energy is much larger than their potential energy)

4. The time between gas molecule collisions is much larger than the duration of the collisions.

So, a few asusumptions here, but what does this mean in real life - or when does a real gas follow the behaviour of an ideal gas.

• Well, if the volume of the vessel must be much larger then the volume of the gas molecules, that means our gas should have a LOW DENSITY (if the gas density or pressure is higher, the volume of gas molecules relative becomes significant relative to the vessel volume).

• If the kinetic energy of the molecules needs to be much larger than the potential, we should have a HIGH TEMPERATURE (if the molecules move much slower, the effects of intermolecular attraction are more significant).

This Crash Course Chemistry video talks through how if we have a real gas which does not perfectly obey these assumptions we can modify our ideal gas equation to model this real-life behaviour.

## Temperature

We know that as we heat a substance, the molecules move faster. In the previous section we stated:

Temperature is proportional to the average kinetic energy of the molecules of a substance.

We can use our mathematical models and assumptions stated earlier to derive a mathematical equation relating temperature and average kinetic energy. The derivation is not required, but is an interesting application in making use of these assumptions - find it here.

This new equation is given as:

This introduces a new constant, Boltzmann's constant, that is useful when working out the average kinetic energy of each molecule. Unsurprisingly that will give us a very small number (think how small the kinetic energy of one molecule would be).

I like to think of Boltzmann's constant as the molecular version of the Gas Constant. Our Gas Law constant describes the macroscopic behaviour of gases, whereas the Boltzmann constant is used when looking at the molecular behaviour. These two constants are linked by the following equation:

Note, this equation is NOT given in this form in your formula booklet. It is hidden within another equation in your formulae for Sub-Topic 3.2. Can you spot it hidden below? Sneaky, eh?

I find these kinetic energy ideas quite hard to visualise, so here is a nice illustration to get our heads around this stuff. Going back to our previous PHET simulation - select the third tab, Energy.

In your vessel add in some blue molecules and some smaller red molecules.

On the drop down bar on left you can select a speed distribution graph and a kinetic energy distribution graph.

Both these gases are at the same temperature, which means they have the same average kinetic energy. You can see that the red molecules move much faster on average because they have a much smaller mass.

## Video Lessons

 Khan Academy Boltzmann's Constant Kinetic Theory Maxwell-Boltzmann Distribution Science Shorts Kinetic Model of Gases Chris Doner Gas Laws and Ideal Gases Applying the Ideal Gas Law IB Specific

## Resources

 IB Physics Topic 3 Notes IB-Physics.net Chapter 3 Summary IB Revision Notes Isaac Physics Kinetic Models of Gases Partial Pressure not relevant Mr. G 3.2 Teaching Notes 3.2 Student Notes Physics and Maths Tutor Thermal Definitions Thermal Detailed Notes Thermal Key Points PDF Thermal Flashcards A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 3: Add Qs Topic 3: Add Qs MS Topic 3: MCQs CUP Website Link Freely available online Grade Gorilla 3.2 (Ideal Gases) MCQ Topic 3 (Thermal) End Quiz Quick IB Specific Mixed MCQs Mr. G 3.2 Formative Assessment Topic 3 Summary Qs IB Specific Questions Physics and Maths Tutor Kinetic Theory (AQA 2) Kinetic Theory MS (AQA 2) MCQ Thermal (AQA 2) MCQ Thermal MS (AQA 2) A-Level Qs: overlapping content

Definitions and Key Words : Chapter 3

A set of Quizlet flashcards of the key words and definitions for this chapter is provided here.

## IB Questions

A question by question breakdown of the IB papers by year is shown below to allow you to filter questions by topic. Hopefully you have access to many of these papers through your school system. If available, there may be some links to online sources of questions, though please be patient if the links are broken! (DrR: If you do find some broken links, please contact me through the site)

Questions on this topic (Section 3) are shown in yellow.

Use this grid to practice past IB questions topic by topic. You can see from the colours how similar the question topic breakdown is year by year. The more you can familiarise yourself with the IB question style the better - eventually you will come to spot those tricks and types of questions that reappear each year.

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