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Essential Question

How does particle motion explain gas behavior?

We can’t see gas particles, but we can measure things like pressure and temperature. By the end of this lesson, you should be able to connect what the particles are doing to what we observe in the lab.

What Is KMT?

Kinetic molecular theory (KMT) is a model that links particle motion to gas properties we can measure.

  • Particle level: individual atoms or molecules moving around a container
  • Macroscopic level: pressure (P), volume (V), temperature (T), and amount (n)

KMT is the bridge between these two levels. On the AP exam, you’ll often explain something you can observe using a particle-level reason.

Constant, Random Motion

  • Particles move in straight lines until they collide with another particle or a wall.
  • They move in all directions at many different speeds. At any moment, a gas contains a range of particle speeds.

Pressure Comes from Collisions

  • Particles hit the container walls, and each hit pushes on the wall.
  • Pressure = force ÷ area (P = F/A)
  • More hits or harder hits → greater pressure. Harder hits happen when particles move faster.

Compressing a Gas

  • Same temperature → same particle speeds
  • Smaller volume → more wall hits each second
  • More hits → greater pressure

Cutting the volume from V to ½V gives particles less distance to travel between walls, so they hit the walls more often. This is the particle-level explanation for Boyle’s law: when volume decreases, pressure increases, as long as temperature and moles stay the same.

Kinetic Energy of a Particle

KE = ½mv2

  • KE = kinetic energy (J)
  • m = mass of one particle (kg, not g!)
  • v = speed (m/s)

A faster or heavier particle has more KE. Because v is squared, doubling the speed makes the kinetic energy four times larger. Since 1 J = 1 kg·m2/s2, plugging in kilograms and meters per second gives an answer in joules.

Temperature and Average KE

  • Kelvin temperature is proportional to average KE.
  • Double the K → double the average KE (the speed does not double, because v is squared).
  • Always use kelvins! K = °C + 273

The same gas at 400 K has twice the average kinetic energy of the gas at 200 K, so its particles are faster on average.

Same Temperature, Different Gases

  • Same temperature → same average KE, no matter what the gas is.
  • Lighter particles move faster.

He (4.00 g/mol) and Xe (131.29 g/mol) at 300 K have the same average KE. Since KE = ½mv2, helium’s small mass means a large speed, and xenon’s large mass means a small speed.

Maxwell-Boltzmann Distribution

  • Shows the spread of particle speeds at one temperature (speed on the x-axis, number of particles on the y-axis).
  • Most particles are near the peak, the most probable speed.
  • A few are very slow or very fast. The total area under the curve represents all of the particles.

Maxwell-Boltzmann: Temperature

  • Higher T → peak shifts right and gets lower
  • More fast particles in the high-speed tail
  • Area under the curve stays the same, because the number of particles hasn’t changed

Exam tip: for two curves of the same gas, the flatter curve shifted to the right is at the higher temperature.

Maxwell-Boltzmann: Particle Mass

  • Same T → same average KE
  • Lighter gas: peak farther right, wider curve
  • Heavier gas: slower, taller and narrower curve

He (4.00 g/mol) is about ten times lighter than Ar (39.95 g/mol), so at the same temperature helium’s curve is shifted right and spread out. This looks like heating a gas, but for a different reason, so always check whether the curves show different temperatures or different gases.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Kinetic Energy of a Molecule

An N2 molecule has a mass of 4.65 × 10−26 kg and moves at 515 m/s. Calculate its kinetic energy.

  1. Equation: KE = ½mv2
  2. Knowns: m = 4.65 × 10−26 kg, v = 515 m/s. The mass is already in kilograms, so no conversions are needed.
  3. Square the speed: v2 = (515 m/s)2 = 2.65225 × 105 m2/s2. Square the units too, and keep extra digits until the end.
  4. Substitute: KE = ½(4.65 × 10−26 kg)(2.65225 × 105 m2/s2)
  5. Multiply: KE = ½(1.2333 × 10−20 kg·m2/s2) = 6.17 × 10−21 kg·m2/s2
  6. Answer: KE = 6.17 × 10−21 J (3 significant figures). A tiny number makes sense for the energy of a single molecule.

Example 2: Changing Temperature

A gas sample is heated from 27 °C to 327 °C. By what factor does the average kinetic energy of its particles increase?

  1. Relationship: KEavg ∝ T (in K), so both temperatures must be in kelvins.
  2. Convert: T1 = 27 + 273 = 300 K and T2 = 327 + 273 = 600 K
  3. Set up the ratio: KE2/KE1 = T2/T1 = 600 K / 300 K
  4. Answer: = 2. The average KE doubles. The kelvins cancel, so the factor has no units.
  5. The trap: dividing the Celsius temperatures gives 327/27 ≈ 12, which is wrong. Celsius is not proportional to kinetic energy. Always use Kelvin.

Example 3: Reading M-B Curves

The graph shows Maxwell-Boltzmann distributions for Ne and Kr at 300 K. Curve X has a tall, narrow peak at a low speed, and curve Y is lower and wider with its peak at a higher speed. Which curve represents Kr? Justify your answer.

  1. Compare masses: Kr = 83.80 g/mol and Ne = 20.18 g/mol, so krypton atoms are about four times heavier.
  2. Same temperature: both gases are at 300 K, so they have the same average KE.
  3. Apply KE = ½mv2: with the same KE, a larger m means a smaller v. Krypton atoms move slower on average.
  4. Answer: Kr = Curve X (peak at a lower speed, taller and narrower).

Sample exam justification: Curve X is krypton. Both gases are at the same temperature, so they have the same average kinetic energy. Because krypton atoms have a greater mass, they must have a lower average speed, so krypton’s distribution peaks at a lower speed.

Big Ideas

  • Gas particles are in constant, random motion.
  • Wall collisions cause pressure.
  • Kelvin T is proportional to average KE.
  • M-B curves show the spread of speeds.

Back to the essential question: how fast the particles move tells us the temperature, and the number and force of their collisions with the walls tell us the pressure. Next up, Topic 3.6 looks at what happens when real gases stop behaving the way this model predicts.