Short on time? This page is the recap. For explanations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- A gas is a huge number of atoms moving randomly. They collide with each other and with the walls, and momentum is conserved in each collision.
- Pressure comes from atoms changing momentum when they hit a surface. Many tiny impulses each second add up to a steady force.
- Only the perpendicular velocity component reverses in an elastic bounce. The parallel component is unchanged.
- Pressure exists everywhere inside a gas, not just at the walls.
- Temperature measures the average kinetic energy of the atoms. One atom has no temperature.
- The Maxwell–Boltzmann distribution shows the spread of atomic speeds. You need its shape, not its formula.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| Δp = 2m v cos θ | m in kg, v in m/s, θ from the normal | find the momentum change of one atom in a bounce |
| F⊥ = (collisions per second) × Δp | F in N | turn many impulses into an average force |
| P = F⊥/A | P in Pa = N/m², A in m² | find the pressure on a surface |
| K_avg = (3/2)k_B T = ½m v_rms² | k_B = 1.38 × 10⁻²³ J/K, T in K | link temperature, kinetic energy and speed |
| v_rms = √(3k_B T/m) | m = mass of one atom | compare speeds of gases or temperatures |
| Change | Effect on the speed distribution |
|---|---|
| Higher T, same gas | peak moves to higher speed, curve wider, peak lower, area unchanged |
| Lighter atoms, same T | same shifts as a hotter gas; same K_avg |
Assumptions behind the numbers
- The gas is ideal: atoms are tiny and interact only in collisions.
- Collisions are elastic and walls are smooth.
- Temperatures are in kelvin.
- “Same direction for every atom” calculations are simplified models of a random gas.
Mistakes to avoid
- Δp = mv for a bounce. It is 2mv head-on, because the atom reverses.
- Using sin θ for the perpendicular part. Measure θ from the normal; the perpendicular part is v cos θ.
- Forgetting to convert cm² to m². 1 cm² = 1 × 10⁻⁴ m².
- Doubling T doubles v_rms. It doubles K_avg; v_rms rises by √2.
- Heavier atoms have more kinetic energy at the same T. Same K_avg, lower speed.
- Heating makes the curve taller. The peak gets lower and moves right.
- Using °C in (3/2)k_B T.
Quick self-check
- A helium atom (6.65 × 10⁻²⁷ kg) hits a wall head-on at 1200 m/s and rebounds elastically. What is the magnitude of its momentum change? (1.60 × 10⁻²³ kg·m/s, which is 2mv)
- What is the average kinetic energy of an atom in a gas at 400 K? (8.28 × 10⁻²¹ J)
- A gas is heated from 100 K to 900 K. By what factor does v_rms change? (×3, since √9 = 3)
Next: practice questions.