Kinetic Molecular Theory
Revisit distribution spread and area before interpreting collision-energy tails.
AP Chemistry Topic 3.5 · CED printed p. 64 / PDF p. 71. This teaching, practice and feedback were added for the AP edition; original author material remains separately traceable.
Before you start
- Ideal gas law
- Mass and squared speed
- Graph areas
Pressure and temperature come from particle motion
The ideal kinetic molecular model treats gas particles as tiny relative to their separation, moving randomly with negligible interparticle attractions between elastic collisions. Collisions with walls transfer momentum and cause pressure. At fixed amount and volume, increasing temperature increases speeds on average, increasing impact and pressure. At fixed temperature, adding particles increases the rate of wall collisions without changing the species’ equilibrium speed distribution.
For an individual particle, translational kinetic energy is , where v is speed. For a distribution, average energy uses mean squared speed: . It is not ½m times the square of the average velocity vector, which can be zero for random motion. Kelvin temperature is proportional to mean translational kinetic energy in the ideal model. Equal-temperature gases have equal mean translational energy, even when their masses and typical speeds differ.
A temperature does not give every particle the same speed
A Maxwell–Boltzmann distribution shows a spread of speeds or kinetic energies. Its horizontal axis must specify which of those quantities it displays. Heating the same gas shifts a speed distribution toward larger speeds and broadens it. The peak becomes lower if total area is held constant; that does not mean particles vanished. Area over a speed interval gives the fraction in that interval, and the most probable speed is not identical to the mean or root-mean-square speed.
At a common temperature, lighter molecules have higher typical speeds because the same mean energy is shared with a smaller mass. On a kinetic-energy-axis plot, ideal monatomic translational energy distributions at the same temperature match regardless of molecular mass; on a speed-axis plot they do not. The distinction makes a useful AP data-representation check.
Probability density / (s m−1)
Equal total probability area; different peak heights
Let's work through it
Work it through
At the same temperature, compare He (4.00 g mol−1) and Ne (20.18 g mol−1) and then predict the effect of heating He from 300 K to 600 K at fixed volume.
- Equal temperature gives equal mean translational kinetic energy, not equal speed. From m〈v²〉 equal, representative root-mean-square speed scales as 1/√M.
- The He-to-Ne rms-speed ratio is √(20.18/4.00) = 2.25. He’s speed distribution lies farther toward high speeds.
- Doubling He’s Kelvin temperature doubles mean translational energy and multiplies rms speed by √2 ≈ 1.41, not by 2.
- At fixed n,V, pressure doubles. The normalized distribution broadens with the same total area.
Make the picture move
Interactive concept model
Connect pressure to particles hitting the walls.
Predict first. Hold two quantities fixed, then change the third. Predict pressure before reading the output. Does doubling temperature double characteristic speed?
PV = nRT; at fixed identity, characteristic speed ∝ √T
- P(V), selected n and T
- Labeled selected or reference state
On a small screen, scroll inside the graph or diagram to read its labels. The values and explanation also appear below.
At 0.5 mol, 300 K, and 15 L, the ideal pressure is 0.821 atm. At fixed n and T, pressure varies inversely with volume.
Read the data and comparisons
| Volume (L) | Pressure (atm) | n (mol) | T (K) |
|---|---|---|---|
| 5 | 2.46 | 0.5 | 300 |
| 10 | 1.23 | 0.5 | 300 |
| 15 | 0.821 | 0.5 | 300 |
| 20 | 0.615 | 0.5 | 300 |
| 30 | 0.41 | 0.5 | 300 |
Macroscopic / data. Pressure 0.821 atm is the computed macroscopic outcome.
Particle view. Arrows show a characteristic speed scale, not identical speeds in a real gas. The diagram uses representative counts rather than individual molecules in a mole.
Symbolic view. P = (0.5 mol)(0.08206 L atm mol⁻¹ K⁻¹)(300 K)/(15 L).
Why it works. At the same temperature, reducing volume increases the rate of wall collisions. Increasing the amount increases particles per volume. Increasing temperature raises average translational kinetic energy and the typical momentum transferred in a collision. The gas law uses absolute temperature.
Conditions and limits. Ideal gas: negligible particle volume and intermolecular attractions, equilibrium temperature, fixed gas identity. R = 0.08206 L atm mol⁻¹ K⁻¹. Dots are representative particles; box width qualitatively tracks volume without a quantitative geometric scale. This still picture does not simulate motion, collisions, or a velocity distribution. Real gases depart from ideal behavior, especially at high pressure or low temperature.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
At equal temperature, does a heavier ideal gas have greater average translational kinetic energy?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
At equal temperature, does a heavier ideal gas have greater average translational kinetic energy?
Worked answer and reason
No. The mean translational energy is set by Kelvin temperature; the heavier gas has lower typical speeds.
Mass and speed compensate in ½m〈v²〉.
Retrieval 2
Does a lower heated-distribution peak show fewer total particles?
Worked answer and reason
Not if the total area is preserved. The distribution has broadened.
Counts or probability depend on area, not peak height alone.
Retrieval 3
What does tripling a particle’s instantaneous speed do to its kinetic energy at fixed mass?
Worked answer and reason
It multiplies kinetic energy by 9.
Kinetic energy depends on the square of speed.
Catch the tempting mistake
- Every molecule at 300 K has one speed.
- Temperature describes a distribution and mean energy.
- Doubling Kelvin temperature doubles all characteristic speeds.
- Mean energy doubles; rms speed scales with the square root of temperature.
- Average random-motion velocity can be inserted into the energy formula.
- The average vector may cancel to zero; use the mean of squared speeds for mean kinetic energy.
Put it to work
Give the chemical reason as well as the calculation. Worked solutions are on a separate page.
- Unit 3 problem 12 · ap application · data
- Unit 3 problem 13 · ap synthesis · multiple
Scientific calculator workspace
Use this workspace for arithmetic, logarithms and elementary functions. Keep intermediate precision and report final values with appropriate units and significant figures. Graphing and general equation solving need a suitable approved tool.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
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For the teacher
- Always ask students to identify whether the graph axis is energy or speed.
- Use area and peak-height comparisons to distinguish distribution spread from number change.
Source objectives and scope
Source edition: ap-chemistry-ced-fall2024-local-b5dfe867. See this topic in the course outline →
- 3.5.A
- Explain the relationship between the motion of particles and the macroscopic properties of gases with: i. The kinetic molecular theory (KMT). ii. A particulate model. iii. A graphical representation.
- 3.5.A.1
- The kinetic molecular theory (KMT) relates the macroscopic properties of gases to motions of the particles in the gas. The Maxwell-Boltzmann distribution describes the distribution of the kinetic energies of particles at a given temperature.
- 3.5.A.2
- All the particles in a sample of matter are in continuous, random motion. The average kinetic energy of a particle is related to its average velocity by the equation: EQN: KE = ½ mv2.
- 3.5.A.3
- The Kelvin temperature of a sample of matter is proportional to the average kinetic energy of the particles in the sample.
- 3.5.A.4
- The Maxwell-Boltzmann distribution provides a graphical representation of the energies/velocities of particles at a given temperature.
Science references for this teaching
- AP Chemistry Course and Exam Description, effective Fall 2024 (supplied source) — Exact topic-numbered learning objectives, essential knowledge, science practices and AP exclusions; supplied PDF SHA256 b5dfe8677ef3d88c613865d2e2a3e8d6125d652e2b24c71ef1e8ce4e011094f0.
- OpenStax Chemistry 2e §9.5, Kinetic-Molecular Theory — Distribution interpretation, temperature and mean squared speed.
Glossary and deeper reading · 3 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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Ideal gas law
The ideal gas law relates pressure, volume, amount, and absolute temperature in an idealized gas model. It treats particles as having negligible volume and no intermolecular attractions. For example, at fixed amount and temperature, doubling the container volume halves the ideal gas pressure.
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Chemical equilibrium
Chemical equilibrium is a dynamic state in which forward and reverse reaction rates are equal. Macroscopic composition remains constant in a closed system under fixed conditions, but reactant and product concentrations need not be equal. For example, a reversible reaction can reach equilibrium with much more reactant than product.
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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.