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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 9 · Topic 9.1

Particle Motion Behind Temperature and Pressure

Build the macroscopic ideas of temperature and pressure from microscopic particle motion. The central story is that temperature tracks average translational kinetic energy, while pressure comes from momentum transferred during countless collisions with a container wall.

Learning Goals

  • State the assumptions of the ideal-gas particle model and identify when they become unreliable.
  • Use momentum change and collision rate to explain gas pressure.
  • Derive and interpret \(PV=\frac13Nm\overline{v^2}\).
  • Connect absolute temperature to average translational kinetic energy and rms speed.
  • Predict how pressure, kinetic energy, and rms speed change when \(N\), \(V\), or \(T\) changes.

1. The Ideal-Gas Microscopic Model

An ideal gas is a useful model, not a claim that real molecules literally have no size or forces.

  • A very large number of particles move continuously in random directions.
  • Particle size is negligible compared with the distance between particles.
  • Particles follow Newton's laws between brief collisions.
  • Intermolecular forces are neglected except during collisions.
  • Collisions with other particles and the walls are elastic, so total kinetic energy is conserved during each collision.

Model limit: real gases depart most strongly from ideal behavior at high density or low temperature, when particle volume and attractions can no longer be ignored.

2. Three Different Quantities

QuantityParticle-level meaning
Temperature \(T\)Sets the average translational kinetic energy per particle. Use kelvin in proportional reasoning.
Pressure \(P\)Average normal force per area produced by particle-wall momentum transfers.
Internal energy \(U\)Total microscopic energy of the sample. For a monatomic ideal gas, it is the sum of translational kinetic energies.

Key distinction: temperature is not “the total kinetic energy.” Two samples can have the same temperature but different internal energies because they contain different numbers of particles.

3. From One Collision to Pressure

Place a particle of mass \(m\) in a cubic container of side length \(L\), so \(V=L^3\). Let \(v_x\) be the component of velocity perpendicular to the right wall.

  1. An elastic collision reverses the particle's x-momentum. The wall receives impulse of magnitude \(2m|v_x|\).
  2. The particle travels to the opposite wall and back before striking the same wall again, so the time between those impacts is \(\Delta t=2L/|v_x|\).
  3. The particle's average force on that wall is impulse divided by time: \(F_i=mv_{x,i}^{,2}/L\).
  4. Add the forces from \(N\) particles and divide by wall area \(A=L^2\).
\(P=\frac{F}{A}=\frac{m}{V}\sum_{i=1}^{N}v_{x,i}^{,2}=\frac{Nm\overline{v_x^2}}{V}\)

Random, isotropic motion has no preferred direction, so \(\overline{v_x^2}=\overline{v_y^2}=\overline{v_z^2}=\frac13\overline{v^2}\). Therefore,

\(PV=\frac13Nm\overline{v^2}=\frac23N\overline K\)

The factor \(1/3\) does not mean only one third of the particles strike a wall. It comes from sharing the mean squared speed among three perpendicular velocity components.

4. Temperature as an Energy Scale

Compare the microscopic pressure result with the ideal-gas equation \(PV=Nk_{\mathrm B}T\):

\(\overline K=\frac12m\overline{v^2}=\frac32k_{\mathrm B}T\)

At the same temperature, light and heavy ideal-gas particles have the same average translational kinetic energy. The lighter particles must therefore move faster on average.

Here \(k_{\mathrm B}=1.380649\times10^{-23}\,\mathrm{J/K}\). The relation uses absolute temperature; Celsius values cannot be inserted into proportional ratios.

5. Root-Mean-Square Speed

Because molecular velocities point in every direction, the ordinary average velocity of a gas at rest is approximately zero. Squaring first prevents opposite directions from canceling.

\(v_{\mathrm{rms}}=\sqrt{\overline{v^2}}=\sqrt{\frac{3k_{\mathrm B}T}{m}}=\sqrt{\frac{3RT}{M}}\)

Use molecular mass \(m\) with \(k_{\mathrm B}\), or molar mass \(M\) in kilograms per mole with \(R\). The rms speed is a useful energy-linked speed, but it is not the same as the mean speed or the most probable speed.

Do not confuse linear and square-root scalingFor a fixed particle type, \(\overline K\propto T\), but \(v_{\mathrm{rms}}\propto\sqrt{T}\). Doubling kelvin temperature doubles average kinetic energy and multiplies rms speed by only \(\sqrt2\).

6. Macro–Micro Bridge

ModelRelationshipWhat it reveals
Ideal-gas equation\(PV=Nk_{\mathrm B}T=nRT\)Connects four measurable state variables.
Kinetic pressure\(P=\frac13\frac NVm\overline{v^2}\)Pressure rises with number density and mean squared speed.
Energy–temperature link\(\overline K=\frac32k_{\mathrm B}T\)Temperature depends on average energy per particle, not total sample energy.
RMS speed\(v_{\mathrm{rms}}=\sqrt{3RT/M}\)At the same temperature, smaller molar mass means greater characteristic speed.

7. Worked Example: Molecular Speed

Estimate the rms speed and average translational kinetic energy of an \(\mathrm{N_2}\) molecule at \(300\,\mathrm K\). Use \(M=0.0280\,\mathrm{kg/mol}\).

\(v_{\mathrm{rms}}=\sqrt{\frac{3(8.314)(300)}{0.0280}}\approx5.17\times10^2\,\mathrm{m/s}\)
\(\overline K=\frac32(1.380649\times10^{-23})(300)\approx6.21\times10^{-21}\,\mathrm J\)

Reasonableness check: the speed is hundreds of meters per second, while the energy of one molecule is tiny because molecular mass is tiny.

8. Worked Example: Heating at Fixed Volume

A sealed rigid container holds an ideal gas at \(100\,\mathrm{kPa}\) and \(300\,\mathrm K\). It is heated to \(450\,\mathrm K\). Find the new pressure and the rms-speed ratio.

\(P_2=P_1\frac{T_2}{T_1}=(100\,\mathrm{kPa})\frac{450}{300}=150\,\mathrm{kPa}\)
\(\frac{v_{\mathrm{rms},2}}{v_{\mathrm{rms},1}}=\sqrt{\frac{T_2}{T_1}}=\sqrt{1.5}\approx1.225\)

Faster particles strike the walls more frequently and deliver more momentum per collision. Pressure rises by 50%, while rms speed rises by about 22.5%.

9. AP Graph and Experiment Reasoning

  • For fixed \(N\) and \(V\), a graph of \(P\) versus \(T\) in kelvin is linear with slope \(Nk_{\mathrm B}/V\).
  • A graph of \(P\) versus \(\overline{v^2}\) is linear when particle type and number density are fixed.
  • A graph of \(P\) versus \(v_{\mathrm{rms}}\) is curved because \(P\propto v_{\mathrm{rms}}^2\).
  • To test the fixed-volume prediction, use a rigid sealed container, vary temperature, wait for thermal equilibrium, and plot pressure against kelvin temperature.

Evidence limit: a nonzero pressure intercept can indicate sensor offset, incomplete equilibrium, a leak, or model breakdown; do not immediately call it new physics.

10. Common Traps

  • Using Celsius ratios: \(40^\circ\mathrm C\) is not twice \(20^\circ\mathrm C\). Convert both to kelvin.
  • Confusing speed with velocity: random velocities cancel in the mean, but rms speed is positive.
  • Equating pressure and temperature: pressure also depends on number density \(N/V\).
  • Assuming faster means more energetic for every comparison: at equal temperature, light particles are faster but have the same average translational kinetic energy as heavy particles.
  • Assuming compression heats the gas automatically: temperature depends on the process. Isothermal compression raises pressure mainly by increasing collision frequency per area.
Mastery Check

1. A monatomic ideal gas has its absolute temperature doubled while \(N\) and \(V\) remain fixed. What happens to \(P\), \(\overline K\), and \(v_{\mathrm{rms}}\)?

Show reasoning and answer

From \(P=Nk_{\mathrm B}T/V\), pressure doubles. From \(\overline K=3k_{\mathrm B}T/2\), average kinetic energy doubles. Since \(v_{\mathrm{rms}}\propto\sqrt T\), rms speed is multiplied by \(\sqrt2\), not 2.

2. Helium and neon gases are at the same temperature. Which has greater average translational kinetic energy, and which has greater rms speed?

Show reasoning and answer

The average translational kinetic energies are equal because each is \(3k_{\mathrm B}T/2\). Helium has the greater rms speed because \(v_{\mathrm{rms}}\propto1/\sqrt M\).

3. An ideal gas is compressed isothermally. Explain the pressure increase without claiming that particles move faster.

Show reasoning and answer

Isothermal means temperature and therefore the speed distribution remain unchanged. The smaller volume increases number density and shortens the average travel distance to a wall, so wall collisions occur more frequently per unit area and pressure rises.

Optional Investigation

Use the free PhET Gas Properties simulation as a virtual experiment. First lock volume and particle number, record pressure at several kelvin temperatures, and test whether the graph is linear. Then compare light and heavy particles at one temperature and explain why their speed distributions differ even though their average translational kinetic energies match.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.