2.4.2 Charles's Law

Introduction

Boyle's Law describes what happens when pressure changes. But what happens when temperature changes instead? That is where Charles's Law comes in.

Charles's Law describes the relationship between the volume of a gas and its absolute temperature, while the pressure and the amount of gas stay the same.

Definition

Charles's Law: It states that for a fixed amount of gas at constant pressure, the volume of a gas is directly proportional to its absolute temperature.

In simple words: If you heat a gas, its volume increases. If you cool a gas, its volume decreases — as long as the pressure and the amount of gas stay the same.

Main Idea

• Direct relationship: When temperature goes up, volume goes up.
• When temperature goes down, volume goes down.
• The ratio of volume to temperature is constant (at constant pressure and fixed amount of gas).
• Temperature must be in Kelvin (K), not Celsius (°C).

Core Concept

The relationship can be written mathematically as:

V ∝ T

This means:

V = kT

Rearranging:

VT = k

VT = constant

Where:

• V = volume of the gas
• T = absolute temperature (in Kelvin)
• k = a constant (for a fixed amount of gas at constant pressure)

Key point: If the absolute temperature is doubled, the volume is doubled. If the absolute temperature is halved, the volume is halved.

Mathematical Form

The mathematical statement of Charles's Law is:

V ∝ T

V = kT

VT = k

VT = constant

For two different sets of conditions (with the same amount of gas at constant pressure):

V₁T₁ = V₂T₂

Where:

• V₁ = initial volume
• T₁ = initial absolute temperature (in K)
• V₂ = final volume
• T₂ = final absolute temperature (in K)

Units:

• Volume is usually measured in dm³, cm³, or m³.
• Temperature must be in Kelvin (K).

Converting Celsius to Kelvin:

T(K) = T(°C) + 273

Example: 0°C = 273\ K 100°C = 373\ K

Kinetic Particle Theory Interpretation

Effect of Temperature on Volume

What happens when temperature increases:

• When the temperature of a gas increases, the kinetic energy of the gas particles increases.
• This means the particles move faster.
• To maintain constant pressure, the volume of the container must increase to allow the particles to travel further between collisions with the container walls.
• This keeps the pressure constant.

What happens when temperature decreases:

• If the temperature decreases, the kinetic energy of the particles decreases.
• The particles move slower.
• The volume must decrease to maintain constant pressure.

Simple explanation: Think of a balloon. When you heat the air inside, the particles move faster and push harder on the walls. To keep the pressure the same, the balloon expands. When you cool the air, the particles slow down, and the balloon shrinks.

Constant Pressure

• The volume changes to balance the increased or decreased frequency and force of collisions due to temperature changes.
• This maintains constant pressure.

Key point: At constant pressure, the volume adjusts so that the pressure stays the same, even though the temperature (and therefore particle speed) has changed.

Formula / Mathematical Form

V ∝ T

V = kT

VT = k

VT = constant

V₁T₁ = V₂T₂

Important: Temperature must be in Kelvin (K).

Solved Example

Example: A gas occupies a volume of 300\ cm³ at a temperature of 27°C. If the temperature is increased to 127°C at constant pressure, what will be the new volume?

Given: V₁ = 300\ cm³ T₁ = 27°C = 27 + 273 = 300\ K T₂ = 127°C = 127 + 273 = 400\ K

Required: V₂ = ?

Formula: V₁T₁ = V₂T₂

Substitution: 300300 = V₂400

Calculation: 1 = V₂400 V₂ = 400\ cm³

Final Answer: The new volume is 400\ cm³.

Check: Temperature increased from 300 K to 400 K (ratio 4:3). Volume increased from 300 cm³ to 400 cm³ (same ratio). ✓

Important Points

• - Charles's Law applies to a fixed amount of gas at constant pressure.
• - Volume is directly proportional to absolute temperature.
• - VT = constant.
• - The formula for two sets of conditions is V₁T₁ = V₂T₂.
• - Temperature must be in Kelvin (K).
• - When temperature increases, volume increases (particles move faster, need more space).
• - When temperature decreases, volume decreases (particles move slower, need less space).
• - At constant pressure, the volume adjusts to keep the pressure the same.

Remember

• - Charles's Law → volume and temperature are directly related.
• - VT = constant.
• - V₁T₁ = V₂T₂.
• - Temperature ↑ → Volume ↑.
• - Temperature ↓ → Volume ↓.
• - Pressure must be constant.
• - Amount of gas must be fixed.
• - Temperature must be in Kelvin (K).
• - T(K) = T(°C) + 273.

Common Mistakes

Quick Revision

Charles's Law states that for a fixed amount of gas at constant pressure, the volume of a gas is directly proportional to its absolute temperature.

• Mathematically:
• V ∝ T
• V = kT
• VT = k
• VT = constant
• V₁T₁ = V₂T₂

Kinetic Particle Theory Interpretation:

• Effect of Temperature on Volume: When the temperature of a gas increases, the kinetic energy of the gas particles increases. This means the particles move faster. To maintain constant pressure, the volume of the container must increase to allow the particles to travel further between collisions with the container walls, thus keeping the pressure constant.
• Constant Pressure: The volume changes to balance the increased or decreased frequency and force of collisions due to temperature changes, maintaining constant pressure.

Important: Temperature must be in Kelvin (K).

60-Second Revision

• Charles's Law: V ∝ T at constant pressure and fixed amount of gas.
• VT = constant
• V₁T₁ = V₂T₂
• Temperature ↑ → Volume ↑
• Temperature ↓ → Volume ↓
• Kelvin only! T(K) = T(°C) + 273.
• Particles move faster when heated → volume expands.
• ⚠️ Warning: always convert temperature to Kelvin before using Charles's Law.