Kinetic Molecular Theory: Study Guide (Chemistry 3.5)

From a law to a model

In Topic 3.4 you used PV = nRT to connect pressure, volume, temperature and amount of gas. That law describes what you measure. It does not say why gases behave that way. The kinetic molecular theory (KMT) answers the “why”. It links the large-scale (macroscopic) properties you measure to the motion of the tiny particles you cannot see.

“Kinetic” means “to do with motion”. The central idea is simple: all particles of matter are in continuous, random motion. This is true in solids and liquids as well as gases. In a gas the particles are far apart and move freely, so the model is easiest to apply.

The assumptions of the model

KMT describes an ideal gas with five assumptions:

1. Constant, random motion. Gas particles move in straight lines until they hit another particle or a wall. Their directions are random.
2. Particle volume is negligible. The particles are tiny compared with the space between them, so almost all of the gas’s volume is empty space.
3. No attractions or repulsions. Particles do not pull on or push each other, except at the instant of a collision.
4. Elastic collisions. When particles collide, they can swap energy, but the total kinetic energy does not change. No energy is lost as heat to the walls.
5. Temperature measures average kinetic energy. The average kinetic energy of the particles is proportional to the Kelvin temperature.

Assumptions 2 and 3 are never exactly true for real gases. In Topic 3.6 you will see when they break down and what happens then.

Pressure from collisions

When a particle hits a wall, it pushes on the wall and bounces back. Billions of these impacts every second add up to a steady force. Pressure is the force of these collisions per unit area of wall. Two things decide the pressure:

• How often particles hit each square centimetre of wall (collision frequency).
• How hard each particle hits (which depends on its speed and mass).

Every gas law from Topic 3.4 follows from these two ideas.

Change (others held constant)	What happens to the particles	Effect on pressure
Raise T (fixed V, n)	Particles move faster: they hit the walls more often and harder	P increases (P ∝ T)
Reduce V (fixed T, n)	Same speeds, but less wall area and less distance to travel, so more hits per unit area per second	P increases (P ∝ 1/V)
Add more gas (fixed V, T)	More particles, so more hits per second	P increases (P ∝ n)
Raise T (fixed P, n), for example a balloon	Faster particles push the walls outward until the collision rate per unit area falls back	V increases (V ∝ T)
Mix two gases (fixed V, T)	Particles do not attract or repel, so each gas hits the walls as if it were alone	P_total = P_A + P_B + …

Notice one point in the second row: squeezing a gas at constant temperature does not make the particles move faster. Only a change in temperature changes the average kinetic energy.

A particle diagram

In a particle diagram, keep the number of particles the same when only temperature changes, give the particles random directions, and show a range of arrow lengths in both boxes. Some particles in the hot gas can still be slow.

Temperature and kinetic energy

The kinetic energy of one moving particle is

KE = ½mv², where m is the particle’s mass and v its speed.

Assumption 5 says the average KE of the particles is proportional to the Kelvin temperature. Two results follow, and you will use them often:

• Same temperature → same average kinetic energy, whatever the gas. At 25 °C, helium atoms and sulfur dioxide molecules have the same average KE.
• Kelvin temperature doubles → average KE doubles. Because KE depends on v², the average speed rises by a factor of only √2 ≈ 1.41.

This only works in kelvin. On the Kelvin scale, 0 K means zero particle motion in this model, so the numbers are proportional. On the Celsius scale they are not: 20 °C is not “twice as hot” as 10 °C.

Background (not required): for one mole of an ideal gas, the average kinetic energy is (3/2)RT. At 300 K that is (3/2)(8.314 J mol⁻¹ K⁻¹)(300 K) = 3.74 kJ mol⁻¹, the same for every gas.

Comparing speeds of different gases

If two gases are at the same temperature, their average kinetic energies are equal:

½m₁v₁² = ½m₂v₂², so v₁ / v₂ = √(m₂ / m₁)

The particle mass is proportional to the molar mass, so you can use molar masses in the ratio. A particle four times heavier moves, on average, half as fast.

Worked example 1: two gases at the same temperature

Question. A flask at 25 °C contains a mixture of methane, CH₄, and sulfur dioxide, SO₂. (a) Compare the average kinetic energies of the two kinds of molecule. (b) Which molecules move faster on average, and by what factor?

1. (a) Both gases are at the same temperature, 298 K. The average kinetic energy depends only on the Kelvin temperature, so the average KE of a CH₄ molecule equals the average KE of an SO₂ molecule.
2. (b) Molar masses: M(CH₄) = 12.01 + 4(1.008) = 16.04 g mol⁻¹; M(SO₂) = 32.06 + 2(16.00) = 64.06 g mol⁻¹.
3. Equal average KE means the lighter molecule must be faster: v(CH₄) / v(SO₂) = √(64.06 / 16.04) = √3.99 = 2.00.

Answer. Equal average kinetic energies; methane molecules move about 2.0 times faster on average than sulfur dioxide molecules.

Check. SO₂ is about 4 times heavier, and √4 = 2. Background: the root-mean-square speeds at 298 K are about 681 m s⁻¹ for CH₄ and 341 m s⁻¹ for SO₂, which gives the same ratio. A common error is to say the speed ratio is 4 (the mass ratio) instead of 2.

The Maxwell–Boltzmann distribution

Collisions swap energy between particles all the time, so at any moment some particles are slow, most are moderate and a few are very fast. A Maxwell–Boltzmann distribution is a graph of how many particles (or what fraction) have each speed or kinetic energy at one temperature.

How to read and sketch these graphs:

• Start at the origin. Almost no particles are at rest at any moment, so the curve starts at zero.
• One peak, then a long tail to the right. The peak is the most likely speed (or energy). The tail shows a few very fast particles. The curve is not symmetrical.
• Area = all the particles. For the same sample, the area under the curve stays the same at every temperature.
• Higher temperature: peak moves right (faster), peak gets lower and the curve gets wider. It must be lower because the same area is spread over a wider range. For O₂, about 4% of molecules are faster than 800 m s⁻¹ at 300 K but about 25% at 600 K (model values).
• Same temperature, different gases (speed axis): the heavier gas has a taller, narrower curve with its peak at a lower speed (Figure 3).
• Same temperature, different gases (kinetic energy axis): the curves are identical, because the energy distribution depends only on temperature.

Worked example 2: heating a gas in a rigid container

Question. A sealed steel container of oxygen gas is heated from 27 °C to 327 °C. The pressure at 27 °C is 1.20 atm. (a) By what factor does the average kinetic energy of the O₂ molecules change? (b) By what factor does their average speed change? (c) Use KMT to explain why the pressure rises, and find the new pressure. (d) Describe how the Maxwell–Boltzmann speed distribution changes.

1. Convert to kelvin. T₁ = 27 + 273 = 300 K; T₂ = 327 + 273 = 600 K.
2. (a) Average KE ∝ T, so the factor is 600 / 300 = 2.00. (Using Celsius, 327 / 27 = 12.1, would be badly wrong.)
3. (b) KE ∝ v², so v ∝ √T. The factor is √2.00 = 1.41. The molecules have the same mass at both temperatures, so only speed can change.
4. (c) The volume and number of molecules are fixed. Faster molecules hit the walls more often and with more force on each impact, so the force per unit area rises. P₂ = 1.20 atm × (600 K / 300 K) = 2.40 atm.
5. (d) At 600 K the curve’s peak moves to a higher speed (from about 395 to about 558 m s⁻¹, model values), the peak is lower, the curve is broader, and a larger fraction of molecules have high speeds. The area under the curve is unchanged because the number of molecules is unchanged (Figure 2).

Check. The pressure doubles while the average speed rises by only 1.41 times. That is consistent: both the frequency of collisions and the force of each collision increase, and each goes up by a factor of about 1.41, giving 1.41 × 1.41 = 2.

Common misconceptions

• “Heavier gases have more kinetic energy.” At the same temperature, the average kinetic energy is the same for every gas. Heavier particles simply move more slowly.
• “Compressing a gas speeds up the particles.” At constant temperature the average speed does not change. The pressure rises because collisions with each unit of wall area become more frequent.
• “All particles in a gas have the same speed.” There is always a spread of speeds, shown by the Maxwell–Boltzmann distribution. Temperature fixes the average KE, not each particle’s.
• “Doubling the temperature doubles the speed.” Doubling the Kelvin temperature doubles the average KE; speed rises by √2.
• Using °C in ratios. Average KE is proportional to the Kelvin temperature only.
• Drawing the hot curve taller. At higher temperature the peak is lower and wider, because the area (number of particles) stays the same.
• Drawing a symmetrical bell curve. The distribution starts at zero and has a long tail on the high-speed side.
• “Particles in solids are not moving.” All particles of matter are in continuous random motion; in solids they vibrate about fixed positions.

Where this leads

KMT assumes particles have no volume and no attractions. In Topic 3.6 you will use those assumptions to explain when and why real gases do not obey PV = nRT: see Deviation from Ideal Gas Law. The Maxwell–Boltzmann distribution comes back in Unit 5, where it shows why only a fraction of collisions have enough energy to react, and in Unit 9, where a wider spread of energies means higher entropy. Now try the practice questions, then use the revision notes and the checklist.