States of Matter: Question 5

Worked solution

Part (a): Assumptions of the ideal gas model

The kinetic theory model of an ideal gas makes two key simplifying assumptions:

1. The volume occupied by the gas particles themselves is negligible compared with the total volume of the container.
2. There are no intermolecular forces of attraction or repulsion between particles, and all collisions between particles (and with the container walls) are perfectly elastic (no kinetic energy is lost).

Part (b): Conditions of greatest deviation from ideal behaviour

A real gas deviates most from ideal gas behaviour at high pressure and low temperature.

Part (c): Explaining the deviation

Effect of high pressure: at high pressure, the gas particles are forced much closer together, so the container volume becomes much smaller relative to the number of particles present. The particles’ own volume, which the ideal gas model assumes is negligible, now makes up a significant fraction of the total volume, so assumption 1 breaks down.

Effect of low temperature: at low temperature, the particles have less average kinetic energy, so they move more slowly. This means the (normally very brief and weak) intermolecular attractive forces between particles have more time to act and a proportionally larger effect on particle motion. The particles are pulled together rather than moving and colliding completely independently, so assumption 2 (no intermolecular forces) breaks down.

Overall effect: because both of the simplifying assumptions used to derive $pV = nRT$ no longer hold under these conditions, the real gas’s measured pressure and volume no longer match the values that the ideal gas equation predicts. For example, intermolecular attraction tends to lower the pressure the gas exerts below the ideal prediction, while the particles’ own volume tends to make the gas resist compression more than the ideal equation predicts, especially as the gas approaches the temperature and pressure at which it would liquefy.

Final answers

• (a) Negligible particle volume compared with the container; no intermolecular forces between particles (perfectly elastic collisions)
• (b) High pressure and low temperature
• (c) At high pressure the particles’ own volume becomes significant compared with the container volume; at low temperature, weaker particle motion allows intermolecular attractive forces to have a significant effect, both broken assumptions cause pV to depart from the value predicted by $pV = nRT$