AP® Chemistry review sheet from Aim for Five (aimforfive.com/chem/units/3/3-5)
Unit 3 · Topic 3.5
3.5 Kinetic Molecular Theory
Kinetic molecular theory explains gas behavior with tiny particles in constant random motion. The Kelvin temperature is proportional to the particles' average kinetic energy, so at the same temperature lighter particles move faster, and a Maxwell-Boltzmann distribution shows the spread of particle speeds.
Key terms
- kinetic molecular theory
- average kinetic energy
- Kelvin temperature
- Maxwell-Boltzmann distribution
- particle speed
The model
Kinetic molecular theory (KMT) is the particle model behind the ideal gas law. Its main assumptions for an ideal gas:
- Gas particles are in constant, random motion.
- The particles themselves take up a negligible volume compared with the container.
- Particles don't attract or repel each other except during collisions.
- Collisions are elastic: kinetic energy can be passed between particles, but the total isn't lost.
- The average kinetic energy of the particles is proportional to the Kelvin temperature.
Temperature and kinetic energy
A particle's kinetic energy is KE = ½mv², where m is its mass and v is its speed. The Kelvin temperature is proportional to the average kinetic energy of the particles, so doubling the Kelvin temperature doubles the average kinetic energy.
At the same temperature, all gases have the same average kinetic energy. Since KE = ½mv², a lighter particle must move faster to have the same kinetic energy. Helium atoms at room temperature move, on average, about 3.2 times faster than argon atoms, which are about 10 times heavier.
KMT explains pressure too. Pressure comes from particles hitting the container walls. Faster particles hit harder and more often, so raising the temperature at constant volume raises the pressure.
KMT also explains the simple gas laws. Squeeze a gas into a smaller volume at constant temperature, and particles hit the walls more often, so the pressure rises. Heat a gas in a flexible container at constant pressure, and the faster particles push the walls outward until the pressure from their collisions matches the outside pressure again, so the volume grows.
The Maxwell-Boltzmann distribution
Not every particle has the same speed. A Maxwell-Boltzmann distribution is a graph of the number (or fraction) of particles on the y-axis against speed or kinetic energy on the x-axis. It starts at zero, rises to a peak (the most common speed) and has a long tail toward high speeds.
The area under the curve represents all the particles, so it stays the same if the number of particles doesn't change.
| Change | Effect on the speed distribution |
|---|---|
| Higher temperature | peak shifts right (faster) and gets lower; curve spreads out |
| Lower temperature | peak shifts left and gets taller and narrower |
| Lighter gas, same temperature | peak at higher speed, curve broader and lower |
| Heavier gas, same temperature | peak at lower speed, curve narrower and taller |
Why it matters later
In kinetics (topic 5.5), you'll use the same distribution plotted against energy to explain why a small rise in temperature can speed up a reaction a lot: it raises the fraction of particles with enough energy to react.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Comparing two gases at the same temperature
Samples of He and Ar are at 300 K. Compare (a) the average kinetic energies of their atoms and (b) their average speeds.
Show the solutionHide the solution
- Step 1: (a) Average KE depends only on the Kelvin temperature. Both are at 300 K, so their average kinetic energies are equal.
- Step 2: (b) With equal KE, ½m(He)·v(He)² = ½m(Ar)·v(Ar)², so v(He)/v(Ar) = √(m(Ar)/m(He)).
- Step 3: √(39.95 ÷ 4.003) = √9.98 = 3.16.
- Step 4: He atoms move about 3.16 times faster on average because they're lighter.
Answer: (a) Equal average kinetic energies. (b) He atoms move about 3.16 times as fast as Ar atoms on average.
- Example 2Calculator allowed
Doubling the temperature (classic trap)
A gas is heated from 100 °C to 200 °C. Does the average kinetic energy of its particles double?
Show the solutionHide the solution
- Step 1: Average kinetic energy is proportional to the Kelvin temperature, not the Celsius temperature.
- Step 2: 100 °C = 373.15 K and 200 °C = 473.15 K.
- Step 3: 473.15 ÷ 373.15 = 1.27, so the average kinetic energy increases by about 27%, not 100%.
Answer: No. It increases by a factor of about 1.27, because the Kelvin temperature only rises from 373 K to 473 K.
- Example 3
Describing a distribution change
Describe how the Maxwell-Boltzmann speed distribution of a sample of N₂ changes when it is heated from 300 K to 600 K.
Show the solutionHide the solution
- Step 1: Higher temperature means higher average kinetic energy, so more particles move at high speeds.
- Step 2: The peak moves to a higher speed, and the curve spreads out over a wider range of speeds.
- Step 3: The number of particles is unchanged, so the area under the curve stays the same; spreading out means the peak gets lower.
Answer: The curve shifts to higher speeds, flattens and broadens, with a lower peak and the same total area.
Common mistakes
- Saying heavier gases have more kinetic energy at the same temperature. Average KE is the same; heavier particles are slower.
- Using Celsius when comparing kinetic energies.
- Drawing the high-temperature curve taller. It's lower and wider, because the area stays the same.
- Thinking all particles in a sample move at the same speed.
On the exam
- Expect to compare two Maxwell-Boltzmann curves and say which is at the higher temperature or which belongs to the lighter gas. Justify using kinetic energy and the shape of the curve.
- When explaining a gas property, connect what the particles are doing (speed, collisions with walls) to what is measured (pressure, temperature).
Connected topics
Videos
Check yourself
4 questions on 3.5 Kinetic Molecular Theory. Pick an answer to see if you got it, and why.
Two identical flasks at 300 K contain equal numbers of moles of helium and argon. Which of the following correctly compares the gases?
A sample of N₂ gas is heated from 300 K to 600 K. Compared with the Maxwell-Boltzmann distribution of molecular speeds at 300 K, the distribution at 600 K
A rigid, sealed steel container of nitrogen gas is heated. According to kinetic molecular theory, which of the following best explains why the pressure increases?
A sample of argon gas in a sealed container is warmed from 300 K to 400 K. Its Maxwell-Boltzmann distribution of speeds is plotted at both temperatures, with the number of atoms on the vertical axis. Which feature of the two curves is the same?
0 of 4 answered