Ideal Gases

Statement

• At constant temperature and for a fixed amount of an ideal gas:

P × V = constant

• Equivalently, between two states:

p₁ × V₁ = p₂ × V₂

• where:
  • p₁, p₂ = pressures in Pa
  • V₁, V₂ = volumes in m³ (or any consistent unit, as long as both match)
• p and V are inversely proportional: doubling the pressure halves the volume; halving the pressure doubles the volume

Why it works microscopically

• At constant temperature the molecules have the same average kinetic energy, so each individual collision with the wall carries the same average force
• Shrink the container to half its volume and the molecules have to travel only half as far between wall collisions
• They therefore hit each wall twice as often per second, and the pressure doubles
• The same number of molecules, moving at the same average speed, in half the space, gives twice the pressure

Checking the answer makes sense

• A useful sanity check after any gas-law calculation:
  • if the volume has shrunk (compression), expect the new pressure to be larger than the original
  • if the gas has been heated at fixed volume, the new pressure should be larger than before
  • if you get the opposite trend, you've probably substituted the temperatures the wrong way round, or forgotten to convert °C to K