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General Chemistry I · Core Concept

Kinetic Molecular Theory

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On this page 8 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Worked example
  6. Key takeaway
  7. Quick check
  8. Study tools

In 30 seconds

The models a gas as a huge collection of tiny particles in constant, random motion that exert pressure by colliding with container walls. depends only on temperature, so at a given temperature lighter molecules move faster — quantified by the urms = 3RT/M. predicts that lighter gases effuse and diffuse faster (inversely proportional to the square root of molar mass). Real gases deviate from these ideal predictions at high pressure and low temperature, and the corrects for finite molecular size and intermolecular attractions.

Why this matters

and Graham's law underpin gaseous in the lungs: oxygen and carbon dioxide exchange across the alveolar membrane at rates influenced by their molecular masses. Clinically, this is why conditions that thicken the alveolar–capillary barrier slow gas exchange. In the laboratory, Graham's law is the basis of isotope enrichment by effusion (e.g., separating 235UF6 from 238UF6) and of leak detection using fast, light helium. The van der Waals equation matters in designing gas-storage systems and understanding why real gases (unlike ideal ones) can be liquefied — attractive forces, absent in the ideal model, are what allow condensation.

The college version

1. The Postulates of Kinetic Molecular Theory

KMT describes an ideal gas through five assumptions:

  1. A gas consists of a large number of tiny particles (atoms or molecules) in constant, random, straight-line motion.
  2. The volume of the particles themselves is negligible compared with the total volume of the gas.
  3. Particles exert no forces on one another except during collisions; there are no attractive or repulsive forces.
  4. Collisions between particles (and with container walls) are perfectly elastic — no kinetic energy is lost.
  5. The average kinetic energy of the particles is proportional to the absolute (Kelvin) temperature.

The same postulates "explain" the empirical gas laws: pressure comes from wall collisions, temperature from average kinetic energy, and Boyle's/Charles's laws from how speed and collision frequency respond to volume and temperature changes.

2. Root-Mean-Square Speed and Temperature

Because particles in a sample have a distribution of speeds, chemists summarize the sample with the root-mean-square (rms) speed:

urms = 3RTM

where R = 8.314 J·mol-1·K-1, T is the temperature in kelvin, and M is the molar mass in kilograms per mole (kg/mol). This equation shows two key ideas: rms speed rises with the square root of temperature, and falls as the square root of molar mass increases. Average kinetic energy per mole is KE = 32RT, depending only on temperature — so at a fixed temperature, all gases have the same average kinetic energy, and lighter ones simply move faster.

3. Effusion, Diffusion, and Real Gases

Effusion is the escape of a gas through a tiny hole into a vacuum; diffusion is the gradual mixing of gases as particles wander through one another. Graham's law relates their rates to molar mass:

r1r2 = M2M1

Lighter gases effuse and diffuse faster. Real gases deviate from ideal behavior at high pressure (particles crowded so their own volume matters) and low temperature (slow particles feel intermolecular attractions). The van der Waals equation corrects the ideal gas law:

(P + an2V2)(V - nb) = nRT

The a term accounts for attractive forces (reducing pressure), and the b term accounts for the finite volume of molecules. Each gas has its own a and b constants.

How it works

  1. Model the gas as independent particles in random, elastic motion.
  2. Use urms = 3RT/M (with M in kg/mol) to estimate particle speed.
  3. Recognize that at fixed temperature all gases share the same average kinetic energy, so lighter = faster.
  4. For effusion or diffusion, apply Graham's law using molar masses.
  5. For real gases at high pressure or low temperature, apply the van der Waals equation with the gas's a and b constants.

Common confusions

Do not confuseWithDifference
EffusionDiffusionEffusion = escape through a tiny hole; diffusion = mixing through space
Root-mean-square speedAverage speedrms is a specific (larger) average weighted by squared speeds; both follow T/M
g/mol in rms formulakg/mol in rms formulaThe formula requires kg/mol so units cancel to m/s
Ideal gasReal gasIdeal ignores size and attractions; real gases deviate at high P, low T
a termb terma corrects for attractions (pressure); b corrects for molecular volume

Memory aids

"Light = Fast, Hot = Fast" — speed grows with temperature and shrinks with mass, captured by urms = 3RT/M. For Graham's law, remember "heavy is slow" (rate is inversely proportional to the square root of molar mass).

Quick review

Topic Recap

The kinetic molecular theory models an ideal gas as non-interacting point particles in constant random motion whose collisions are elastic, explaining the empirical gas laws. Temperature measures average kinetic energy, so lighter molecules move faster (urms = 3RT/M), and Graham's law predicts lighter gases effuse and diffuse faster. Real gases break the ideal model at high pressure and low temperature, which the van der Waals equation corrects with the a (attraction) and b (volume) terms.

Knowledge Check

  1. Which KMT postulate explains why gas pressure increases with temperature at constant volume?
  2. At the same temperature, which gas has the greater rms speed: Ne (20.2 g/mol) or Ar (39.9 g/mol)?
  3. Calculate the rms speed of He (4.003 g/mol) at 298 K.
  4. Which gas effuses faster, and by what factor: He (4.0 g/mol) or SO₂ (64 g/mol)?
  5. Under what conditions do real gases deviate most from ideal behavior?

Answers and Rationales

  1. Particles move faster at higher temperature, so they collide with the walls harder and more often, raising pressure. (Average kinetic energy is proportional to Kelvin temperature.)
  2. Ne. Lighter molar mass at the same temperature means higher rms speed (urms ∝ 1/M).
  3. 1.36 × 10³ m/s. M = 0.004003 kg/mol; urms = 3(8.314)(298)/0.004003 ≈ 1360 m/s.
  4. He, by a factor of 4. 64/4.0 = 16 = 4; helium effuses four times faster.
  5. High pressure and low temperature. High pressure makes molecular volume significant; low temperature lets attractions dominate.
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

Picture a swarm of bees inside a jar, flying randomly and bumping into the walls. The theory of gases says a gas is exactly that: tiny particles (the bees) flying in straight lines until they crash into each other or the walls, and every crash against the wall is what we feel as pressure. The hotter the gas, the faster the bees fly; and if two different kinds of bees are at the same temperature, the lighter ones zip around faster than the heavy ones. If you poke a tiny hole in the jar, the fast, light bees escape first — that's why a light gas "leaks" (effuses) faster than a heavy one. This picture stops being exact when the bees are so big or so crowded that they can't be treated as points, and when they start sticking to each other — which is exactly what happens to real gases under high pressure or very low temperature.

Simple Example

At the same temperature, hydrogen molecules (molar mass 2.0 g/mol) move much faster than oxygen molecules (32 g/mol). Graham's law predicts hydrogen effuses 32/2 ≈ 4 times faster than oxygen through the same small opening.

Worked example

Worked Example 1 — Root-mean-square speed. Calculate the rms speed of O₂ molecules at 25.0 °C. (Molar mass of O₂ = 32.00 g/mol = 0.03200 kg/mol.)

T = 25.0 + 273.15 = 298.15 K

urms = 3(8.314 J·mol-1·K-1)(298.15 K)0.03200 kg·mol-1 = 2.32 × 105 m2/s2 ≈ 482 m/s

This is over 1000 mph — typical for small molecules at room temperature.

Worked Example 2 — Graham's law. Compare the effusion rates of H₂ (2.016 g/mol) and O₂ (32.00 g/mol) through the same opening.

rH2rO2 = MO2MH2 = 32.002.016 = 15.87 ≈ 3.98

Hydrogen effuses about 4 times faster than oxygen.

Worked Example 3 — Temperature dependence of speed. If the rms speed of N₂ is 515 m/s at 300 K, what is it at 1200 K?

Because urms ∝ T:

u1200 = 515 m/s × 1200 K300 K = 515 m/s × 2.00 = 1030 m/s

Quadrupling the Kelvin temperature doubles the rms speed.

Common setup errors. (1) Using molar mass in g/mol instead of kg/mol in the rms-speed formula — the units won't cancel to m/s. (2) Using Celsius temperatures. (3) Forgetting that Graham's law has molar mass under the square root inverted (lighter gas has the larger rate). (4) Confusing effusion (through a hole) with diffusion (mixing through a medium).

Key takeaways

  • High yield: Average kinetic energy depends only on temperature: KE = 32RT.
  • High yield: urms = 3RT/M — use molar mass in kg/mol.
  • High yield: Graham's law: lighter gas effuses faster by Mheavy/Mlight.
  • KMT assumes negligible particle volume, no intermolecular forces, and elastic collisions.
  • Real gases deviate from ideality at high pressure and low temperature.
  • The van der Waals b term corrects for molecular volume; the a term corrects for attractions.
  • At a given temperature, all gases have the same average kinetic energy regardless of identity.
  • Effusion is through a tiny hole; diffusion is mixing through space.

Quick check

1 question here. Answers stay hidden until you check.

Question 1 of 1

According to kinetic molecular theory, the average kinetic energy of gas molecules depends only on:

Choose an answer, then check it.

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Study tools & related lessonsYou’ll learn to · Key vocabulary · Related

You’ll learn to

  • State the postulates of the kinetic molecular theory (KMT) of gases.
  • Relate average kinetic energy and root-mean-square speed to Kelvin temperature and molar mass.
  • Distinguish effusion from diffusion and apply Graham's law.
  • Explain when and why real gases deviate from ideal behavior, and describe the van der Waals equation.

Key vocabulary

Kinetic molecular theory
Model of gases as tiny particles in constant random motion
Elastic collision
Collision that loses no kinetic energy
Root-mean-square speed
3RT/M, a representative molecular speed
Average kinetic energy
32RT per mole; depends only on temperature
Effusion
Gas escaping through a tiny hole
Diffusion
Gas spreading through another medium
Graham's law
Rates ∝ 1/M
Real gas
A gas that deviates from ideal behavior
van der Waals equation
Ideal gas law corrected for size (b) and attractions (a)

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