STATES OF MATTER - properties of gases and liquids (fluids) and solids

18. The kinetic particle theory of gases and Boyles Law gas calculations using the ideal gas equation P₁V₁ = P₂V₂ - exam practice questions

18, The particle model of a gas - and gas pressure-volume calculations using Boyle's Law

• Explanation of Boyle's Law

• All particles have mass and their movement gives them kinetic energy and momentum.

• The particles in a gas are in constant random motion - random direction, variety of velocities and kinetic energies.

• When the fast moving gas particles collide with a surface, their millions of impacts create a force that we measure as gas pressure - the total force of impacts per unit area.

• The particles collide with the container surface completely at random and impact at every angle, BUT, the effect is to create a net force at right angles to the surface - gas pressure!

• The greater the number of collisions per unit area of surface, the greater the pressure, assuming the gas volume and temperature are kept constant.

• If the temperature is kept constant and the volume increased, the impacts are more spread out and less frequent per unit area, so the gas pressure decreases.

• Conversely, if a gas is compressed into a smaller volume at constant temperature, the number of impacts per unit area increases, so the pressure increases.

• From measurements of volumes and pressure of gases at constant pressure, a numerical inverse law can be formulated - Boyle's law.

• pressure x volume = a constant (at constant temperature)

• pV = constant (at constant temperature)

• The standard units of pressure and volume are:

• p = pressure in pascals (Pa), V = volume (m³)

• You can connect two pressure and two volumes by the simple equation

• p₁ x V₁ = p₂ x V₂

  • where 1 represent the original conditions of the gas in terms of pressure and volume,

  • and 2 the final situation if an enforced change of p₁ or V₁ is made.

• Examples of simple gas calculations

• (i) 5 m³ volume of a gas at a pressure 101 300 Pa was compressed to a volume of 2.8 m³.

• • Calculate the final pressure.

  • p₁ x V₁ = p₂ x V₂

  • rearranging gives p₂ = (p₁ x V₁) / V₂

  • p₂ = (101 300 x 5) / 2.8 = 180893 Pa

• (ii) 10m³ of gas at a pressure of 100 000 Pa was compressed to a pressure of 300 000 Pa.

• • Calculate the final volume of the gas.

  • p₁ x V₁ = p₂ x V₂

  • rearranging gives V₂ = (p₁ x V₁) / p₂

  • V₂ = (100 000 x 10) / 300 000 = 3.33 m³

More on gas pressure and volume calculations

AND some practice questions for you to do, answers at end of page!

Boyle's Law for volume and gas pressure

• The particle theory of gas pressure was explained above so this section concentrates on the gas law calculations involving pressure and volume.

• Boyle's Law states that for given mass of gas at a constant temperature (^oC or K), the product of the pressure multiplied by the volume is a constant.

• p x V = constant

• Therefore, for initial values of p₁ and V₁, which change to final values of p₂ and V₂, the following equation applies ...

• p₁ x V₁ = p₂ x V₂ (for fixed amount of gas at constant temperature)

• or p₂ = p₁ x V₁/V₂ or V₂ = p₁ x V₁/p₂

• The graph shows how the pressure and volume vary according to Boyles Law at two different temperatures.

• At lower temperatures the volume and pressure values are lower (see next section).

• You can use any volume or pressure units you like as long as both pressures and both volumes have the same units.

• Using particle theory and simple arithmetical values to explain Boyles Law.

  • If a gas is compressed to half its original volume the concentration or density of the gas is doubled. Therefore there will be twice as many collisions with the surface causing twice the impact effect i.e. double the pressure.

  • If the volume of a gas is increased by a factor of three, the concentration is reduced by the same factor, so the chance of particle collision with the container walls is similarly reduced, so the pressure decreases by a factor of three.

• Gases e.g. oxygen for hospitals, can be stored under high pressure enabling reasonably efficient storage. Because the internal pressure in the cylinder is so much greater than the external pressure, on fitting a valve, a large volume of gas can be released to flow slowly under controlled conditions for a patients respiration.

• Examples of Boyle's Law calculations (constant temperature assumed)

• Ex. Q 1

  • 240cm³ of air at a pressure of 100kPa in a bicycle pump is compressed to a volume of 150cm³.

  • What is the pressure of the compressed air in the pump?

  • -

• Ex. Q2

  • 10 m³ of butane gas at 1.2 atm was required to be stored at 6 atm pressure. To what volume must the gas be compressed to give the required storage pressure?

  • -

• Ex. Q3

  • A 100 cm³ gas syringe containing 80 cm³ of gas that was compressed to 60 cm³. If atmospheric pressure is 101325 Pa, and the temperature remains constant, what is the pressure of the gas in the syringe after compression?

  • -

• Ex. Q4

  • In hospital the gas pressure in a 100 dm³ cylinder of oxygen is 5.52 atm (5.52 x atmospheric pressure). What volume of gas can be released slowly to a patient on releasing it to a residual atmospheric pressure 1.01 atm?

  • -

ANSWERS

Key points about Boyle’s Law & Kinetic Particle Theory

Re-cap: Kinetic Particle Theory of Gases

• Gas particles move in constant random motion.
• Collisions with container walls create pressure.
• If volume decreases (same number of particles, same temperature):
  • Particles have less space.
  • Collisions with walls become more frequent.
  • Pressure increases.
• If volume increases:
  • Collisions are less frequent.
  • Pressure decreases.

Boyle’s Law

• Statement: At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume.
• Equation:
  P₁V₁ = P₂V₂
• Relationship:
  1 represents the initial pressure and volume, and 2 represents the final pressure and volume

Student Exam Tips when using Boyle's Law

• Always state assumptions: temperature constant, fixed mass of gas.
• Use the correct units:
  • Pressure in Pa or kPa.
  • Volume in m³ or cm³ (be consistent).
• Rearrange P₁V₁ = P₂V₂ equations carefully:
• Graphs:
  • Pressure versus Volume → curve (inverse).
  • Pressure versus 1/Volume → straight line.
• In multiple-choice, check if the question asks for direct or inverse proportionality.

Typical Misconceptions about Boyle's Law

• ❌ Thinking Boyle’s Law applies when temperature changes — it only holds at constant temperature.
• ❌ Forgetting to convert units (e.g., mixing cm³ and m³).
• ❌ Assuming pressure is caused by particle collisions with each other — it’s collisions with container walls.
• ❌ Believing pressure and volume are directly proportional — they are inversely proportional.
• ❌ Misinterpreting graphs: pressure versus volume is not linear, but pressure versus 1/volume is.

Quick Recap on Boyle's Law

• Kinetic particle theory explains Boyle’s Law: smaller volume → more collisions → higher pressure.
• Equation: P₁V₁ = P₂V₂
• Exam boards require knowledge of particle theory, inverse proportionality, and calculations.
• Tips: watch units, state assumptions, interpret graphs correctly.
• Misconceptions: forgetting temperature condition, mixing units, misunderstanding collision causes.

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• Ex. Q 1

  • 240cm³ of air at a pressure of 100kPa in a bicycle pump is compressed to a volume of 150cm³.

  • What is the pressure of the compressed air in the pump?

  • p₁ x V₁ = p₂ x V₂ , rearranging to scale up for the new higher pressure

  • p₂ = p₁ x V₁/V₂  = 100 x 240/150 = 160 kPa

• Ex. Q 2

  • 10 m³ of butane gas at 1.2 atm was required to be stored at 6 atm pressure. To what volume must the gas be compressed to give the required storage pressure?

  • p₁ x V₁ = p₂ x V₂ , rearranging to scale down for the new lower volume

  • V₂ = p₁ x V₁/p₂ = 1.2 x 10/6 = 2.0 m³

• Ex. Q 3

  • A 100 cm³ gas syringe containing 80 cm³ of gas that was compressed to 60 cm³. If atmospheric pressure is 101325 Pa, and the temperature remains constant, what is the pressure of the gas in the syringe after compression.

  • p x V = constant

  • p₁ x V₁ = p₂ x V₂

  • p₂ = p₁ x V₁/V₂

  • p₂ = 101325 x 80/60 = 135100 Pa

• Ex. Q 4

  • In hospital the gas pressure in a 100 dm³ cylinder of oxygen is 5.52 atm (5.52 x atmospheric pressure). What volume of gas can be released slowly to a patient on releasing it to a residual atmospheric pressure 1.01 atm?

  • p x V = constant

  • V₂ = p₁ x V₁/p₂

  • V₂ = p₁ x V₁/p₂ = 5.52 x 100/1.01 = 547 dm³  (3 s.f.)