Thermal Expansion: Solids, Liquids, Gases & Applications

TAKEAWAYS:

• Thermal expansion means objects grow bigger when heated and shrink when cooled.
• Solids expand linearly (length) or volumetrically (volume), with coefficients α (linear) and γ (volume).
• Liquids expand only in volume, measured by γ, and their expansion depends on the container.
• Gases expand freely, following Charles’s Law (V ∝ T at constant pressure).
• Applications include bridges, railway tracks, thermostats, and liquid-in-glass thermometers.
• Problems use the formula ΔL = αL₀ΔT, ΔV = γV₀ΔT, or V/T = constant for gases.

1. What is Thermal Expansion?

When an object is heated, its particles gain kinetic energy and move faster. This increased motion pushes particles apart, making the object expand. Conversely, cooling makes it contract.

Key Idea:

• Linear expansion = change in length (ΔL).
• Area expansion = change in surface area (ΔA).
• Volume expansion = change in volume (ΔV).

2. Thermal Expansion in Solids

Solids expand in all three dimensions when heated. The change depends on:

• Original length (L₀) or volume (V₀).
• Temperature change (ΔT).
• Material property: Coefficient of linear expansion (α) or volume expansion (γ).

Formulas:

1. Linear Expansion: $$\Delta L=\alpha L_0\Delta T$$

   • ΔL = change in length (m)
   • α = coefficient of linear expansion (K⁻¹ or °C⁻¹)
   • L₀ = original length (m)
   • ΔT = temperature change (K or °C)

2. Area Expansion: $$\Delta A=2\alpha A_0\Delta T$$ (Area expansion is twice the linear expansion.)

3. Volume Expansion: $$\Delta V=\gamma V_0\Delta T\text{or}\gamma =3\alpha$$

   • γ = coefficient of volume expansion (K⁻¹ or °C⁻¹)
   • V₀ = original volume (m³)

Example 1: Railway Track Expansion

A steel railway track is 100 m long at 20°C. If the temperature rises to 40°C, how much does it expand? (Given: α for steel = 12 × 10⁻⁶ K⁻¹)

Solution:

• ΔT = 40°C − 20°C = 20°C
• ΔL = αL₀ΔT = (12 × 10⁻⁶)(100)(20) = 0.024 m = 2.4 cm

Answer: The track expands by 2.4 cm.

3. Thermal Expansion in Liquids

Liquids only expand in volume (no fixed shape). Their expansion depends on:

• The liquid’s coefficient of volume expansion (γ).
• The container’s expansion (since liquids take the container’s shape).

Apparent vs. Real Expansion:

• Real expansion (γ): Expansion if the container did not expand.
• Apparent expansion (γ_app): Observed expansion because the container also expands.

Formula: $${\gamma }_{app}={\gamma }_{liquid}-{\gamma }_{container}$$

Example 2: Mercury Thermometer

A mercury thermometer has a bulb volume of 0.5 cm³ at 0°C. If γ for mercury = 182 × 10⁻⁶ K⁻¹, how much does it expand when heated to 100°C?

Solution:

• ΔT = 100°C − 0°C = 100°C
• ΔV = γV₀ΔT = (182 × 10⁻⁶)(0.5)(100) = 0.0091 cm³

Answer: Mercury expands by 0.0091 cm³.

4. Thermal Expansion in Gases

Gases expand freely in all directions. Their behavior follows Charles’s Law (at constant pressure):

$$\frac{V_1}{T_1}=\frac{V_2}{T_2}$$

• V = volume (m³)
• T = temperature in Kelvin (K) (not °C!)

Example 3: Balloon in a Hot Air Balloon

A gas balloon has a volume of 500 L at 300 K. If heated to 350 K, what is its new volume?

Solution: $$\frac{500}{300}=\frac{V_2}{350}⟹V_2=\frac{500\times 350}{300}=583.33\text{ L}$$

Answer: The balloon expands to 583.33 L.

flowchart TD
    A["Gas Heated"] -->|"Volume Increases"| B["Charles's Law: V ∝ T"]
    B --> C["If P is constant, V/T = constant"]
    C --> D["Example: Balloon expands"]

5. Applications of Thermal Expansion

6. Problems and Solutions

Problem 1 (NEB-style):

A brass rod is 2 m long at 20°C. If heated to 120°C, how much does it expand? (α for brass = 19 × 10⁻⁶ K⁻¹)

Solution:

• ΔT = 120°C − 20°C = 100°C
• ΔL = αL₀ΔT = (19 × 10⁻⁶)(2)(100) = 0.0038 m = 3.8 mm

Answer: 3.8 mm

Problem 2 (NEB-style):

A mercury thermometer reads 0°C when the bulb volume is 0.1 cm³. At 100°C, the volume is 0.1091 cm³. Find γ for mercury.

Solution:

• ΔV = 0.1091 − 0.1 = 0.0091 cm³
• γ = ΔV / (V₀ΔT) = 0.0091 / (0.1 × 100) = 0.00091 K⁻¹ = 910 × 10⁻⁶ K⁻¹

Answer: γ = 910 × 10⁻⁶ K⁻¹

Exam Tip

1. Memorize formulas:

   • ΔL = αL₀ΔT
   • ΔV = γV₀ΔT (γ = 3α for solids)
   • V₁/T₁ = V₂/T₂ (gases)

2. Units matter!

   • Always use Kelvin (K) for gas problems.
   • ΔT in °C or K works the same (since it’s a difference).

3. Real vs. apparent expansion:

   • For liquids, subtract the container’s expansion if asked.

4. Common mistakes:

   • Forgetting to convert °C to K for gases.
   • Using α instead of γ for volume problems.

5. NEB loves:

   • Railway track/bridge expansion.
   • Thermometer problems (mercury/alcohol).
   • Bimetallic strip applications.

Gaps in railway tracks prevent buckling due to thermal expansion. (Image: Thomas Nugent, CC BY-SA 2.0, via Wikimedia Commons)