(a) Amonton's law expresses the relationship between pressure and temperature. Use Charles's law and Boyle's law to derive the proportionality relationship between P and T.

Verified step by step guidance

1. Start by recalling Charles's Law, which states that the volume of a gas is directly proportional to its temperature at constant pressure: \( V \propto T \) or \( V = kT \), where \( k \) is a constant.
2. Recall Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume at constant temperature: \( P \propto \frac{1}{V} \) or \( PV = k' \), where \( k' \) is a constant.
3. Combine the two laws by substituting the expression for \( V \) from Charles's Law into Boyle's Law: \( P(kT) = k' \).
4. Simplify the equation to express the relationship between pressure and temperature: \( P \propto T \) or \( P = k''T \), where \( k'' \) is a new constant.
5. Conclude that Amonton's Law, which is also known as Gay-Lussac's Law, states that the pressure of a gas is directly proportional to its temperature when volume is held constant.

Key Concepts

Amonton's Law

Amonton's Law, also known as the pressure-temperature law, states that the pressure of a gas is directly proportional to its absolute temperature when the volume is held constant. This relationship implies that as the temperature increases, the kinetic energy of gas molecules increases, leading to higher pressure if the volume does not change.

Charles's Law

Charles's Law describes how the volume of a gas is directly proportional to its absolute temperature when pressure is held constant. This means that if the temperature of a gas increases, its volume will also increase, provided the pressure remains unchanged. This law is essential for understanding how temperature affects gas behavior.

Boyle's Law

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when the temperature is held constant. This means that if the volume of a gas decreases, its pressure increases, assuming the temperature remains constant. This law is crucial for analyzing the relationships between pressure, volume, and temperature in gas systems.