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Kinetic Theory of Gases: Pressure, Temperature and rms Speed

Kinetic theory explains gas behaviour by picturing a gas as tiny, fast, randomly moving molecules that bounce elastically and do not pull on each other. Each hit on a wall reverses the molecule's velocity and hands the wall momentum 2mvₓ. Adding the hits of all molecules gives the pressure P = ⅓ n m v̄² = ⅓ ρ v̄². Comparing with PV = N k T shows that the average kinetic energy of a molecule is (3/2) k T: temperature is a measure of the average kinetic energy of the molecules. The root mean square speed is v_rms = √(3RT/M) = √(3kT/m), so lighter gases move faster at the same temperature.

🎬 Step-by-step story

  1. Kinetic theory pictures a gas as tiny balls that move at random, pull on nothing, and bounce off walls without losing energy.
  2. Follow one red molecule. When it hits the right wall, its x-velocity flips from +vₓ to −vₓ. The wall gets momentum 2mvₓ.
  3. Now all molecules hit the walls again and again. The average push per area is the pressure: P = ⅓ n m v̄².
  4. Heat the gas. Molecules speed up, and their average kinetic energy grows as (3/2) k T. Temperature is a measure of this energy.
  5. Speed depends on mass. At the same temperature light H₂ molecules move 4 times faster than heavy O₂: v_rms = √(3RT/M).
  6. Your turn: change the temperature and the gas. Read the rms speed and kinetic energy.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why do we ignore the size of molecules?

The gaps between gas molecules are about 10 times their size, so their own volume is a tiny part of the container. Ignoring it keeps the model simple and works well at low pressure.

Why does the wall get 2mvₓ and not mvₓ?

The molecule arrives with +mvₓ and leaves with −mvₓ. The total change is 2mvₓ, and by Newton's third law the wall gets an equal and opposite share.

Where does the one-third in P = ⅓ n m v̄² come from?

Molecules move at random, so their speed squared is shared equally among x, y and z. Only the x part pushes the right wall, which is one-third of v̄².

Does a heavy gas have more energy than a light gas at the same temperature?

No. Each molecule's average KE is (3/2)kT for any gas. A heavy molecule simply moves slower to carry the same energy.

Why is rms speed used instead of the normal average speed?

Pressure and kinetic energy depend on v², so the natural average is the mean of v². Its square root is v_rms. Also, the average velocity (with direction) is zero.

Do all molecules have the same speed?

No. Speeds are spread out; a few are very slow and a few very fast. v_rms, v̄ and v_p are three different ways to describe this spread.

Assumptions of kinetic theory

To build a simple model, we assume:

  1. A gas is made of a very large number of identical molecules.
  2. The molecules are tiny: their own size is negligible compared with the gaps between them.
  3. They move in all directions at random, with many speeds.
  4. They do not attract or repel each other, except during collisions.
  5. Collisions (with each other and with walls) are perfectly elastic: kinetic energy is not lost.
  6. A collision takes almost no time compared with the time between collisions.
  7. Newton's laws apply to each molecule. Gravity on them is ignored.

Pressure of an ideal gas: the derivation

Take a cube of side l with N molecules, each of mass m.

Step 1: one hit

A molecule with x-velocity vₓ hits the right wall and comes back with −vₓ. Its momentum change is −2mvₓ, so the wall receives 2mvₓ.

Step 2: how often

It goes across and back (2l) before hitting the same wall again. Time between hits = 2l/vₓ. Hits per second = vₓ/2l.

Step 3: force from one molecule

Force = momentum per second = 2mvₓ × vₓ/2l = mvₓ²/l.

Step 4: all molecules

F = (m/l) Σ vₓ². Pressure P = F/l² = (m/l³) Σ vₓ² = (m N/V) × average of vₓ².

Step 5: share out directions

Motion is random, so the average of vₓ² = average of v_y² = average of v_z² = ⅓ v̄² (mean square speed). So

P = ⅓ n m v̄² = ⅓ ρ v̄², where n = N/V and ρ = nm is the density.

So PV = ⅓ N m v̄² = ⅔ × (total translational kinetic energy).

Kinetic meaning of temperature

Compare PV = ⅓ N m v̄² with the gas equation PV = N kB T:

⅓ m v̄² = kB T, so average KE of one molecule = ½ m v̄² = (3/2) kB T.

rms speed

The root mean square (rms) speed is found in three steps: square all speeds, take their mean, then take the square root.

vrms = √(v̄²) = √(3kBT/m) = √(3RT/M) = √(3P/ρ)

Other averages: mean speed v̄ = √(8RT/πM) and most probable speed vp = √(2RT/M). Order: vp < v̄ < vrms.

Try it: the 3D and at home

Board exam focus

Key formulas and definitions

Worked examples

1. Four molecules have speeds 2, 4, 6 and 8 m/s. Find their rms speed.

Step 1: squares: 4, 16, 36, 64. Step 2: mean = 120/4 = 30. Step 3: square root. Answer: v_rms = √30 ≈ 5.48 m/s (the plain mean is 5 m/s).

2. Find the average kinetic energy of a gas molecule at 27 °C. (k = 1.38 × 10⁻²³ J/K)

Step 1: T = 300 K. Step 2: KE = (3/2) k T = 1.5 × 1.38 × 10⁻²³ × 300. Answer: KE ≈ 6.21 × 10⁻²¹ J.

3. Find the rms speed of oxygen molecules at 300 K. (M = 32 g/mol, R = 8.31)

Step 1: M = 0.032 kg/mol. Step 2: v_rms = √(3 × 8.31 × 300 / 0.032) = √(233719). Answer: v_rms ≈ 483 m/s.

4. At what temperature will the rms speed of a gas be double its value at 27 °C?

Step 1: v_rms ∝ √T. Step 2: to double v, T must become 4 times. Step 3: T = 4 × 300 = 1200 K. Answer: 1200 K = 927 °C.

5. Compare the rms speeds of hydrogen (M = 2) and oxygen (M = 32) at the same temperature.

Step 1: v_H/v_O = √(M_O/M_H). Step 2: = √(32/2) = √16. Answer: hydrogen is 4 times faster.

6. A gas has density 1.25 kg/m³ at a pressure of 1.0 × 10⁵ Pa. Find the rms speed of its molecules.

Step 1: v_rms = √(3P/ρ). Step 2: = √(3 × 10⁵ / 1.25) = √(240000). Answer: v_rms ≈ 490 m/s.

7. Find the total translational kinetic energy of 2 mol of helium at 400 K. (R = 8.31)

Step 1: KE for n moles = (3/2) n R T. Step 2: = 1.5 × 2 × 8.31 × 400. Answer: KE ≈ 9.97 × 10³ J.

Common mistakes

Practice quiz

1. In kinetic theory, collisions of molecules are assumed to be:
2. Pressure of an ideal gas is P =
3. Average kinetic energy of a gas molecule is:
4. If the kelvin temperature of a gas is made 4 times, its rms speed becomes:
5. At the same temperature, which gas has the highest rms speed?

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is the kinetic theory of gases?

It explains gas pressure and temperature by treating a gas as many tiny molecules in random motion that collide elastically.

What is the formula of rms speed?

v_rms = √(3RT/M) = √(3kT/m) = √(3P/ρ).

What is the kinetic interpretation of temperature?

Temperature is proportional to the average kinetic energy of the molecules: ½ m v̄² = (3/2) kT.

Where this is taught

Ukraine10 класMolecular physics and thermodynamics
Ukraine10 класMolecular physics and thermodynamics
CBSE (India)Class 11Behaviour of Perfect Gases and Kinetic Theory of Gases
England (GCSE, A level)Year 133.6 Further mechanics and thermal physics
USA (Common Core, NGSS, AP)Grade 11Properties of Substances and Mixtures
USA (Common Core, NGSS, AP)Grade 12Thermodynamics
Japan高校2年Various motions
Russia10 классMolecular kinetic theory
Russia10 классMolecular physics
China高三Selective 3 Ch.1 Molecular kinetic theory

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