Thermal Expansion

Overview

Most materials expand when heated and contract when cooled. This occurs because increased temperature causes atoms to vibrate with greater amplitude, increasing the average distance between them.

Linear Expansion

For solids, the change in length is proportional to original length and temperature change:

$$\Delta L = \alpha L_0 \Delta T$$ $$L = L_0(1 + \alpha \Delta T)$$

Where:

• $\Delta L$ = change in length
• $L_0$ = original length
• $\alpha$ = coefficient of linear expansion (1/K or 1/°C)
• $\Delta T$ = temperature change

Coefficients of Linear Expansion

Material	$\alpha$ (×10⁻⁶ /°C)
Aluminum	24
Brass	19
Copper	17
Glass (ordinary)	9
Glass (Pyrex)	3.2
Iron/Steel	12
Lead	29
Concrete	12
Invar	0.9

Area Expansion

For a flat surface:

$$\Delta A = 2\alpha A_0 \Delta T = \gamma A_0 \Delta T$$ $$A = A_0(1 + 2\alpha \Delta T)$$

Where $\gamma = 2\alpha$ is the area expansion coefficient

Volume Expansion

For solids:

$$\Delta V = 3\alpha V_0 \Delta T = \beta V_0 \Delta T$$ $$V = V_0(1 + \beta \Delta T)$$

Where $\beta = 3\alpha$ is the volume expansion coefficient

For liquids ($\beta$ is given directly):

$$\Delta V = \beta V_0 \Delta T$$

Volume Expansion Coefficients (Liquids)

Liquid	$\beta$ (×10⁻⁴ /°C)
Alcohol (ethyl)	11
Gasoline	9.5
Mercury	1.8
Water (20°C)	2.1

Anomalous Expansion of Water

Water has unusual behavior:

• Contracts when heated from 0°C to 4°C
• Maximum density at 4°C
• Expands when heated above 4°C
• Ice is less dense than water (floats)

This is crucial for aquatic life in cold climates!

Thermal Stress

When expansion is constrained, thermal stress develops:

$$\text{Stress} = \sigma = Y\alpha\Delta T$$ $$\text{Force} = F = YA\alpha\Delta T$$

Where $Y$ = Young's modulus

Bimetallic Strip

Two metals with different $\alpha$ bonded together:

• Bends when temperature changes
• Used in thermostats

The radius of curvature:

$$R \approx \frac{t}{(\alpha_2 - \alpha_1)\Delta T}$$

Where $t$ = thickness of the strip

Examples

Example 1: Railroad Track

A 10 m steel rail installed at 20°C. Find expansion at 40°C.

$$\Delta L = \alpha L_0 \Delta T = (12 \times 10^{-6})(10)(20) = 2.4 \times 10^{-3} \text{ m} = 2.4 \text{ mm}$$

Example 2: Ring and Rod

An aluminum ring has inner diameter 5.000 cm at 20°C. A brass rod has diameter 5.010 cm at 20°C. At what temperature will the ring fit the rod?

Need: $\Delta L_{Al} = \Delta L_{\text{brass}} + 0.010$ cm

$$\alpha_{Al} \times 5.000 \times \Delta T = \alpha_{\text{brass}} \times 5.010 \times \Delta T + 0.010$$ $$(24 - 19 \times 1.002) \times 10^{-6} \times \Delta T = \frac{0.010}{5}$$ $$\Delta T = \frac{0.002}{4.96 \times 10^{-6}} \approx 403°\text{C}$$ $$T = 20 + 403 = 423°\text{C}$$

Example 3: Area Expansion

A copper sheet 2 m × 3 m is heated from 20°C to 120°C.

$$\Delta A = 2\alpha A_0 \Delta T = 2 \times 17 \times 10^{-6} \times 6 \times 100 = 0.0204 \text{ m}^2 = 204 \text{ cm}^2$$

Example 4: Volume of Mercury

A mercury thermometer contains 0.5 cm³ at 20°C. Find volume at 100°C.

$$\Delta V = \beta V_0 \Delta T = 1.8 \times 10^{-4} \times 0.5 \times 80 = 0.0072 \text{ cm}^3$$ $$V = 0.5072 \text{ cm}^3$$

Example 5: Thermal Stress

A steel rod fixed between rigid walls at 20°C. Find stress at 60°C. ($Y = 200$ GPa)

$$\sigma = Y\alpha\Delta T = 200 \times 10^9 \times 12 \times 10^{-6} \times 40$$ $$\sigma = 96 \times 10^6 \text{ Pa} = 96 \text{ MPa}$$

Example 6: Apparent vs Real Expansion

Gasoline in an aluminum tank. If tank expands, how much does gasoline level rise?

Apparent expansion coefficient:

$$\beta_{\text{apparent}} = \beta_{\text{liquid}} - \beta_{\text{container}}$$ $$\beta_{\text{apparent}} = 9.5 \times 10^{-4} - 3 \times 24 \times 10^{-6} = 9.5 \times 10^{-4} - 0.72 \times 10^{-4}$$ $$\beta_{\text{apparent}} = 8.78 \times 10^{-4} \text{ /°C}$$

Applications

• Expansion joints in bridges and buildings
• Gaps in railroad tracks
• Thermostat switches (bimetallic strips)
• Mercury/alcohol thermometers
• Shrink fitting of metal parts