Kinetic Theory of Temperature and Pressure
Connect microscopic momentum changes with pressure, temperature, and a distribution of speeds.
AP Physics 2 Topic 9.1 · College Board CED printed p. 27, 28 / PDF p. 34, 35 · See this topic in the course outline
A gas can push on every side of a container even though its atoms have no organized motion. Start with a single collision, then average over an enormous number of collisions. That connects microscopic momentum to measurable pressure and temperature.
Before you start
- Momentum is a vector; impulse changes momentum.
- Elastic collisions conserve kinetic energy and total momentum.
- Pressure is force per area; kinetic energy is one half mv².
A wall feels the momentum changes of atoms
Atoms travel in many directions and collide both with one another and with their container. During an elastic collision between two atoms, forces are equal and opposite: the momentum lost by one atom is gained by the other. Total vector momentum and total kinetic energy of that pair are conserved, even though each atom’s speed can change.
At a smooth fixed wall, resolve velocity into components perpendicular and parallel to the wall. If the perpendicular component changes from +vₓ to −vₓ, the atom has . Its parallel momentum stays the same. The wall receives the opposite impulse, +2mvₓ. A wall clamped to Earth is not an isolated atom–wall system; include its support and Earth when invoking conservation of total momentum. The wall’s immense mass makes its recoil speed negligible.
Average many impulses over a time interval: . More collisions per second or larger impulses per collision produce a larger force. In two-dimensional atom–atom collisions, conserve x and y momentum separately; only their vector sum is guaranteed to be conserved.
Pressure is normal force per area, throughout the gas
Pressure uses the force perpendicular to a surface, not a force tangent to it: . For a gas, F⊥ is the sum of the normal forces from its many atoms, averaged over enough collisions for the reading to be steady. One pascal is one newton per square meter. Doubling area at unchanged pressure doubles the force on that area.
Imagine placing a very small pressure sensor inside the gas. Atoms collide with each face, just as they collide with the container wall. Pressure therefore exists throughout the gas. At equilibrium with negligible gravity effects it is uniform; the balanced forces on opposite faces do not mean the pressure is zero. Temperature and number density affect the collision rate and impulse, linking atomic motion to this local pressure.
Temperature measures an average, not a single atom’s speed
Random microscopic motion is different from the bulk motion of the whole container. For a classical ideal gas, absolute temperature T in kelvin is proportional to the average translational kinetic energy per atom: . Here k_B = 1.38 × 10⁻²³ J/K and m is one atom’s mass. The root-mean-square speed is a way to encode the average of squared speeds; it is not the average velocity. In an equilibrium gas the vector average velocity can be zero while rms speed is large.
At the same temperature, two ideal gases have the same average kinetic energy per atom. The lighter atoms move faster: rms speed is proportional to √T and inversely proportional to √m. To double rms speed for one species, raise kelvin temperature by a factor of four. A Celsius ratio cannot be used because 0 °C does not mean zero average kinetic energy.
Reading a Maxwell–Boltzmann distribution
An equilibrium gas contains slow atoms, fast atoms, and many speeds in between. A Maxwell–Boltzmann speed distribution plots how much of the population lies in each speed interval. The area over an interval is the fraction in that interval; the total area is one when the vertical axis is a normalized fraction density. An energy distribution tells the same population story with kinetic energy on the horizontal axis.
For the same species at higher temperature, the most likely speed shifts right, the distribution broadens, the high-speed tail extends, and the peak gets lower if the total area is held fixed. More atoms have large energies, and average energy increases. Do not read the tallest point as the speed of every atom or confuse the mode with rms speed. AP Physics 2 requires reasoning about these features, not memorizing the distribution’s functional form; collision analysis stays in one or two dimensions.
A two-dimensional collision: solve each component, then check energy
Two identical ideal-gas atoms each have mass 2.0 × 10⁻²⁶ kg. Atom A initially has velocity (200, 300) m/s and atom B is at rest. After a short collision, A has velocity (200, 0) m/s. Find B’s velocity and check whether this specified outcome is elastic. Treat the two-atom pair as isolated during the collision.
Draw x to the right and y upward, and use the following component table. Velocities are not added as scalar speeds.
Velocity-component representation; all entries in m/s Atom and state v_x v_y Arrow direction A before +200 +300 up and right B before 0 0 stationary A after +200 0 right B after unknown unknown to be determined Why this step? A component representation keeps the two independent momentum constraints visible; it bridges the one-dimensional collision method to two dimensions.
, so vBx = 0. Separately, , so vBy = +300 m/s.
Why this step? The pair has negligible external impulse, so total momentum is conserved separately in each direction. Equal masses cancel, but velocity vectors must not be replaced by their magnitudes.
; .
Why this step? Momentum conservation alone does not prove elasticity. Sum both atoms’ kinetic energies and compare the before and after totals.
Answer and check
B leaves with velocity (0, +300) m/s, an upward arrow. A leaves to the right. The kinetic-energy totals match, so this specified outcome is elastic. Each atom’s individual momentum and kinetic energy can change even though the pair’s totals are conserved.
From one collision to pressure
Atoms of mass 5.0 × 10⁻²⁶ kg strike a wall with normal speed 400 m/s and rebound elastically. There are 3.0 × 10²² such collisions each second over 0.020 m². Find average wall force and pressure.
Choose the atoms and wall. For each atom, = 4.0 × 10⁻²³ kg·m/s.
Why this step? The normal momentum reverses sign; the wall receives an equal opposite impulse.
F⊥ = (4.0 × 10⁻²³)(3.0 × 10²²) = 1.2 N.
Why this step? Impulse per collision times collisions per second gives impulse per time, which is force.
P = 1.2/0.020 = 60 Pa.
Why this step? Pressure divides the normal force by the area receiving those collisions.
Answer and check
The average force is 1.2 N and pressure is 60 Pa. The tiny impulse of one atom becomes a measurable force because the collision rate is enormous.
A kelvin temperature gives atomic energy and rms speed
An ideal helium gas is at 300 K. Take one atom’s mass as 6.64 × 10⁻²⁷ kg. Find average kinetic energy and rms speed, then predict the rms speed at 1200 K.
Use Kavg = (3/2)(1.38 × 10⁻²³)(300) = 6.21 × 10⁻²¹ J.
Why this step? Temperature is an absolute measure of average translational energy per atom.
vrms = √(2Kavg/m) = √[(1.242 × 10⁻²⁰)/(6.64 × 10⁻²⁷)] ≈ 1.37 × 10³ m/s.
Why this step? The energy relation uses the mean of v², so take a square root after dividing by the atom’s mass.
At 1200 K, vrms changes by √(1200/300) = 2.
Why this step? At unchanged species mass, rms speed scales with the square root of kelvin temperature.
Answer and check
Kavg = 6.21 × 10⁻²¹ J; vrms ≈ 1.37 km/s at 300 K and 2.74 km/s at 1200 K. The speed doubles, while average energy quadruples.
Make the model respond
Focused Physics 2 model
Ideal gas: pressure and molecular speed
Predict first. Predict how doubling temperature or volume changes pressure. Which change affects molecular speed?
PV = nRT; vᵣₘₛ = √(3RT/M); n = 0.500 mol of helium
- Selected T = 300 K
- Reference T = 300 K
- Selected or reference point
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
At 300 K and 12 L, P = 103.9 kPa and vᵣₘₛ = 1,367 m/s.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Amount n | 0.5 | mol | Fixed; helium molar mass 0.004002602 kg/mol |
| Temperature | 300 | K | |
| Volume | 0.012 | m³ | |
| Pressure | 1.039 × 10^5 | Pa | |
| Rms speed | 1,367 | m/s | |
| Monatomic internal energy | 1,871 | J |
At fixed amount and temperature, larger volume lowers the collision rate per area and pressure varies inversely with volume. At fixed volume, pressure is proportional to absolute temperature. The rms speed depends on temperature and molecular mass, not volume. A warmer equilibrium speed distribution has a higher characteristic speed and greater width; the arrows here only illustrate a speed scale.
Assumptions and limits. Dilute ideal monatomic helium in thermal equilibrium; pointlike particles with negligible interaction energy except elastic collisions. n = 0.500 mol and M = 0.004002602 kg/mol stay fixed. Container and arrow sizes are schematic. The model does not calculate a Maxwell–Boltzmann distribution or nonideal-gas corrections.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
A gas has a pressure of 80 kPa. Are atoms pressing only on its walls? What average force acts on a 2.0 × 10⁻⁴ m² sensor face inside it?
Worked answer and reason
Pressure exists inside the gas. F⊥ = PA = (8.0 × 10⁴)(2.0 × 10⁻⁴) = 16 N on each face. Opposite face forces may cancel.
Local pressure is normal momentum transfer per area; a net force of zero is compatible with nonzero pressure.
Question 2
At equal temperature, atom B has four times the mass of atom A. Compare average kinetic energies and rms speeds.
Worked answer and reason
The average kinetic energies are equal; vB,rms = vA,rms/2.
Equal T fixes average kinetic energy. The greater mass requires a smaller speed squared to give that same energy.
Question 3
You heat a sealed sample without changing its species. Sketch how a normalized speed distribution changes and explain the peak height.
Worked answer and reason
The peak shifts toward larger speed, the curve broadens, and the peak lowers. The total area remains one because it still represents the whole population.
A higher average kinetic energy increases characteristic speeds, while a spread over more speed intervals requires a lower normalized peak.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- At thermal equilibrium all atoms move at the same speed.
- A distribution of speeds remains. Equal temperature means equal average translational kinetic energy, not identical atom speeds.
- A heavier gas must be hotter if it has more total kinetic energy.
- Temperature tracks average energy per atom. Total energy also depends on how many atoms there are.
- Pressure only exists where gas hits a real wall.
- A small imagined surface anywhere in a gas experiences momentum exchange from atoms on both sides; pressure is a local property.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
- Problem 1 · vector components; numerical pressure; explanatory argument
- Problem 2 · distribution sketch; ratio reasoning
- Problem 15 · conduction rate; symbolic first-law derivation; microscopic scaling; entropy
Scientific calculator workspace
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Scientific calculator
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For the teacher
- Work through the two-atom component-table example before assigning q01. Ask students to write two separate momentum equations and then perform the total kinetic-energy check; momentum conservation alone does not establish an elastic collision.
- Ask students to draw incoming and outgoing velocity components before calculating an impulse. The normal component reverses at an ideal smooth wall; the tangential component does not.
- For two gases at the same temperature ask separately about average kinetic energy, rms speed, and total internal energy. These are different questions.
- Use qualitative distribution sketches. Do not require the functional formula of the Maxwell–Boltzmann distribution or three-dimensional collision calculations.
Source objectives and scope
- 9.1.A — Describe the pressure a gas exerts on its container in terms of atomic motion within that gas.
- 9.1.B — Describe the temperature of a system in terms of the atomic motion within that system.
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.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 Course and Exam Description, Unit 9 (PDF pages 34–42; printed pages 27–35) — All Unit 9 learning objectives, essential knowledge clauses, and AP scope boundaries; source equations checked in the supplied pinned PDF.
- OpenStax University Physics Volume 2, Molecular Model of an Ideal Gas — Supporting open-textbook reading for ideal-gas assumptions, absolute pressure, and state-variable graphs; real-gas extensions in the source are beyond this unit’s AP requirements.
- OpenStax University Physics Volume 2, First Law of Thermodynamics — Supporting open-textbook reading for energy accounting; authoring translates any work-by convention into the AP work-on convention.
- OpenStax University Physics Volume 2, Entropy — Optional conceptual supporting reading for qualitative entropy and reversibility; quantitative entropy calculations in the source are outside this AP treatment.
Glossary and deeper reading · 15 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Linear momentum
Linear momentum is a vector describing an object’s motion and equals mass times velocity in the nonrelativistic model. The vector sum of momenta is conserved when the net external impulse on the system is zero. For example, two identical carts moving equally fast in opposite directions have zero total linear momentum but nonzero total kinetic energy.
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Temperature
Temperature is a state quantity that determines the direction of net energy transfer through thermal contact. For a classical ideal gas it is proportional to the average translational kinetic energy per particle, rather than to the total energy of the sample. For example, a small hot metal sample can have a higher temperature but less total internal energy than a large cooler sample.
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Vector
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes. Vectors add by adding corresponding components rather than simply adding their magnitudes. For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast. The zero vector has zero magnitude and no unique direction.
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Impulse
Impulse is the change in linear momentum produced by a force over a time interval. It equals the signed vector area under a force-versus-time graph and is measured in newton-seconds. For example, a padded landing spreads the stopping impulse over more time, reducing the average force for the same momentum change.
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Elastic collision
An elastic collision conserves the total kinetic energy of the colliding system. Total momentum is also conserved when the system has negligible net external impulse, although each object’s momentum and energy can change. For example, ideal gas atoms redistribute their speeds in elastic collisions without changing the pair’s total kinetic energy.
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Kinetic energy
Kinetic energy is energy associated with motion relative to a chosen reference frame. For a nonrelativistic translating object it is one half of its mass times speed squared. For example, doubling a cart’s speed quadruples its translational kinetic energy when its mass stays fixed.
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Force
A force describes an interaction that can change an object’s momentum. It is a vector measured in newtons, so both size and direction matter when forces are added. For example, a rope can pull a cart while the floor exerts friction in another direction.
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Velocity
Velocity describes how position changes with time and includes direction. Instantaneous velocity describes one moment, while average velocity is displacement divided by elapsed time. For example, a car turning a corner can change velocity while maintaining the same speed.
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Vector component
A vector component describes the part of a vector along a chosen axis. Its signed value changes when the axes change, even though the physical vector remains the same. For example, a diagonal launch velocity can be resolved into horizontal and vertical components that share the same elapsed time.
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Normal force
A normal force is a contact force perpendicular to the local surface of contact. Its magnitude is set by the motion and the other forces, so it need not equal the object’s weight. For example, a wall’s normal force on a leaning ladder points horizontally rather than upward.
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Gravity
Gravity is the attractive interaction between masses in the Newtonian model. For spherical bodies or point-mass approximations, its strength decreases with the square of the separation between their centers. For example, Earth’s gravity provides the inward force that keeps a satellite in orbit.
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Ideal gas
An ideal gas model treats particles as having negligible volume and no sustained intermolecular forces between brief elastic collisions. The equilibrium state obeys the ideal gas law using absolute pressure and kelvin temperature. For example, a dilute gas far from condensation often approximates this model better than a dense gas near a phase change.
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Kelvin temperature scale
The kelvin scale measures thermodynamic temperature from absolute zero. A temperature interval of one kelvin has the same size as one degree Celsius, but their zero points differ. For example, temperature ratios in the ideal gas law must use kelvin values rather than Celsius values.
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Root-mean-square speed
Root-mean-square speed is the square root of the mean of the particles’ squared speeds. It represents a kinetic-energy average and is different from both average speed and vector-average velocity. For example, an equilibrium gas can have zero vector-average velocity while its rms speed is substantial.
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Mass density
Mass density is mass per unit volume of a material or sample. It is commonly measured in kilograms per cubic meter and can vary with temperature or pressure. For example, equal-volume samples of water and oil generally have different masses because their densities differ.