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

Example — a fixed mass of gas occupies 4.0 × 10⁻³ m³ at a pressure of 1.5 × 10⁵ Pa. Without changing the temperature, the gas is squeezed down to 1.0 × 10⁻³ m³. Calculate the new pressure.

• Use p₁ × V₁ = p₂ × V₂
• p₂ = (p₁ × V₁) / V₂ = (1.5 × 10⁵ × 4.0 × 10⁻³) / (1.0 × 10⁻³)
• p₂ = (600) / (1.0 × 10⁻³) = 6.0 × 10⁵ Pa
• Volume fell to a quarter of the original; pressure rose to four times, exactly the inverse proportionality predicted

Example — a syringe holds 8.0 cm³ of air at atmospheric pressure (1.0 × 10⁵ Pa). The nozzle is sealed and the plunger is pushed in until the air occupies 3.0 cm³. The temperature is unchanged. Calculate the new pressure.

• Use p₁ × V₁ = p₂ × V₂ (volume units can stay in cm³ provided both sides use the same unit)
• p₂ = (1.0 × 10⁵ × 8.0) / 3.0 = (8.0 × 10⁵) / 3.0 = 2.67 × 10⁵ Pa

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