Physics / A2 level
Link a macroscopic gas law to random particle motion.
The ideal-gas equation relates pressure, volume, amount of gas and absolute temperature. An ideal model neglects particle volume and intermolecular forces except during brief elastic collisions.
Pressure comes from momentum changes when particles strike the walls. The particles have a distribution of speeds, while mean translational kinetic energy is proportional to thermodynamic temperature. Real gases depart from ideal behaviour, especially near condensation.
The ideal-gas relationship pV = nRT uses absolute pressure, volume, amount in moles and temperature in kelvin. Holding two relevant quantities fixed reveals familiar proportionalities: at fixed amount and temperature, pressure is inversely proportional to volume. State the controlled conditions before using a ratio. Gauge pressure must be converted to absolute pressure if the equation requires the full pressure relative to vacuum.
The same model can be written pV = NkT, with N the number of molecules. The constants are connected through R = Nₐk, and molecule count is nNₐ. Mixing n with k, or N with R, introduces a huge scale error. A mole is an amount of substance rather than a volume; different gases at the same conditions can contain equal numbers despite having different masses.
Gas particles move randomly and change momentum in collisions with the container. Their average force on a wall divided by area gives pressure. The ideal kinetic model treats molecules as negligible in size, neglects long-range interactions and assumes elastic collisions. Root-mean-square speed concerns the mean of speed squared, not simply the arithmetic mean speed, because momentum-transfer calculations involve squared velocity components.
Mean translational kinetic energy per molecule is 3kT/2 in this model. Doubling absolute temperature doubles that mean energy and multiplies rms speed by √2, not by two. Molecules still have a distribution of energies rather than identical speeds. Real gases depart from the ideal assumptions when intermolecular attraction or molecular volume becomes important, especially at high density or near condensation.
The particles strike the walls more frequently in the smaller available space. Mean particle kinetic energy remains fixed by the temperature, but momentum transfer per wall area increases.
Equation of state for an ideal gas.
Per molecule in an ideal gas.
An ideal gas initially has p = 200 kPa and V = 2 L. At unchanged temperature and amount, V becomes 4 L. What is p?
pV is constant, so p₂ = 200 × 2/4 = 100 kPa.
In the simulation keep particle count and temperature fixed, vary volume and plot pressure against inverse volume.
Track collisions and vary one controlled quantity at a time. These models support kinetic reasoning; their simplified particles and controls do not reproduce every real-gas interaction or thermodynamic process.
Particle collisions with the walls
Changes of particle momentum produce an average wall force.
100 kPa
pV is constant, so p₂ = 200 × 2/4 = 100 kPa.
It doubles.
Mean translational kinetic energy is proportional to T.