Physics HL · Chapter 9: The Gas Laws
9.2 Pressure and the Ideal-Gas Model
Derive pressure as molecular momentum transfer and formalize the assumptions of the ideal-gas model.
Estimated time: 30 minutes
Pressure as Momentum Transfer per Unit Area
Pressure is defined using the normal force component on an area, not the tangential component.
At a macroscopic level this definition seems static, but its microscopic origin is dynamic. Gas molecules repeatedly collide with walls and reverse part of their momentum. Every collision imparts an impulse, and the total impulse rate over wall area appears macroscopically as pressure.
A key interpretation is that pressure rises when collisions become more frequent, when each collision changes momentum more strongly, or both. This gives immediate qualitative predictions: shrinking volume increases collision frequency, raising temperature increases typical molecular speeds, and increasing molecule number increases total collision events.
Molecular Collisions and Normal Momentum Change
For an elastic wall collision, the normal velocity component reverses sign, doubling normal momentum change magnitude.
This relation is local to one particle-wall event, but kinetic theory sums the effect over vast populations. The ideal-gas model assumes collisions are elastic and collision time is negligible relative to free-flight time. Those assumptions let us aggregate momentum exchange cleanly into closed-form pressure relationships.
Newton's third law is central here. If the wall exerts force on a molecule to reverse its normal momentum, the molecule exerts equal and opposite force on the wall. Pressure is simply that wall-side force per area averaged over many collisions.
Ideal-Gas Assumptions and Their Meaning
- Molecules are treated as point particles with negligible own volume.
- Between collisions, molecules move according to ordinary mechanics.
- Intermolecular forces are negligible except during collisions.
- Collisions between molecules and with walls are elastic.
- Molecules have a distribution of speeds rather than one shared speed.
Each assumption suppresses one possible complication so we can isolate dominant behavior. Ignoring molecular volume simplifies available-space geometry. Ignoring intermolecular forces removes potential-energy coupling. Elastic collisions preserve kinetic-energy bookkeeping in collisions. The result is a model simple enough to solve but rich enough to explain major gas-law behavior.
When Real Gases Behave Approximately Ideally
Real gases are closest to ideal behavior when density is low and temperature is moderate or high relative to condensation conditions. Low density means particles are far apart, weakening intermolecular-force influence. Higher temperature means kinetic energy dominates weak attraction effects, reducing relative impact of non-ideal interactions.
In practical terms, many classroom and engineering gases near atmospheric pressure are well approximated by ideal equations. But compressing strongly or cooling toward phase-change regions causes visible deviation that must be modeled with correction factors or non-ideal equations of state.
Simulation: Pressure from Collisions
Use Boyle-mode piston compression to watch how reduced volume increases collision-driven pressure at fixed amount and temperature.
Ideal Gas Law Lab
Active law interpretation
Boyle mode keeps n and T fixed, so pressure responds inversely to volume (P*V = constant).
Container micro-view (animated gas particles + piston)
P-V map with isotherms
State diagnostics
P
325.2 kPa
3.21 atm
T
320.0 K
46.9 deg C
V
9.00 L
9.0000e-3 m^3
Microscopic metrics
RMS speed c_rms: 524.6 m s^-1
Mean molecular kinetic energy: 6.627e-21 J
Density estimate: 3.544 kg m^-3
Monatomic internal energy estimate: 4389.8 J
Model validity note
Ideal-model range: this state is in a typical low-density, moderate-temperature regime where PV = nRT is usually reliable.
Try this workflow: hold n fixed, then switch between Boyle/Charles/Gay-Lussac modes and verify each ratio form before returning to full-state PV = nRT checks.
Test Yourself
A sealed rigid container of fixed volume is held at constant temperature, and the amount of gas is doubled. In the ideal model, what happens to pressure?