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
Figure 1: Basic Thermodynamics
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 (º F). 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 (º C). Water freezes at exactly 0º C and
boils at 100º C. 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
two. The freezing point of water is 0º C and 32º F. A change in temperature of nine degrees Fahrenheit
corresponds to a change of five degrees Celsius, so that, for instance, 41º F is equivalent to 5º C. In
general, we can relate any temperature of yº F to any temperature of xº C 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 –273º C, 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 indepth, 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):
3𝑘𝐵 𝑇
𝑣𝑟𝑚𝑠 = √
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:
𝑣𝑟𝑚𝑠 2 ∝ 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
Tells us about the average speed of a particle.
Internal Thermal Energy
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 3𝑛𝑅𝑇
=
2
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”
𝐽
𝑅 = 8.31
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=
𝑚𝑣 2
2
𝑣𝑟𝑚𝑠 2 =
𝐾=
3𝑘𝐵 𝑇
𝑚
𝑚 3𝑘𝐵 𝑇
(
)
2 𝑚
Equation which tells you how much kinetic energy each particle has:
𝑲=
𝟑𝒌𝑩 𝑻
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