15.1.2 Ideal Gases

• An ideal gas is one which obeys the relation:

pV ∝ T

• Where:
  • p = pressure of the gas (Pa)
  • V = volume of the gas (m³)
  • T = thermodynamic temperature (K)

• The molecules in a gas move around randomly at high speeds, colliding with surfaces and exerting pressure upon them

Gas molecules move about randomly at high speeds

• Imagine molecules of gas free to move around in a box
• The temperature of a gas is related to the average speed of the molecules:
  • The hotter the gas, the faster the molecules move
  • Hence the molecules collide with the surface of the walls more frequently

• Since force is the rate of change of momentum:
  • Each collision applies a force across the surface area of the walls
  • The faster the molecules hit the walls, the greater the force on them

• Since pressure is the force per unit area
  • Higher temperature leads to higher pressure

• If the volume V of the box decreases, and the temperature T stays constant:
  • There will be a smaller surface area of the walls and hence more collisions
  • This also creates more pressure

• Since this equates to a greater force per unit area, pressure in an ideal gas is therefore defined by:

The frequency of collisions of the gas molecules per unit area of a container

Boyle’s Law

• If the temperature T is constant, then Boyle’s Law is given by:

• This leads to the relationship between the pressure and volume for a fixed mass of gas at constant temperature:

P₁V₁ = P₂V₂

Charles's Law

• If the pressure P is constant, then Charles’s law is given by:

V ∝ T

• This leads to the relationship between the volume and thermodynamic temperature for a fixed mass of gas at constant pressure:

Pressure Law

• If the volume V is constant, the the Pressure law is given by:

P ∝ T

• This leads to the relationship between the pressure and thermodynamic temperature for a fixed mass of gas at constant volume:

Step 1: State the known values

• • Volume, V = 4.5 × 10⁻³ m³
  • Initial pressure, P₁ = 6.2 × 10⁵ Pa
  • Initial temperature, T₁ = 30°C = 303 K
  • Initial temperature, T₂ = 40°C = 313 K

Step 2: Since volume is constant, state the pressure law

Step 3: Rearrange to make P₂ the subject

Step 4: Substitute in known values and calculate P₂