Unit 9 · Topic 9.1 · about 30 minutes · 7 parts · 11 scored checks, 2 of them multiple choice

AP Physics 2: Algebra-Based 9.1: Kinetic Theory of Temperature and Pressure

Q: What does the temperature of a gas measure?

The average kinetic energy of its atoms, Kavg = 32kBT, with T in kelvins. It describes the whole gas, since its atoms have a spread of speeds.

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Unit 9: 15 to 18% of the multiple-choice section

What you'll be able to do

Describe the pressure a gas exerts on its container in terms of the collisions of its atoms, and explain why pressure exists everywhere in the gas.
Describe the temperature of a gas in terms of the average kinetic energy of its atoms, find rms speeds, and read how the spread of atomic speeds changes with temperature.

Key vocabulary

Term	What it means	The exam trap
Kelvin scale	The absolute temperature scale, in kelvins (K), which starts from absolute zero: $T = T_{C} + 273$ .	A kelvin is the same size as a Celsius degree, so a temperature change is the same number on both scales, but a temperature itself is not.
Root-mean-square speed	The square root of the average of the squared speeds of the atoms in a gas, $v_{rms}$ : the speed an atom with the average kinetic energy would have.	It describes the whole gas, so do not read it as the speed of any one atom you pick out.
Boltzmann constant	The constant $k_{B} = 1.38 \times 10^{- 23}$ J/K that links a temperature to the energy of single atoms.	It works per atom, so use it with a number of atoms, never with a number of moles.

An atom moves up and to the right, strikes a wall at 40 degrees from straight on, and bounces off up and to the left at the same speed. Beside it, each velocity is split into two parts. The part perpendicular to the wall points into the wall before the bounce and out of it after, with the same size; the part along the wall points up both times, unchanged. — In an elastic bounce, only the part of the velocity perpendicular to the wall reverses.

Part 1 Think first A tank in the sun

Commit to a guess first. Being wrong now makes the right idea stick better.

Predict first

A sealed steel tank of helium sits in the sun all afternoon, and the helium inside warms from 20 °C to 50 °C. No gas gets in or out. What changed about the helium atoms?

In short. Kinetic theory explains a gas's pressure as the push of its atoms colliding with surfaces, $P = \frac{F_{⟂}}{A}$ , and its temperature as the average kinetic energy of its atoms, $K_{avg} = \frac{3}{2} k_{B} T$ .

Only the part of each force perpendicular to a surface adds to the pressure, and pressure exists everywhere in the gas, not only at the walls.
At one temperature every gas has the same average kinetic energy per atom, so lighter atoms move faster: $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$ , with $T$ in kelvins.
Atoms have a spread of speeds. Heating a gas moves the peak of the distribution to higher speeds and makes the curve lower and wider.

Part 2 Learn it Atoms in motion

2a. Temperature is average kinetic energy

The atoms of a gas fly about in every direction. The temperature of the gas is a measure of the average kinetic energy of those atoms:

K_{avg} = \frac{3}{2} k_{B} T = \frac{1}{2} m v_{rms}^{2}

Here $k_{B} = 1.38 \times 10^{- 23}$ J/K is the Boltzmann constant, $m$ is the mass of one atom, and $T$ is the absolute temperature in kelvins: $T = T_{C} + 273$ . At 0 °C atoms still move fast, so a Celsius temperature never goes into this equation.

$v_{rms}$ is the root-mean-square speed: square each atom's speed, average the squares, then take the square root. It is the speed an atom with exactly the average kinetic energy would have.

The equation has no mass in its middle term, so every gas at the same temperature has the same average kinetic energy per atom. Since that energy is $\frac{1}{2} m v_{rms}^{2}$ , lighter atoms move faster: $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$ .

For helium at 300 K, with $m = 6.64 \times 10^{- 27}$ kg, $K_{avg} = 6.21 \times 10^{- 21}$ J and $v_{rms} = 1370$ m/s.

Quick check · Calculate

Helium atoms have a mass of $6.64 \times 10^{- 27}$ kg. Treat helium as an ideal gas and take $k_{B} = 1.38 \times 10^{- 23}$ J/K. At what temperature is the rms speed of the atoms in a sample of helium 1000 m/s? Give your answer in kelvins, to the nearest 1 K.

Your answer K

2b. A spread of speeds

Atoms in a gas do not all move at $v_{rms}$ . Collisions keep trading energy between them, so at any instant some are nearly still and a few are very fast.

The Maxwell-Boltzmann distribution is a graph of how many atoms have each speed, for a gas at one temperature. You will not need its formula, only how its features follow the temperature:

It starts at zero at zero speed, rises to a peak at the most common speed, and has a long tail at high speeds.
The area under the curve stands for the number of atoms, so for a given sample it does not change.
Heating the gas moves the peak to a higher speed. Since the area stays the same, a curve that spreads out to higher speeds must also get lower.
At the same temperature, a gas of heavier atoms has its peak at a lower speed, with a taller, narrower curve.

So a temperature describes the whole crowd of atoms, never one atom.

Quick check · Match

Tap a change or feature of a Maxwell-Boltzmann speed distribution, then what it means.

2c. Pressure comes from collisions

Atoms in a gas collide with each other and with the walls of their container. Treat an atom bouncing off a fixed wall as an elastic collision, so it leaves at the same speed. As the picture at the top of this lesson shows, the bounce reverses the part of the velocity perpendicular to the wall and leaves the part along the wall unchanged.

Reversing that perpendicular part changes the atom's momentum by $2 m v_{⟂}$ , all of it perpendicular to the wall. A change in momentum needs a force, so the wall pushes the atom outward, and by Newton's third law the atom pushes the wall outward just as hard. Conservation of momentum describes collisions between pairs of atoms the same way.

Billions of these short pushes every second add up to a steady force. Pressure is the total of the perpendicular parts of those forces divided by the area they act on:

P = \frac{F_{⟂}}{A}

Pressure is measured in pascals: 1 Pa = 1 N/m². The air around you is at about $1.0 \times 10^{5}$ Pa. A force along the surface adds nothing to the pressure.

Quick check · Decide and justify

Two identical atoms strike a flat, fixed wall at the same speed and bounce off elastically. Atom A moves straight at the wall. Atom B arrives at 60° from the straight-on direction. A student says the two collisions give the wall the same push, because the atoms have the same mass and the same speed and both bounce off elastically. Is the student right?

Which reason supports your answer?

2d. Pressure everywhere in the gas

Pressure is not only at the walls. Picture a thin sheet held anywhere inside the gas. Atoms strike it from both sides, just as often and just as hard as they strike a wall, so the gas presses on both faces. Pressure exists at every point in the gas.

That is why a pressure gauge pushed into the middle of a tank reads the same as one on the wall, once the gas has settled and if gravity is small enough to ignore. It is also why the air presses on all sides of your hand, not only on the side facing the sky.

Pressure has no direction of its own. It acts on any surface you put in the gas, and the force it makes on that surface is perpendicular to the surface, with size $F_{⟂} = P A$ . On a diagram, pressure is a number at a point, never an arrow.

Quick check · Multiple choice

A sealed tank holds a gas that has settled, with every part at the same temperature, and gravity can be ignored. A small, thin pressure sensor is held in the middle of the tank, far from the walls, and an identical sensor is fixed to one wall. How do their readings compare?

Common mistakes

The mistake	Why it is wrong	Do this instead
Using a Celsius temperature in $K_{avg} = \frac{3}{2} k_{B} T$ .	The average kinetic energy is proportional to the absolute temperature. At 0 °C the atoms still move at hundreds of meters per second.	Add 273 to a Celsius temperature before it goes into any kinetic theory formula.
Taking a ratio of squared speeds as the ratio of the speeds.	Kinetic energy goes as $v^{2}$ , so energies and masses fix $v_{rms}^{2}$ , not $v_{rms}$ .	Find the ratio of the squared speeds, then take its square root: four times the temperature gives twice the speed.
Treating pressure as a force, or as something found only at the walls.	Pressure is a perpendicular force per unit area, and a settled gas has it at every point.	Multiply a pressure by an area to get the force on a surface, and never draw pressure as an arrow.

Part 3 Watch it done How fast are argon atoms?

Predict first, then watch each step, then finish the problem yourself.

A cylinder holds argon gas at 27 °C. Each argon atom has a mass of $6.63 \times 10^{- 26}$ kg. Treat the argon as an ideal gas and take $k_{B} = 1.38 \times 10^{- 23}$ J/K. What is the rms speed of the argon atoms?

Predict first

Before you open the first step, predict the rms speed of the argon atoms.

Step 1 · Change the temperature to kelvins
$T = 27 + 273 = 300$ K.
Step 2 · Find the average kinetic energy
$K_{avg} = \frac{3}{2} k_{B} T = \frac{3}{2} (1.38 \times 10^{- 23}) (300)$ That is $6.21 \times 10^{- 21}$ J, the average for one atom. Some atoms have more and some less.
Why this step?
Why must the temperature be in kelvins?
The average kinetic energy is proportional to the absolute temperature, which is zero only where the atoms would have no kinetic energy at all. At 0 °C argon atoms still move fast, but a Celsius temperature would give them no energy, and at 27 °C it would give far too little.
Your step · Calculate

Finish it. Argon atoms, each of mass $6.63 \times 10^{- 26}$ kg, in a gas at 300 K have an average kinetic energy of $6.21 \times 10^{- 21}$ J. What is their rms speed? Give your answer to the nearest 1 m/s.

Your answer m/s

Part 4 Try it Lighter, hotter, faster

Mass and temperature both set how fast atoms move. Predict first: is argon at 1200 K faster than neon at 600 K, slower, or the same speed? An argon atom has about twice the mass of a neon atom.

Try it · Rank it

Four samples of ideal gas are listed with the mass of one atom, in atomic mass units (u), and the temperature. Sample A: helium, 4 u, at 300 K. Sample B: neon, 20 u, at 600 K. Sample C: argon, 40 u, at 300 K. Sample D: argon, 40 u, at 1200 K. Rank the samples by the rms speed of their atoms.

Fastest first

Give each one a place, 1 for first. Give two the same place if they are equal.

Sample A
Sample B
Sample C
Sample D

Your ranking: none yet

Now go from pressure to force on a wall.

Try it · Calculate

A box holds a gas at a pressure of $1.2 \times 10^{5}$ Pa. One flat wall of the box measures 0.30 m by 0.50 m. What is the size of the total force the gas inside exerts on that wall? Give your answer in kilonewtons, to the nearest 1 kN.

Your answer kN

Part 5 Explain it Helium and argon at one temperature

On the exam this is handwritten, so answer on paper first. Nothing you type here leaves this page. Then check your answer against the model and each scoring point.

Free response is 50% of the exam score. This one is shaped like Question 4, Qualitative/Quantitative Translation.

Free-response practice · Question 4 style

Two identical rigid containers hold the same number of atoms at the same temperature $T$ . Container 1 holds helium, whose atoms have mass $m$ . Container 2 holds argon, whose atoms have mass $10 m$ . Treat both gases as ideal.

(a) Without using equations, explain how the average kinetic energy of a helium atom compares with that of an argon atom, and which atoms typically move faster.

(b) Derive an expression for the ratio $\frac{v_{H e}}{v_{A r}}$ of the rms speeds of the two kinds of atom.

(c) On paper, draw one helium atom and one argon atom, each moving at its gas's rms speed, hitting a wall head-on and bouncing straight back elastically. Draw an arrow for each atom's momentum before and after, with lengths in proportion. Do this by hand before you open the model. Then use your drawing and your answer to (b) to explain whether a student is correct who claims that the argon exerts the lower pressure, because its atoms move more slowly and so hit the walls less often.

Your answer stays on this page. It is never sent or saved.

Model answer

(a) Both gases are at the same temperature, and temperature measures the average kinetic energy of the atoms, so a helium atom and an argon atom have the same average kinetic energy. An argon atom has ten times the mass, so to have the same kinetic energy it must move more slowly. Helium atoms are typically faster.

(b) $\frac{1}{2} m v_{H e}^{2} = \frac{3}{2} k_{B} T = \frac{1}{2} (10 m) v_{A r}^{2}$ , so $v_{H e}^{2} = 10 v_{A r}^{2}$ and $\frac{v_{H e}}{v_{A r}} = \sqrt{10} \approx 3.2$ .

(c) Compare your drawing with the model below. A head-on bounce reverses the momentum, so the wall receives $2 m v$ each hit. The argon atom's momentum is $(10 m) \frac{v_{H e}}{\sqrt{10}} = \sqrt{10} m v_{H e}$ , so its arrows are about 3.2 times as long as the helium atom's. The student is incorrect. Argon atoms do hit the walls less often, by the same factor of $\sqrt{10}$ from (b), since they cross the container more slowly. But each hit pushes $\sqrt{10}$ times as hard. The two factors cancel, so the pressures are equal, which agrees with (a): equal average kinetic energies.

Tap to open the picture full size

One correct drawing. Yours needs equal and opposite arrows for each atom, with the argon arrows about 3.2 times as long.

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Explains without equations that equal temperatures mean equal average kinetic energies, so the heavier argon atoms move more slowly.
Derives $\frac{v_{H e}}{v_{A r}} = \sqrt{10}$ from $\frac{1}{2} m v_{rms}^{2} = \frac{3}{2} k_{B} T$ .
Draws equal and opposite momentum arrows before and after each bounce, with the argon arrows about 3.2 times the helium arrows.
States that the claim is incorrect: argon hits less often by the factor found in (b), but each hit delivers that much more momentum, so the pressures are equal.

Part 6 Check yourself Two samples of neon

Both case questions use the graph.

Two samples of neon gas hold the same number of atoms. One is at 300 K and the other at 900 K. The graph shows how many atoms in each sample have each speed. The curves are labeled X and Y, and the graph does not say which is which.

A graph of how many atoms have each speed, against speed in meters per second from 0 to 2500, for two samples of neon with the same number of atoms. Curve Y rises steeply to a tall, narrow peak near 500 m/s and falls to almost nothing by about 1500 m/s. Curve X rises more gently to a lower, broader peak near 860 m/s and has a long tail reaching past 2000 m/s. The vertical axis has no numbers. — Speed distributions of two neon samples with the same number of atoms.

Case question 1 · Multiple choice

Using the graph of samples X and Y: which sample is at 900 K, and what in the graph shows it?

Case question 2 · Calculate

Using the graph of samples X and Y: the rms speed of the atoms in the 300 K sample is 609 m/s. What is the rms speed of the atoms in the 900 K sample? Give your answer to the nearest 1 m/s.

Your answer m/s

The rest stand on their own and do not use the case.

Check yourself · Fill the statement

Tap a blank, then an answer.

An atom that bounces off a wall changes its , so the wall pushes on the atom and the atom pushes back on the wall. Only the part of each atom's force adds to the gas's pressure on the wall. The pressure is the total of those parts divided by .

Check yourself · Spot the error

A student compares sample P, helium at 300 K, with sample Q, argon at 600 K. An argon atom has 10 times the mass of a helium atom. Tap the first line that goes wrong.

Part 7 Keep it Hold on to it

This is the first lesson of the course, so there is nothing earlier to bring back yet. From the next lesson on, two questions from earlier lessons appear here.

In three lines

Pressure is the push of atoms colliding with a surface: the perpendicular parts of their forces divided by the area, $P = \frac{F_{⟂}}{A}$ , and it exists everywhere in the gas.
Temperature measures the average kinetic energy of the atoms, $K_{avg} = \frac{3}{2} k_{B} T = \frac{1}{2} m v_{rms}^{2}$ , with $T$ in kelvins, so at one temperature lighter atoms move faster.
Atoms have a spread of speeds; heating a gas moves the peak of its speed distribution to higher speeds and makes the curve lower and wider, with the same area.

Exam strategy for this topic

Section I, Multiple Choice: 42 questions in 85 minutes.
Section II, Free Response: 4 questions in 95 minutes.
This lesson's Explain it is shaped like free-response question 4, Qualitative/Quantitative Translation.
A four-function, scientific, or graphing calculator is allowed on both sections of the exam.

When a question compares gases, first ask what is the same. The same temperature means the same average kinetic energy per atom, whatever the gas; then $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$ sorts out the speeds.

Questions students ask

What causes gas pressure?

Atoms colliding with a surface. Each bounce changes an atom's momentum, so the surface pushes on the atom and the atom pushes back. The perpendicular parts of those forces, added up and divided by the area, are the pressure.

What does the temperature of a gas measure?

The average kinetic energy of its atoms, $K_{avg} = \frac{3}{2} k_{B} T$ , with $T$ in kelvins. It describes the whole gas, since its atoms have a spread of speeds.

What is the rms speed of a gas?

The square root of the average of the squared speeds: the speed an atom with the average kinetic energy would have, $v_{rms} = \sqrt{\frac{3 k_{B} T}{m}}$ .

Do I need the formula for the Maxwell-Boltzmann distribution?

No. You need its features: it starts at zero, rises to a peak and has a long tail at high speeds, and heating the gas moves the peak to higher speeds and spreads the curve out, with the same area underneath.

Beyond the AP Exam

College Board sets this limit on what the AP Exam asks about this topic. Anything past it is not on the exam.

AP Physics 2 only expects students to perform qualitative and quantitative analysis of collisions in one and two dimensions. Students are not expected to know the functional form of the Maxwell-Boltzmann distribution but are expected to be familiar with how features of the distribution are related to the temperature of the gas.

Course alignment, for teachers

AP Physics 2: Algebra-Based topic 9.1, Unit 9: Thermodynamics.

Skill 1.A: Create diagrams, tables, charts, or schematics to represent physical situations.
Skill 2.C: Compare physical quantities between two or more scenarios or at different times and locations in a single scenario.
Skill 3.B: Apply an appropriate law, definition, theoretical relationship, or model to make a claim.
Skill 3.C: Justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

This lesson maps all 7 essential knowledge statements of the topic to the part that teaches each one: 7 are checked by a scored question. The map is in the lesson file.