Chemistry: Atoms First 2e · Gases

The Kinetic-Molecular Theory

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On this page 9 sections

In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

The gas laws of Topic 2 describe what gases do — pressure rises when you compress a gas, volume grows when you heat it — but not why. The (KMT) supplies the explanation: a gas is a huge collection of tiny particles in constant, random motion. It is a model, a simplified picture that treats particles as hard spheres with negligible volume and no attractions between them.

That model explains every empirical gas law with one idea: gas pressure is the summed force of countless molecular collisions with container walls, and temperature measures the of the molecules. This topic states the five postulates, quantifies the temperature–energy link, introduces molecular speed distributions, and shows how the model explains Boyle's, Charles's, Avogadro's, and Dalton's laws. It also previews where the model breaks down — the subject of Topic 6.

Why this matters

The KMT is the "why" behind every gas law. If you understand the model, you never have to memorize whether pressure and volume are directly or inversely related — you reason it out from collisions.
It explains diffusion and effusion. Light molecules move faster than heavy ones at the same temperature, which is why helium leaks from a balloon faster than air.
It explains everyday phenomena. Tires heat up on a long drive because faster molecules hit the walls harder; weather balloons expand at altitude because fewer collisions mean lower outside pressure.
It sets up non-ideal behavior. Real gases have volume and attract each other, so the KMT's assumptions mark exactly where the ideal gas law fails (Topic 6).
Exams: KMT postulates, average kinetic energy, and root-mean-square speed calculations are standard items.

The college version

Core Concepts

The five postulates

Gases are composed of tiny particles (molecules or single atoms) separated by distances far larger than their diameters.
The particles are in constant, random, straight-line motion, colliding with each other and with container walls.
Collisions are perfectly elastic — kinetic energy is transferred between particles but none is lost overall.
There are no attractive or repulsive forces between particles. Each moves independently.
The average kinetic energy is directly proportional to the absolute (kelvin) temperature, and the particles' own volume is negligible compared with the container.

Postulates 4 and 5 are the ideal assumptions. Real gases violate both — the more they violate them, the less ideal they behave.

Temperature measures average kinetic energy

5 has a quantitative form. For one mole of any gas:

$KE = 32 RT$

where KE is the average translational kinetic energy per mole (joules), R = 8.314 J mol^-1K^-1, and T is (kelvin). Two consequences: at the same temperature, every gas has the same average kinetic energy per mole — heavy molecules simply move more slowly — and at 0 K, molecular motion would stop, which is why gas calculations require kelvin, not Celsius.

Molecular speeds: the Maxwell–Boltzmann distribution

Molecules do not all move at one speed. Speeds spread out in the : a few very slow, a few very fast, most near the middle. Heating shifts the whole distribution to higher speeds and flattens it.

The speed used in calculations is the root-mean-square speed:

$u rms = \sqrt 3RT M$

with M in kilograms per mole — a critical detail, because grams per mole will not cancel to m s^-1.

Explaining the gas laws with the model

Boyle's law (inverse P–V at constant T): squeezing gas into a smaller volume means molecules hit a smaller wall area more often — pressure rises.
Charles's law (direct V–T at constant P): heating speeds molecules up; the container must expand to keep collision frequency per area constant.
Avogadro's law (direct V–n): more molecules mean more collisions; volume must grow to hold pressure constant.
Dalton's law: with no interparticle forces, each gas collides with the walls as if alone, so total pressure is the sum of partial pressures.

Where the model stops working

At high pressure, particles are crowded and their volume is no longer negligible; at low temperature, molecules move slowly enough that attractions matter. Both make real gases deviate from ideal predictions — the next topic.

Common Confusions

Do Not Confuse	With	Difference
Temperature	Heat	Temperature measures average kinetic energy; heat is energy transferred between objects at different temperatures
Average KE per mole	Total KE of the sample	Per mole is 32RT for any gas; total scales with moles
u_rms	Average or most probable speed	Three different measures of the speed distribution; u_rms is the largest
Molar mass in g/mol	Molar mass in kg/mol	u_rms requires kg/mol; g/mol gives wrong units
Kelvin	Celsius	Doubling kelvin doubles average kinetic energy; doubling Celsius does not

Eli explains

The same idea, in plain words

Explain it like I’m 10

Imagine a room full of bouncy balls flying around and smacking the walls. The balls are gas molecules and the walls are the container. The harder and more often they hit, the more "pressure" they make. Heat the room and the balls bounce faster and hit harder — that's why hot gases push harder. Make the room smaller and they hit the walls more often — that's why squeezing gas raises pressure. At the same bounciness, a heavy ball moves slower than a light one, but both carry the same bounce-energy.

Worked example

Example 1: Root-mean-square speed of oxygen

Calculate u_rms for O₂ at 298 K (molar mass 32.00 g/mol).

Step 1 — Write the formula before substituting:

$u rms = \sqrt 3RT M$

Step 2 — Convert molar mass to kg/mol:

$M = 32.00 g mol \times 1 kg 1000 g = 0.03200 kg mol -1$

Step 3 — Substitute R = 8.314 J mol^-1K^-1, T = 298 K:

$u rms = \sqrt 3(8.314)(298) 0.03200 = \sqrt 232, 200 \approx 482 m s -1$

Dimensional check: J mol^-1K^-1 · Kkg mol^-1 = kg m²s^-2kg = m²s^-2, whose square root is m s^-1. ~482 m/s (~1700 km/h) is reasonable for oxygen at room temperature.

Example 2: Average kinetic energy per mole

What is the average translational kinetic energy of one mole of any gas at 298 K?

Step 1 — Write the formula:

$KE = 32 RT$

Step 2 — Substitute:

$KE = 32 (8.314 J mol -1 K -1)(298 K) = 3.72 \times 103 J mol -1 = 3.72 kJ mol -1$

No molar-mass term appears, so helium and xenon have identical average kinetic energy per mole at the same temperature — the central prediction of postulate 5.

Example 3: Comparing speeds of hydrogen and oxygen

How many times faster do H₂ molecules (2.016 g/mol) move than O₂ molecules (32.00 g/mol) at the same temperature?

Step 1 — Write the ratio formula (the 3RT cancels):

$u rms (H 2) u rms (O 2) = \sqrt M(O 2) M(H 2)$

Step 2 — Substitute (units cancel):

$u rms (H 2) u rms (O 2) = \sqrt 32.00 2.016 = \sqrt 15.87 \approx 3.98 \approx 4$

Hydrogen travels about four times faster than oxygen at the same temperature — the molecular explanation for Graham's law (Topic 4).

Key takeaways

Five postulates: tiny particles, constant random motion, elastic collisions, no interparticle forces, average kinetic energy ∝ T (kelvin).
Average kinetic energy per mole: KE = 32RT — depends only on temperature, not on the gas's identity.
Root-mean-square speed: u_rms = 3RT/M with M in kg/mol.
Lighter molecules move faster: u_rms ∝ 1/M.
Pressure comes from collisions; temperature measures average kinetic energy — these two ideas explain all four gas laws.
Ideal gas law fails at high pressure (particle volume) and low temperature (attractions) → Topic 6.

Check yourself

6 review questions from the chapter. Try each one, then open the answer.

State the five postulates of the kinetic-molecular theory.
Show answer
(1) Gases are tiny particles far apart; (2) particles in constant random motion; (3) elastic collisions; (4) no interparticle forces; (5) average kinetic energy ∝ absolute temperature, particle volume negligible.
Why must temperature be in kelvin for KE = 32RT and u_rms = 3RT/M?
Show answer
The formulas express proportionality to absolute temperature, which starts at absolute zero; Celsius values are offset (0 °C = 273 K), so ratios and products come out wrong.
Calculate u_rms for helium atoms (M = 4.003 g/mol) at 300 K.
Show answer
M = 0.004003 kg/mol; u_rms = 3(8.314)(300)/0.004003 = 1.869 × 10⁶ ≈ 1370 m s^-1.
Two gases at the same temperature: which has greater average kinetic energy per mole — the heavier, the lighter, or neither?
Show answer
Neither — average kinetic energy per mole depends only on temperature.
Use the KMT to explain why compressing a gas at constant temperature raises its pressure.
Show answer
A smaller volume means molecules travel a shorter distance between wall hits, so collision frequency per unit wall area rises — pressure increases.
At what conditions does the KMT fail, and which postulate breaks down?
Show answer
High pressure (postulate 5's negligible-volume assumption fails) and low temperature (postulate 4's no-forces assumption fails because attractions matter at slow speeds).

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Kinetic-molecular theory: Model of gas particles as tiny, fast, non-interacting hard spheres in random motion
Postulate: A basic assumption at the start of a model
Elastic collision: Collision in which kinetic energy is conserved overall
Average kinetic energy: Mean translational energy per mole, 32RT
Maxwell–Boltzmann distribution: Spread of molecular speeds at a given temperature
Root-mean-square speed (urms): Representative average molecular speed, 3RT/M
Absolute temperature: Temperature in kelvin, K = ^°C + 273.15

Sources & references

openstax.org — Chemistry Atoms First 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

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