Question

The pressure of gaseous \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) increases more rapidly with temperature than predicted by the ideal gas equation even though \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) behaves like an ideal gas. Explain.

Short answer

The pressure increases due to \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) dissociating into \(2\mathrm{AlCl}_3\), increasing gas molecules.

Step-by-step solution

Understanding the Problem

We need to understand why the pressure of \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) increases more rapidly with temperature than predicted by the ideal gas law, even though it behaves like an ideal gas.

Considering the Ideal Gas Law

The ideal gas law is given by the equation \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. This relationship assumes that the gas's behavior is unaffected by intermolecular forces or changes in molecular structure.

Analyzing Chemical Behavior

The key to the question lies in the chemical behavior of \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\). Upon heating, \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) undergoes dissociation into \(2\mathrm{AlCl}_3\) molecules. This reaction increases the number of gas particles in the system, leading to a more rapid increase in pressure.

Connecting Dissociation to Pressure Increase

When \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) dissociates, the number of moles \(n\) increases because 1 mole of \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) produces 2 moles of \(\mathrm{AlCl}_3\). According to the ideal gas law, if \(n\) increases while \(V\) and \(T\) are constant, \(P\) must also increase. This is why the pressure rises more rapidly than expected under the assumption of a non-dissociating gas.

Key concepts

Pressure and Temperature Relationship

When we talk about the behavior of gases, the ideal gas law equation \(PV = nRT\) is at the core. In this equation, \(P\) represents pressure, \(V\) signifies volume, \(n\) indicates the number of moles, \(R\) is the ideal gas constant, and \(T\) stands for temperature in Kelvin.
This equation means that if the number of moles and the volume remain constant, pressure is directly proportional to temperature. So, as the temperature increases, pressure should increase proportionally—according to the simple direct relationship.

• Larger temperature increases lead to larger pressure increases.
• This scenario assumes no change in molecular structure or intermolecular forces.

However, real gases may deviate from this straightforward relationship due to various factors, one being molecular dissociation, affecting the relationship significantly.

Dissociation of Gases

Dissociation is a chemical reaction where molecules split into smaller particles. In the case of \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\), when heated, it dissociates into two \(\mathrm{AlCl}_3\) molecules.
This process adds more particles to the system.

• Initially: One molecule of \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\).
• After dissociation: Two molecules of \(\mathrm{AlCl}_3\).

This increase in the number of gas particles changes the number of moles \(n\) in the ideal gas equation, leading to an increase in pressure. Thus, even though \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\) behaves ideally, the dissociation results in a more rapid pressure rise compared to a non-dissociating gas.

Chemical Behavior of Gases

The chemical behavior of gases can significantly influence the relationships predicted by the ideal gas law.
This influence stems from both physical and chemical properties. A gas like \(\mathrm{Al}_{2} \mathrm{Cl}_{6}\), which undergoes dissociation, demonstrates this concept by behaving differently under changing thermal conditions.

• Compared to gases that do not undergo such reactions, dissociative gases change their composition at higher temperatures.
• The splitting increases the gas's ability to expand and its pressure's responsiveness to temperature increases.

Understanding these behaviors is crucial for accurately predicting gas behavior in practical scenarios. This insight points out why some gases deviate from the ideal gas law under certain conditions while maintaining the assumption of ideality in molecular interactions.