Recall and use the equation p V = constant for a fixed mass of gas at constant temperature, including a graphical representation of this relationship

Published by Patrick Mutisya · 14 days ago

IGCSE Physics 0625 – 2.1.2 Particle Model

2.1.2 Particle Model

Objective

Recall and use the equation \$pV = \text{constant}\$ for a fixed mass of gas at constant temperature, and represent this relationship graphically.

Key Concepts

  • The particle model treats gases as a large number of tiny particles moving randomly.

  • At a given temperature, the average kinetic energy of the particles is constant.

  • For a fixed mass of gas, if the temperature does not change, the product of pressure (p) and volume (V) remains constant:


    \$pV = k\$

    where k is a constant for that particular amount of gas at that temperature.

  • This relationship is known as Boyle’s Law.

Derivation (Brief)

  1. Assume a fixed mass of gas in a sealed container.

  2. When the volume is decreased, particles collide with the walls more frequently, increasing pressure.

  3. Conversely, increasing the volume reduces collision frequency, decreasing pressure.

  4. Because temperature (and thus average kinetic energy) is unchanged, the product pV does not vary.

Using the Equation

To find a missing pressure or volume when temperature is constant:

\$p1 V1 = p2 V2\$

Example:

  • Initial state: p₁ = 100 kPa, V₁ = 2.0 L

  • Final volume: V₂ = 3.0 L

  • Find final pressure p₂:


    \$p2 = \frac{p1 V1}{V2} = \frac{100 \times 2.0}{3.0} = 66.7\ \text{kPa}\$

Tabular Illustration

For a fixed amount of gas at 300 K, the constant k = 200 kPa·L. The table shows how pressure varies with volume.

Volume (L)Pressure (kPa)Product pV (kPa·L)
1.0200200
2.0100200
4.050200
5.040200

Graphical Representation

The relationship pV = constant is a hyperbola when pressure is plotted against volume. The curve passes through all points that satisfy the constant product.

Suggested diagram: A graph of pressure (vertical axis) versus volume (horizontal axis) showing a smooth hyperbolic curve that approaches the axes but never touches them. Mark two points, e.g., (2 L, 100 kPa) and (4 L, 50 kPa), to illustrate the constant product.

Common Mistakes to Avoid

  • Assuming temperature changes when using \$pV = \text{constant}\$. The law only applies at constant temperature.

  • Mixing units (e.g., using m³ for volume and kPa for pressure without conversion). Keep units consistent.

  • Treating the constant k as a universal value; it depends on the amount of gas and the temperature.

Practice Questions

  1. A gas occupies 1.5 L at a pressure of 120 kPa. If the volume is increased to 3.0 L at the same temperature, what is the new pressure?

  2. At constant temperature, a gas has a pressure of 80 kPa when its volume is 5.0 L. What volume corresponds to a pressure of 200 kPa?

  3. Explain why the graph of p versus V is a hyperbola and not a straight line.

Summary

For a fixed mass of gas at constant temperature, the product of pressure and volume remains unchanged (\$pV = \text{constant}\$). This principle, known as Boyle’s Law, can be applied algebraically and visualised as a hyperbolic curve on a pressure‑volume graph. Mastery of this relationship enables prediction of how gases behave when confined or expanded without temperature change.