The big idea: A gas is just tiny particles flying around and bouncing off the walls. That one picture explains both pressure and temperature.
Pressure
- Caused by those particles hitting the walls
- Each collision gives the wall a tiny push
- Billions of them together make a steady force on every bit of wall
Temperature
- Measures how fast the particles move
- The hotter the gas, the faster they go
What 'absolute temperature' means: Absolute temperature is measured in kelvin (K), starting from absolute zero (0 K = −273 °C), the coldest possible point where particle motion is least.
To go from Celsius to kelvin, add 273: 27 °C = 300 K. Always use kelvin in the kinetic-model formulas.
The key link — energy and temperature: The average kinetic energy of the particles (the energy of their motion) is proportional to the absolute temperature.
In short: average KE ∝ T (with T in kelvin). Double the temperature in kelvin → double the average kinetic energy.
Animated graph
Watch the graph build step by step in study mode.
Spot it: Faster particles → harder, more frequent wall collisions → higher pressure.
Higher temperature → more average kinetic energy → faster particles. Temperature and average kinetic energy go up together.
Where the pressure comes from
- pressure (Pa)
- density of the gas (kg m⁻³)
- mean square speed (m² s⁻²)
Pressure and mean kinetic energy: Since ρ = Nm/V and Ē_k = ½mv̄², the same result reads p = 2/3 (N/V) Ē_k: more molecules per unit volume, or faster ones, mean more pressure.
Air at 1.2 kg m⁻³ and 1.0 × 10⁵ Pa: v̄² = 3p/ρ = 2.5 × 10⁵ m² s⁻², a root-mean-square speed of about 500 m s⁻¹.
The model and real gases: The kinetic model assumes: a very large number of molecules in random motion; molecules much smaller than the gaps between them (their own volume is negligible); no forces between molecules except during collisions; elastic collisions that take negligible time.
A real gas behaves like this ideal gas at low pressure, high temperature and low density: the molecules are far apart and fast, so their size and the forces between them hardly matter. Near liquefaction (high pressure, low temperature) the model fails.
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The link between average kinetic energy and absolute temperature is given as a formula. kB is the Boltzmann constant — a fixed number (1.38 × 10⁻²³ J K⁻¹) that connects energy to temperature for a single particle.
- average kinetic energy of one particle (J)
- Boltzmann constant (1.38 × 10⁻²³ J K⁻¹)
- absolute temperature, in kelvin (K)
Two things to get right: 1. Always put T in kelvin — add 273 to a Celsius temperature first.
2. This is the average kinetic energy of one particle. It does not depend on the gas's mass or what gas it is — only on the temperature.
A sample of helium is at 27 °C, with kB = 1.38 × 10⁻²³ J K⁻¹. Find the average kinetic energy of one helium particle.
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- internal energy (J)
- number of molecules, amount in moles
- temperature (K)
0.30 m³ of helium at 300 K and 1.0 × 10⁵ Pa is in a rigid container and gains 9.0 kJ of thermal energy. Calculate its rise in temperature.
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How this is tested — this micro is tested mostly as explanation, not just calculation:
Paper 1A
- A quick explain / identify question — e.g. why molecular speed rises when a gas is compressed.
- Or why two gases at the same temperature have the same average kinetic energy.
Paper 2
- Link the particle picture to pressure and temperature in words.
The classic trap: Thinking a heavier gas has more kinetic energy at the same temperature. It does not — at the same temperature every gas has the same average kinetic energy (the heavier particles just move more slowly).
Compression and speed: When a piston pushes in quickly, it does work on the gas. That work goes into the particles' motion, so their average kinetic energy rises — which means a higher temperature and faster particles.
Faster particles also hit the walls harder and more often, so the pressure goes up too.
The words to know
| Word | What it means |
|---|---|
| Ideal gas | A gas that follows the kinetic model exactly: point molecules, no forces except in collisions |
| Mean square speed | The average of v² over all the molecules |
| Internal energy of an ideal gas | Total random kinetic energy: U = 3/2 nRT for a monatomic gas |
| Elastic collision | A collision in which no kinetic energy is lost |
A gas is sealed in a cylinder by a piston. The piston is suddenly pushed in, quickly compressing the gas. Explain, using the kinetic model, why the average speed of the gas molecules increases.
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