Notes: Heat and the Ideal Gas Laws

Level 1: Temperature, Internal Thermal Energy and

Heat

Temperature Thermodynamics is the study of heat and how objects react to heat. Pretty interesting stuff.

Also, kind of the backbone of modern civilization.

Let’s start with something tricky- the difference

between temperature and heat. Temperature

is related to how much the molecules inside the

material are bouncing around. Inside of all

objects, molecules are doing a jiggle dance.

They race around, twirl, and knock into one

another. Basically, they mosh. Temperature tells

us how fast the molecules are moving around.

When the temperature is higher, the molecules

move a lot more, just like a mosh pit starts

thrashing more when a good song comes on.

Key Idea

The higher the temperature, the faster the

molecules inside the object/fluid are moving around.

In this way, you can think of temperature as a molecular speedometer. The higher the

temperature, the faster the average molecule is going inside the material. We symbolize

temperature with a capital T.

Video Demo: This video very clearly illustrates the connection between temperature

and movement inside a fluid/object.

https://www.youtube.com/watch?v=3wZ7pFEMkbE

Measuring Temperature from Sparknotes Physics SAT Prep

Degrees Celsius In the United States, temperature is measured in degrees Fahrenheit (ºF). However, Fahrenheit is

not a metric unit, so it will not show up on SAT II Physics. Physicists and non-Americans usually talk

about temperature in terms of degrees Celsius, a.k.a. centigrade (ºC). Water freezes at exactly 0ºC and

boils at 100ºC. This is not a remarkable coincidence—it is the way the Celsius scale is defined.

SAT II Physics won’t ask you to convert between Fahrenheit and Celsius, but if you have a hard time

thinking in terms of degrees Celsius, it may help to know how to switch back and forth between the

Figure 1: Basic Thermodynamics

two. The freezing point of water is 0ºC and 32ºF. A change in temperature of nine degrees Fahrenheit

corresponds to a change of five degrees Celsius, so that, for instance, 41ºF is equivalent to 5ºC. In

general, we can relate any temperature of yºF to any temperature of xºC with the following equation:

Kelvins In many situations we are only interested in changes of temperature, so it doesn’t really matter

where the freezing point of water is arbitrarily chosen to be. But in other cases, as we shall see when

we study gases, we will want to do things like “double the temperature,” which is meaningless if the

zero point of the scale is arbitrary, as with the Celsius scale.

The Kelvin scale (K) is a measure of absolute temperature, defined so that temperatures expressed in

Kelvins are always positive. Absolute zero, 0 K, which is equivalent to –273ºC, is the lowest theoretical

temperature a material can have. Other than the placement of the zero point, the Kelvin and Celsius

scales are the same, so water freezes at 273 K and boils at 373 K.

Practice Problem Which would be the most comfortable temperature for your bath water?

(A) 0ºC (B) 40 K (C) 110ºC (D) 310 K (E) 560 K

Solution Answer D

Comfortable bath water should be slightly above room temperature. Room temperature is about

20ºC, or 293 K

Practice Problem What temperature change on the Kelvin scale is equivalent to a 10 degree change on the Celsius

scale?

(A) 283 K (B) 273 K (C) 18 K (D) 10 K (E) 0

Answer Answer: D

While temperatures are different on the Celsius and Kelvin scale, the temperature intervals are

identical. 1ºC = 274 K, but 1 Cº = 1 K

Internal Energy It takes energy to move those molecules around. Imagine a cup of water at room temperature.

Now zoom in on one small molecule blazing around inside. That molecule has a certain amount of

kinetic energy. It’s very small (because this is a tiny tiny mass) but it is present. What if we were

to add up all the energies of all the molecules in the glass? We would then know the total amount

of thermal energy, or internal thermal energy, in the cup. Internal thermal energy is the sum

of the energies of all of the molecules moving in a substance. We symbolize internal thermal

energy with a U. Why? I have no idea. At least the units are familiar. Internal energy is measured,

as you might suspect, in Joules (J).

Internal thermal energy is related to temperature, but it is also dependent upon the mass of the

object. A teacup full of hot water would have a higher temperature than a large bucket of water

at room temperature. This means the average molecule inside the teacup is moving around

much faster than the average molecule moving around inside the bucket. The bucket of water

might have a higher internal thermal energy, however, because it has more molecules. Its total

thermal energy could be greater.

Supplemental: Thermal Energy Cartoon. If the previous discussion was

confusing, try watching this old cartoon about the difference between

temperature and internal thermal energy. Cute but not required. Can be helpful

if you are lost, however.

https://www.youtube.com/watch?v=wTi3Hn09OBs

Heat So what is heat? Imagine that you have a warm cup of delicious Starbuck’s coffee sitting on your

table. Unfortunately, it is currently way too hot to drink, maybe 80 degrees Celsius (whew!). The

cup has a lot more energy than its surroundings. The universe doesn’t like this- it wants

everything to be equal, so the extra energy in the cup begins to transfer into the table top and

into the air around it, trying to reach a balance. This is heat. Heat is the transfer of thermal

energy from one object to another. If you were to come up and put your hand on the cup, you

would experience this heat as a burning rush of energy through your poor, tender skin.

The same thing happens when you touch a cold object- only in this case, you have more energy

and the universe is nudging that energy away from you in an attempt to reach thermal

equilibrium- a state where the thermal energies of objects are balanced.

Oh yeah, and we symbolize heat with a capital Q. Why? Because physics is sadistic. Heat is a

type of energy (transfer) so we measure it in Joules (J), too.

We will return to these key ideas in future sections. For now, let’s look at how different materials

react to heat.

Level 2: Linear Expansion Most objects expand when heated and contract when cooled, with water being the most notable

exception (see the last unit, if this is hazy). We call this expansion when objects are heated

thermal expansion. Objects expand (and contract) at different rates, depending upon what they

are made of. We describe how much these objects will expand using a coefficient called the

coefficient of linear expansion, which we symbolize with 𝛼 (alpha). The chart below lists a few

common materials. Objects with a lower coefficient will not expand as much as those with a

higher coefficient.

This coefficient can help us figure

out how much a material will

expand when heated. Imagine a rod

made of a particular metal. If it

experienced a temperature change,

we could calculate its expansion

using the following equation:

∆𝑙 = 𝛼𝑙0∆𝑇

where ∆𝑙 is the change in the

length, 𝛼 is the coefficient of linear

expansion and Δ𝑇 is the change in

temperature (which needs to be in K). lo is the original length of the object.

These are nice mathematically because they are mostly plug n’ chugs.

Practice Problem: Units for the Coefficient of Thermal Expansion Use the equation above to figure out what the units for the coefficient of thermal expansion will

be.

Answer 1/C° or 1/K

Thermal Expansion In The Real World There are lots of practical uses for thermal expansion. One of the best involves bimetals.

Bimetals are basically two metals stuck together (see the example on the right). They can’t

separate from one another. What do you think will happen if you heat up a bimetal?

Engineering Video: Mr. Wizard is a Science BOSS. He’s going to show you a

bimetal and then use it to explain how thermostats work.

https://www.youtube.com/watch?v=6r9UAdb2kDo

This coefficient of thermal expansion doesn’t just effect how much a material expands. It also

effects how much it contracts. Materials with higher coefficients of thermal expansion also

contract more. That means that when the bimetal is cooled we will see it act like this:

Quick question- which material has the higher coefficient of

thermal expansion- invar or brass?

Here’s a piece of thermal expansion that has come in handy many times in the past. Imagine you

a stuck jelly jar. You’ve tried everything, but nothing is working. How could you use thermal

expansion to get it open? Think about the fact that the body of the jar is made of glass and the lid

is usually made of steel. What could you do, given the materials in your kitchen?

Okay, now take a look at the picture below and take a guess what happened. The answer is not

“earthquake”.

Engineers (which some of you will be, very soon) need to think about how the substances they

work with will change when heat is applied. Even huge objects, like bridges, expand and

contract in the heat. Civil engineers (people who design bridges, tunnels, roads…) have

interesting way of dealing with this. Here’s one:

Thermal Expansion with 2D and 3D Objects Now that we’ve talked about how to calculate how much an object lengthens when heated (and

contracts when cooled). What if we are interested in objects with dimensions. For example, a

burner plate around a stove is a square with a hole in it. How would this be effected by heat?

You can think of thermal expansion as enlarging a photograph- the dimensions will stay the same

relative to one another, the object will just become larger.

This is even true of objects with holes in it. The hole will expand as well.

Let’s talk about what happens to the volume of an object as it expands. This is particularly useful

because many of the materials we are going to talk about- such as liquids and gases- don’t have

a length. They are only described by their volume.

This isn’t all that different. Actually, it’s pretty close to having the same equation:

∆𝑉 = 𝛽𝑉𝑜∆𝑇

In this case, we use a different coefficient, the coefficient of volumetric expansion. Each

material has a different value for 𝛼 and 𝛽.

Twu Thermal Video 4: This is the hardest problem I think they could ask you

about these. Try to solve it yourself first, and then watch her solve it. Then, watch

her demo about the ring and ball. It is important to follow this example in order to

answer one of the most typical problems on the AP exam.

https://www.youtube.com/watch?v=Zz3d0tI0FYk#t=34

Key Idea:

When an object expands, it keeps the same relative dimensions. All parts expand.

Practice Problem: Dimensions of Expansion A square steel plate with sides of length 1.00 m has a hole in its center 0.100 m in diameter. If the entire plate is

heated to such a temperature that its sides become 1.01 m long, the diameter of the hole will be

(A) 0.090 m (B) 0.099 m (C) 0.100 m (D) 0.101 m (E) 0.110 m

Answer In linear expansion, every linear dimension of an object changes by the same fraction when

heated or cooled.

Practice Problem: Calculating Thermal Expansion You need to slide an aluminum ring onto a rod. At room temperature (about 20°C), the

internal diameter of the ring is 50.0 mm and the diameter of the rod is 50.1 mm.

(a) Should you heat or cool the ring to make it fit onto the rod? (b) If the coefficient of

linear expansion for aluminum is 2.5 x 10"5 °C~1, at what temperature will the ring just

barely fit onto the rod?

Solution

(a) You should heat the ring. You might initially think that because you want the inner diameter

of the ring to expand, you want the aluminum itself to shrink. However, when a material expands,

it expands in all directions—thus, the hole will expand by the same amount as the surrounding

metal when heated.

(b) In the length expansion equation, AL is the amount by which we need the diameter of the ring

to increase, 0.1 mm. Remember to convert to meters before solving.

0.0001 m = (2.5 x 1O-5°C-')(.O5O m)(A7)

Solving for AT, we find that we need to increase the ring's temperature by 80°C. So the final

temperature of the ring is (20 + 80) = 100°C.

Quick question: Anybody here do Minecraft? I don’t, but when I type in “Thermal Expansion” into

youtube, I get ten million hits about Minecraft. Anyone want to explain to me what this is and if it has

anything to do with what we just learned?

Level 3: Kinetic Molecular Theory Time for a science rerun!

That’s right, we are returning to a beloved topic in chemistry- the

ideal gas laws! If you are a very physics-y person, this was

probably one are of chemistry you thought was actually pretty

interesting. Why? Because it was all about physics. If you have taken

AP Chem, a bunch of this will be quite familiar. Be ready to help

your group get caught up.

By the way, any of you medical/biological types definitely want to

pay attention to this section. It comes up a bunch in treating

people, especially those with difficulty breathing.

We are going to return to ideal gas laws, but we are going to first explore these from a more in-

depth, physic-y fashion. Then, we are going to go past what you learned in chemistry, and

discuss how these laws are actually used to maintain advanced civilization as we know it (no

joke. This stuff is for realz).

Assumptions of Kinetic Theory In order to understand the ideal gas laws, let’s talk about how gases move. Interactions between

molecules can be very complex, but we do have a set of general laws that can simplify these

interactions and make them easier to understand. This easier way of looking at how molecules

bounce around in a gas is called kinetic theory. Kinetic theory makes wildly difficult

mathematical calculations into straight forward algebra problems. There is, however, a catch.

You can only use kinetic theory if we make a few key assumptions. They are listed below.

ASSUMPTIONS OF KINETIC THEORY

(from McGraw AP Review)

• Molecules move in continuous, random motion.

• There are an exceedingly large number of molecules in any container of gas.

• The separation between individual molecules is large.

• Molecules do not act on one another at a distance; that is, they do not exert electrical or

gravitational forces on other molecules.

• All collisions between molecules, or between a molecule and the walls of a container, are

elastic (i.e., kinetic energy is not lost in these collisions).

Spoiler alert! Pretty much all the problems you will deal with in this class will take for granted

that the assumptions above are correct. Gases at room temperature, for example, follow kinetic

theory pretty closely. However (and this sucks) you have to memorize the above. Why? Because

the new AP test is heavily focused on conceptual understanding, and I think that it is likely you

will get a question that asks you to list the assumptions of kinetic theory.

Online Flashcards: Don’t worry, I got your back with this memorization thing.

Here are online flash cards. You can even play matching games until you get

these. Don’t do them too much in class. though. That’s a waste of precious group

time.

Kinetic Theory Once we’re pretty sure that kinetic theory is supported by the above assumptions (which we will

be 99% of the time), we can actually use kinetic theory to solve problems. What does kinetic

theory do? It helps us relate the temperature of the gas to the average velocity of particles in the

gas. It also lets us calculate the internal thermal energy, U, of the gas (if any of these terms is

hazy for you, go back and look at Level 1).

In Level 1, we talked about how temperature is a little like a speedometer for gases- the higher

the temperature, the faster the individual particles inside the gas are moving. We have a

mathematical way of saying this (of course):

𝑣𝑟𝑚𝑠 = √3𝑘𝐵𝑇

𝑚

vrms= average velocity of a particle. The rms refers to a particular statistical way of calculating

the average, which you’ll learn in later physics (and chemistry) classes.

T= temperature (keep in Kelvins) to save yourself a headache.

m= mass of one molecule of gas

kB = This is a constant. It is such an important constant it has a name: The Boltzmann Constant.

It’s actually pretty important in higher level physics, but we won’t get into that here. For now,

you just need to program it into your calculator. I’d put it under “B”. Here it is:

The Boltzmann Constant

kB = 1.3806488 × 10-23 m2 kg s-2 K-1

We keep talking about how this is the average speed. Not all the molecules in the gas are moving

the exact same speed. Some of them are moving a bit slower, some a bit faster. On average,

however, this is how fast they are typically going. All this, of course, supports the key idea we

discussed before:

Key Idea

The higher the temperature, the faster the molecules inside the object/fluid are moving

around.

Notice that the relationship if we look at the proportional relationship between the velocity and

temperature, it isn’t one to one. You can think of it this way:

𝑣𝑟𝑚𝑠2 ∝ 𝑇

That means if you double the velocity of the particles, you are making the temperature four times

as large. Tripling it increases the temperature 9 times…

Earlier, we talked about how the temperature is measure of the molecular speed, but internal

thermal energy tells us how much energy the entire collection of gas has.

Temperature Internal Thermal Energy

Tells us about the average speed of a particle. Tells us about the energy in the whole system

Imagine a balloon. The temperature tells us how fast the average particle in the balloon is

moving. The internal thermal energy tells us how much thermal energy the balloon has as a

whole.

Of course, how much energy the balloon as a whole has is related to how fast each particle is

moving and how much material is in the balloon. If we add up the energy of each of the particles

inside, we’d know how much energy the total balloon has. Here is the relationship expressed

mathematically:

𝑈 =3𝑁𝑘𝑇

2=

3𝑛𝑅𝑇

2

These are two different equations that you need to memorize. Let’s look at each of them.

Use this equation if you are (for some weird reason) using a problem involving the number of

molecules in a system. This equation is used a lot less often.

𝑈 =3𝑁𝑘𝑇

2

U= internal thermal energy (J)

N= the number of molecules in the system (balloon, box, piston..) (unit less- as though they were

counted

T- Temperature (in K)

Use this equation if you are solving a problem involving the number of moles.

𝑈 =3𝑛𝑅𝑇

2

U= internal thermal energy (J)

n= the number of moles in the system (balloon, box, piston..) (mol)

T- Temperature (in K)

R Another constant. Program it into your calculator under “R”

𝑅 = 8.31 𝐽

𝑚𝑜𝑙 ∙ 𝐾𝑒𝑙𝑣𝑖𝑛

Practice Problem The average speed of the atoms of a gas at 100 K is 200 m/s. What would most nearly be the average speed of

the atoms at 300 K?

(A) 67 m/s (B) 140 m/s (C) 200 m/s (D) 350 m/s (E) 600 m/s\

Answer Answer: D

One last equation for kinetic theory. Let’s talk about how we can calculate the energy of a typical

speeding particle inside a system. Remember, when we talk about the internal thermal energy

caused by the system, we are essentially talking about all the kinetic energies of the particles

moving around. If you remember from physics last year:

𝐾 =𝑚𝑣2

2

𝑣𝑟𝑚𝑠2 =

3𝑘𝐵𝑇

𝑚

𝐾 =𝑚

2(

3𝑘𝐵𝑇

𝑚)

Equation which tells you how much kinetic energy each particle has:

𝑲 =𝟑𝒌𝑩𝑻

𝟐

As you would suspect, K is measured in Joules (J).

Level 4: Ideal Gas Laws Okay, now that we have that in order, let’s review the ideal gas laws. I know you had a bunch of

this in chemistry, so I won’t spend too long on it. One thing to keep in mind as you read through

the next section is “STP” stands for standard temperature and pressure- in other words, normal,

everyday conditions. This just means T=0° C=273 K and pressure is our typical pressure of

101300 Pa.

Practice Problem: Solving an Ideal Gas Problem 1 A sample of gas occupies a volume of 1.0 liter at a pressure of 1.0 105 Pa (1 atm of pressure) at a

temperature of 300 K. If the pressure is increased to 3.5 105 Pa, and the temperature is held constant,

what will the volume become?

Solution

Practice Problem: More Ideal Gas A sample of carbon dioxide gas (CO2) at STP has a mass of 20.0 g. What is its volume? Assume carbon dioxide has

a 44g per mol.

Solution

Twu Thermal Video 6: Watch a Practice Problem This is more the level of

difficulty you could expect on the AP test. Don’t worry if it is a little hard- AP

problems are always hard. Solve it yourself first, if you can, and then watch her

solve it.

https://www.youtube.com/watch?v=xBMIaUNZ_jc#t=41

Twu Thermal Video 8: Watch a Practice Problem This type of problem is very

typical of the AP test, and I suspect it will be increasingly common on the newer

AP tests.

https://www.youtube.com/watch?v=uutSKm-Rxvo

Level 5: Heat and Heat Transfer In previous sections, we explored the average velocity of the particles in a gas, and how they

relate to the temperature. We looked at how temperature relates to internal thermal energy. The

one area we’ve neglected is heat, which we discussed briefly in Level 1.

Misconceptions about Heat: Watch this video. Very good and clears up a very

common misconception about heat.

http://www.youtube.com/watch?v=vqDbMEdLiCs

Heat, again, is the transfer of thermal energy. Something feels hot because it is transferring

thermal energy to your body. Something feels cold because it is transferring thermal energy

away from your body. All objects in a set space, as long as they’ve been there for awhile and

allowed to reach equilibrium, will have the same temperature. This does not mean they will all

feel the same, however. Some objects are better able to conduct heat away from your body (as

the video discussed) so they feel cool. Others are more sluggish in this regard, so they feel

warmer.

Because heat is a transfer of energy, we can use Joules for it units. However, it is also often

written in chemistry’s favorite energy units, the calorie.

4.186 𝐽 = 1𝑐𝑎𝑙

The calorie is a very small unit, so we often see it written in kilocalories, which is also called

Calories, but (that’s right) with a capital “C”.

4186 𝐽 = 1 𝑘𝐶 = 1 𝐶

Do you need to memorize this? No. It will be given to you on the test. Not worth you time, just

making you aware. Convert if necessary.

Three Ways of Transferring Heat There are three ways you can transfer heat.

1. Conduction

This is the way you typically think of heat being

transferred- through touch. For example, imagine a

wall, one side in a warm and toasty kitchen, the other

facing a cold, snowy alley. The heat inside the house is

being conducted outward- in other words, the energy

inside the house makes the molecules on that side of the

wall heat up and move around more. This energy is

transmitted through the wall by bouncing particles, until

outer edge of the wall will begin to transfer heat into the

snowy alley.

The equation for heat conduction is here:

𝑄

𝑡=

𝑘𝐴(𝑇1−𝑇2)

𝐿

Q= Heat (J)

t= time

T = Temperature (these are the temperature on either side of the wall)

A= the area of the wall that heat is being conducted through

L= the width of the wall (in the picture, this is shown as x- sorry!)

k= the coefficient of thermal conductivity

The coefficient of thermal conductivity depends on what the conducting material is made of.

Different materials have different values. This was why the hard drive in the video felt colder

than the book in the video- it conducts heat away from your body faster than the book.

Quick question: Would you expect the hard drive to have a higher or lower coefficient of thermal

conductivity than the book? Look at the equation if you get stuck. What should the units for the

coefficient of thermal conductivity be?

2. Convection

Convection Demo and Mini-Lecture This guy has a super soothing voice. Watch

this short video to understand how convection works.

3. Radiation

This deals more with the freaky edges of this class. We will get more into this when we talk about

quantum and nuclear physics in a later unit. For now, you should know that radiation is a type of

heat transfer that does not require a movement- in other words, it can go through space, without

touching anything. Which, if you really think about it, is very strange. An example of radiation

would be the light given off by the sun and the stars.

Supplemental: Meteorology Interested in this stuff? Here is a possible job

opportunity for you. Extra exciting at this point in history, because these days

different parts of the world (like Australia) are experiencing radical weather

changes and desperately need improved weather prediction accuracy.

http://www.youtube.com/watch?v=vqDbMEdLiCs