Gas Laws

The gas laws describe the relationships between pressure (p), volume (V), and temperature (T) for a fixed mass of gas. These laws were discovered experimentally before the kinetic theory provided a molecular explanation. They are fundamental to AQA A-Level Physics and appear regularly in both calculation and explanation questions.

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Temperature Scales

Before studying the gas laws, it is essential to understand that all gas law equations require absolute temperature measured in kelvin (K).

T (K) = θ (°C) + 273.15

For most A-Level calculations, 273 is sufficiently accurate.

Absolute zero (0 K = −273.15 °C) is the lowest possible temperature. At absolute zero:

• Molecules have minimum internal energy.
• All thermal motion ceases (molecules have zero kinetic energy).
• The pressure of an ideal gas would be zero.
• The volume of an ideal gas would be zero.

Absolute zero cannot actually be reached in practice (Third Law of Thermodynamics), but temperatures within a fraction of a kelvin have been achieved in laboratories.

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Boyle's Law

Boyle's Law: For a fixed mass of gas at constant temperature, the pressure is inversely proportional to the volume.

pV = constant (at constant T) p₁V₁ = p₂V₂

Molecular Explanation

At constant temperature, the molecules have the same average kinetic energy. If the volume is halved:

• The molecules are in a smaller space, so they hit the walls more frequently.
• The molecules travel a shorter distance between collisions with the walls.
• The rate of change of momentum at the walls increases.
• Therefore the pressure doubles.

Graphical Representations

• p vs V: A rectangular hyperbola (curved line) — pressure decreases as volume increases. Each curve (isotherm) corresponds to a different temperature; higher-temperature isotherms are further from the origin.
• p vs 1/V: A straight line through the origin with gradient = pV = constant.
• pV vs p: A horizontal straight line (since pV = constant).

Worked Example — Boyle's Law

Question: A gas syringe contains 80 cm³ of gas at a pressure of 1.0 × 10⁵ Pa. The gas is slowly compressed at constant temperature to a volume of 20 cm³. Calculate the new pressure.

Solution:

Using p₁V₁ = p₂V₂:

p₂ = p₁V₁ / V₂ = (1.0 × 10⁵ × 80) / 20 = 4.0 × 10⁵ Pa

The pressure quadruples when the volume is reduced to one quarter, as expected from the inverse proportionality.

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Charles's Law

Charles's Law: For a fixed mass of gas at constant pressure, the volume is directly proportional to the absolute temperature.

V/T = constant (at constant p) V₁/T₁ = V₂/T₂

Molecular Explanation

If the temperature increases at constant pressure:

• The molecules gain kinetic energy and move faster.
• They hit the walls harder and more frequently.
• To maintain constant pressure, the volume must increase so that the molecules travel further between collisions with the walls, reducing the collision rate.
• The volume increases in direct proportion to the absolute temperature.

Graphical Representations

• V vs T (in kelvin): A straight line through the origin (V ∝ T).
• V vs θ (in °C): A straight line that does NOT pass through the origin. If extrapolated back, it crosses the temperature axis at −273 °C (absolute zero).

Worked Example — Charles's Law

Question: A balloon contains 2.0 litres of helium at 20 °C. The balloon is placed in a freezer at −18 °C. Calculate the new volume, assuming the pressure remains constant.

Solution:

Convert temperatures to kelvin: T₁ = 20 + 273 = 293 K T₂ = −18 + 273 = 255 K

Using V₁/T₁ = V₂/T₂:

V₂ = V₁ × T₂/T₁ = 2.0 × 255/293 = 2.0 × 0.8703 = 1.74 litres

The volume decreases because the temperature decreases.

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Pressure Law (Gay-Lussac's Law)

Pressure Law: For a fixed mass of gas at constant volume, the pressure is directly proportional to the absolute temperature.

p/T = constant (at constant V) p₁/T₁ = p₂/T₂

Molecular Explanation