Chemistry 2e · Gases

The Kinetic-Molecular Theory

10 min read

Constants used (R = 8.314 J·mol⁻¹·K⁻¹, 0 K = −273.15 °C, molar masses) are standard published values (NIST, IUPAC); verify against current primary sources before formal citation.

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 the previous topics tell you what gases do — double the temperature and volume doubles; halve the volume and pressure doubles. The (KMT) explains why. Built by Rudolf Clausius, James Clerk Maxwell, and Ludwig Boltzmann in the mid-1800s, the theory models a gas as a huge collection of tiny, rapidly moving particles whose collisions with container walls produce the measured pressure. From a handful of simple assumptions, the theory derives the ideal gas law, explains Boyle's, Charles's, Avogadro's, and Dalton's laws, and predicts molecular speeds. It also explains why real gases deviate from ideal behavior at high pressure and low temperature — and gives chemists a way to calculate how fast molecules actually move, a number that matters in effusion, diffusion, and reaction rates.

Why this matters

Macro to micro: KMT bridges observable quantities (pressure, volume, temperature) and the invisible particle world, turning "gases expand when heated" from a memorized fact into a consequence of faster-moving molecules.
Predicting molecular speeds: The root-mean-square (rms) speed formula tells you how fast molecules around you move — about 500 m/s for O₂ at room temperature, faster than the speed of sound. This directly governs effusion and diffusion rates from the previous topic.
Explaining real-world phenomena: Why does a balloon expand in a hot car? Why does a basketball feel firmer when the air is warm? Why do gases cool as they expand? KMT answers all of these.
Knowing a model's limits: Real gases condense to liquids; ideal gases never do. Understanding KMT's assumptions (no particle volume, no attractions) tells you when the model fails — high pressure, low temperature — which matters for anyone designing aerosol cans or industrial reactors.
Exams: KMT postulates, rms speed calculations, and explaining gas laws with the theory are classic short-answer and calculation items.

The college version

Core Concepts

The five postulates

The kinetic-molecular theory rests on five assumptions about an ideal gas:

Gases consist of a very large number of tiny particles (atoms or molecules) separated by distances much larger than their own sizes; the particles' own volume is negligible compared with the container volume.
Gas particles are in constant, random, straight-line motion, colliding with the container walls and each other.
Collisions are perfectly elastic — kinetic energy is transferred between particles but none is lost on average.
There are no attractive or repulsive forces between particles except during collisions.
The of the particles is directly proportional to the absolute (kelvin) temperature. At the same temperature, every gas — regardless of identity — has the same average kinetic energy.

These assumptions define the ideal gas: point particles, no interactions. Real gases approximate this when molecules are far apart (low pressure) and moving fast (high temperature).

Pressure from collisions

Gas pressure arises from countless particle–wall collisions. Each collision transfers a tiny impulse to the wall; pressure is the total force from all collisions divided by wall area. The theory predicts pressure depends on three factors: more particles (more collisions per second), faster particles (harder, more frequent collisions), and a smaller container (particles hit walls more often). This molecular picture generates the ideal gas law PV = nRT from mechanics — the observable law of gases is derivable from the statistics of moving particles.

Temperature is average kinetic energy

5 is the deepest idea: temperature measures the average translational kinetic energy of the particles. The average kinetic energy per mole of gas is:

$KE = 32 RT$

Because KE = 12mv², and temperature–energy is the same for all gases at a given temperature, a light gas and a heavy gas at the same temperature have the same average kinetic energy but very different speeds — He molecules race around ~2.8× faster than O₂ molecules, as Graham's law showed in the previous topic.

Molecular speeds: the rms speed

The most useful quantitative result of KMT is the root-mean-square speed, the speed of a molecule with the average kinetic energy:

$u rms = \sqrt 3RT M$

where R = 8.314 J/(mol·K), T is kelvin temperature, and M is the molar mass in kg/mol (because J = kg·m²/s²). The rms speed is not the average speed, but it is close and much easier to derive. It is the speed that matters for effusion rates: Graham's law follows directly because effusion rate ∝ u_rms ∝ 1/M.

The Maxwell–Boltzmann distribution

Not all molecules move at the same speed — they follow a spread of speeds called the . Key features: (1) the curve is skewed, with most probable speed < average speed < rms speed; (2) at higher temperature the curve flattens and shifts right — molecules move faster on average, and a larger fraction travel at very high speeds; (3) at the same temperature, lighter molecules have a broader, faster distribution. This distribution explains why evaporation can happen below the boiling point (fast molecules escape the liquid surface) and why reactions speed up with temperature (more collisions carry enough energy).

Explaining the gas laws

KMT accounts for every empirical law in this chapter:

Boyle's law: smaller container → particles travel shorter distances between wall collisions → more collisions per second → higher pressure at constant T.
Charles's law: higher T → faster particles → harder, more frequent collisions → volume must expand to keep pressure constant.
Avogadro's law: more particles → more collisions → volume must grow to keep pressure constant.
Dalton's law: each gas's particles collide with the walls independently; total pressure is the sum of each gas's independent contribution.
Graham's law: effusion rate ∝ average speed ∝ 1/M.

Limits of the model

KMT assumes no particle volume and no intermolecular forces. Both break down at high pressure (particles packed close, their own volume becomes significant) and low temperature (slow molecules spend more time near each other, attractions matter). Real gases then deviate from PV = nRT — the subject of the next topic. KMT is a model, not reality: wonderfully accurate under the right conditions and knowingly wrong outside them.

Common Confusions

Do Not Confuse	With	Difference
Temperature	Heat/energy	Temperature is proportional to average kinetic energy per particle; heat is transferred energy. Two gases can share the same T (same avg KE) with different total energies (different amounts)
rms speed	Average or most probable speed	The distribution has three "average-like" speeds; rms > average > most probable. Exams usually ask for rms
g/mol	kg/mol in the rms formula	M must be kg/mol with R = 8.314 J/(mol·K); using g/mol gives answers ~32× too large
Elastic collisions	Inelastic collisions	In KMT, collisions conserve kinetic energy on average; inelastic collisions (macro scale) lose energy
Ideal gas model	Real gas	The model assumes zero particle volume and zero attractions; real gases deviate at high P and low T
One speed for all molecules	A distribution of speeds	Molecules have a spread of speeds (Maxwell–Boltzmann); "the speed" is an average description
Pressure from static weight	Pressure from collisions	Gas pressure comes from moving-particle collisions, not from the gas "sitting" on the wall (unlike a liquid column)

Eli explains

The same idea, in plain words

Explain it like I’m 10

Imagine a room full of bouncy balls flying everywhere, never slowing down, never sticking together. Every time a ball hits a wall, it gives the wall a tiny push — that push is the gas pressure. Turn up the heat: the balls bounce faster and push harder, so the walls get pushed out — the gas expands. The rules of this bouncy-ball world are the kinetic-molecular theory, and they explain almost everything gases do, as long as the balls are small enough and never stick to each other.

Worked example

Example 1: rms speed of oxygen at room temperature

Calculate the rms speed of O₂ molecules at 298 K. Molar mass of O₂ = 32.00 g/mol.

Step 1 — write the formula:

$u rms = \sqrt 3RT M$

Step 2 — convert molar mass to kg/mol (required for units to work):

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

Step 3 — substitute (R = 8.314 J/(mol·K), and 1 J = 1 kg·m²/s²):

$u rms = \sqrt 3(8.314 J/(mol\cdotK))(298 K) 0.03200 kg/mol = \sqrt 2.32 \times 105 m 2 / s 2 = 482 m/s$

O₂ at room temperature averages ~482 m/s ≈ 1700 km/h — faster than a commercial jet. Using g/mol instead would give ~15,200 m/s, an absurd result that signals the unit error.

Example 2: rms speed comparison — hydrogen versus oxygen

By what factor is the rms speed of H₂ (M = 2.016 g/mol) greater than that of O₂ at the same temperature?

Step 1 — write the ratio (same T, so 3RT cancels):

$u H 2 u O 2 = \sqrt M O 2 M H 2$

Step 2 — substitute molar masses (both in g/mol — the ratio is unitless):

$u H 2 u O 2 = \sqrt 32.00 2.016 = \sqrt 15.87 = 3.98$

H₂ molecules move almost exactly 4× faster than O₂ at the same temperature — the same 1/M relationship as Graham's law.

Example 3: How hot for helium to reach a given speed

At what temperature do He atoms (M = 4.003 g/mol) have an rms speed of 1000 m/s?

Step 1 — write the formula and solve for T:

$u rms = \sqrt 3RT M \to T = M u rms 2 3R$

Step 2 — convert M to kg/mol: 4.003 g/mol = 0.004003 kg/mol.

Step 3 — substitute:

$T = (0.004003 kg/mol)(1000 m/s) 2 3(8.314 J/(mol\cdotK)) = 4003 24.94 = 160.5 K$

Helium reaches 1000 m/s at about −113 °C (160.5 − 273.15 = −112.7 °C). This is why helium balloons lose gas so fast even in cold weather — helium molecules are already speedy at low temperatures.

Key takeaways

Five postulates: tiny particles, constant random motion, elastic collisions, no intermolecular forces, KE ∝ T (kelvin).
Temperature measures average kinetic energy: KE = 32RT per mole — same for all gases at the same T.
rms speed: u_rms = 3RT/M, with M in kg/mol — the most common calculation error is using g/mol.
Maxwell–Boltzmann distribution: spread of speeds; higher T shifts the curve to higher speeds; lighter molecules are faster at the same T.
KMT derives Boyle's, Charles's, Avogadro's, Dalton's, and Graham's laws.
Ideal gas assumptions fail at high pressure and low temperature → real gases deviate (next topic).
For O₂ at 298 K: u_rms = 3(8.314)(298)/0.0320 ≈ 482 m/s — faster than the speed of sound in air.

Check yourself

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

List the five postulates of the kinetic-molecular theory.
Show answer
(1) Gases are many tiny particles with negligible volume; (2) constant random straight-line motion; (3) collisions are perfectly elastic; (4) no attractive/repulsive forces between particles; (5) average kinetic energy ∝ kelvin temperature.
What is the average kinetic energy per mole of any gas at 300 K?
Show answer
KE = 32RT = 32(8.314)(300) = 3.74 × 10³ J/mol = 3.74 kJ/mol — identical for every gas at 300 K.
Calculate the rms speed of N₂ (M = 28.02 g/mol) at 298 K.
Show answer
u_rms = 3(8.314)(298)/(0.02802) = 2.65 × 10⁵ = 515 m/s.
Explain Boyle's law using KMT.
Show answer
At constant temperature molecules keep the same average speed, so compressing the gas into a smaller volume shortens the distance between wall collisions — particles hit walls more often, so pressure rises.
Why does a gas cool when it expands rapidly (e.g., aerosol spray)?
Show answer
Rapid expansion does work on the surroundings, drawing energy from the gas's kinetic energy; slower molecules mean lower temperature — the principle behind refrigeration.
Under what conditions does the ideal gas model fail, and why?
Show answer
At high pressure (particles crowded; their own volume and attractions matter) and low temperature (slow particles linger near each other; attractions dominate); real gases then deviate from PV = nRT.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Kinetic-molecular theory: Model of gases as many tiny, fast, non-interacting particles in random motion
Postulate: A basic assumption of a theory
Elastic collision: Collision in which total kinetic energy is conserved
Average kinetic energy: 32RT per mole; proportional to kelvin temperature
rms speed (urms): Speed of a particle with the average kinetic energy; 3RT/M
Maxwell–Boltzmann distribution: The spread of molecular speeds in a gas at a given temperature
Absolute zero: 0 K (−273.15 °C), where particle motion is minimal

Sources & references

openstax.org — Chemistry 2e

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

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In 30 seconds

Why this matters

The college version

Core Concepts

The five postulates

Pressure from collisions

Temperature is average kinetic energy

Molecular speeds: the rms speed

The Maxwell–Boltzmann distribution

Explaining the gas laws

Limits of the model

Common Confusions

The same idea, in plain words

Worked example

Example 1: rms speed of oxygen at room temperature

Example 2: rms speed comparison — hydrogen versus oxygen

Example 3: How hot for helium to reach a given speed

Key takeaways

Check yourself

Keep learning

Key vocabulary

Related lessons

Sources & references