Unit 10: Thermal Expansion

Introduction to Thermal Expansion

When a substance is heated, its particles gain kinetic energy and tend to move farther apart, resulting in an increase in dimensions. This phenomenon is known as thermal expansion. The expansion can be categorized based on the dimension that changes: length (linear), area (superficial), and volume (cubical). For liquids, expansion is described relative to the container, leading to the concepts of absolute and apparent expansion. Understanding these concepts is essential for applications ranging from engineering design to everyday phenomena like thermometers.

1. Linear Expansion

Definition

Linear expansion refers to the change in length of a solid object when its temperature changes.

Formula

The change in length (\(\Delta L\)) is given by:

where:

• α = coefficient of linear expansion (material constant)
• L₀ = original length at the initial temperature
• ΔT = change in temperature (final – initial) in °C or K

Coefficient of Linear Expansion

Rearranging the formula gives the definition of α:

Units: per degree Celsius (°C⁻¹) or per Kelvin (K⁻¹). The numerical value is the same for both scales because a change of 1 °C equals a change of 1 K.

Measurement Methods

1. Interferometer Method: Uses interference of light beams to measure minute changes in length with high precision (nanometre scale). The sample is placed in one arm of a Michelson interferometer; fringe shifts correspond to ΔL.
2. Micrometer Method: A precision micrometer screw gauge measures the change in length directly. The sample is heated in a controlled environment, and readings before and after heating give ΔL.

Example 1: Linear Expansion of a Steel Rod

New length = 2.000 m + 0.00072 m = 2.00072 m

2. Cubical and Superficial Expansion

Superficial (Area) Expansion

When a solid expands, its surface area also increases. For isotropic materials, the change in area (ΔA) is related to the original area (A₀) and temperature change by:

The coefficient of superficial expansion β is approximately twice the linear coefficient:

Cubical (Volume) Expansion

The change in volume (ΔV) of a solid is given by:

where γ is the coefficient of cubical expansion. For isotropic solids:

Relationship Between Coefficients

Combining the above approximations yields:

These relationships hold well for solids where expansion is uniform in all directions.

Example 2: Area and Volume Expansion of an Aluminium Plate

An aluminium plate has dimensions 0.5 m × 0.3 m at 25 °C. Find the increase in its area and volume (assuming thickness 0.01 m) when heated to 75 °C. (α_Al = 23 × 10⁻⁶ °C⁻¹.)

Solution:

• ΔT = 75 °C – 25 °C = 50 °C
• Original area A₀ = 0.5 m × 0.3 m = 0.15 m²
• β = 2α = 2 × 23 × 10⁻⁶ = 46 × 10⁻⁶ °C⁻¹
• ΔA = β A₀ ΔT = (46 × 10⁻⁶)(0.15)(50) = 3.45 × 10⁻⁴ m² = 0.345 cm²
• Original volume V₀ = A₀ × thickness = 0.15 m × 0.01 m = 0.0015 m³
• γ = 3α = 3 × 23 × 10⁻⁶ = 69 × 10⁻⁶ °C⁻¹
• ΔV = γ V₀ ΔT = (69 × 10⁻⁶)(0.0015)(50) = 5.175 × 10⁻⁶ m³ = 5.18 cm³

3. Liquid Expansion

Absolute vs. Apparent Expansion

Liquids expand when heated, but because they are usually contained in a vessel, the observed rise in liquid level reflects both the liquid’s expansion and the expansion of the container.

• Absolute Expansion (γ_abs): The true fractional increase in volume of the liquid per unit temperature change, assuming the container does not expand.
• Apparent Expansion (γ_app): The observed fractional increase in volume of the liquid relative to the container, i.e., the change in liquid level divided by the original volume and temperature change.
• Container (Vessel) Expansion (γ_vessel): The fractional volume increase of the container material per degree temperature change.

The relationship among them is:

Liquids generally have much larger coefficients of expansion than solids (typically 10⁻³ to 10⁻⁴ °C⁻¹ versus 10⁻⁵ to 10⁻⁶ °C⁻¹ for solids).

Example 3: Determining Absolute Expansion of Mercury

A glass flask (γ_glass ≈ 9 × 10⁻⁶ °C⁻¹) is filled with mercury. When the temperature is increased from 20 °C to 80 °C, the mercury level rises by 0.12 % relative to the flask. Calculate the absolute coefficient of expansion of mercury.

Solution:

1. ΔT = 80 °C – 20 °C = 60 °C
2. Apparent fractional increase = 0.12 % = 0.0012
3. γ_app = (ΔV/V₀)/ΔT = 0.0012 / 60 = 2.0 × 10⁻⁵ °C⁻¹
4. γ_vessel = γ_glass = 9 × 10⁻⁶ °C⁻¹
5. γ_abs = γ_app + γ_vessel = 2.0 × 10⁻⁵ + 9 × 10⁻⁶ = 2.9 × 10⁻⁵ °C⁻¹

Thus, the absolute coefficient of volume expansion of mercury is approximately 2.9 × 10⁻⁵ °C⁻¹.

4. Dulong and Petit Method for Determining Liquid Expansivity

Principle

The Dulong and Petit method (named after the physicists who studied specific heats) provides an experimental technique to measure the absolute coefficient of volume expansion of a liquid by correcting for the expansion of the containing glass vessel.

Apparatus

• A glass flask with a long, narrow stem (to amplify small changes in liquid level).
• The liquid whose expansivity is to be measured.
• A thermometer to monitor temperature.
• A scale or travelling microscope to measure the height of the liquid column in the stem.

Procedure

1. Clean and dry the flask. Fill it completely with the liquid, ensuring no air bubbles.
2. Insert the thermometer through the stopper so that its bulb is immersed in the liquid.
3. Record the initial liquid level (height h₁) in the stem at a known initial temperature T₁ (usually room temperature).
4. Place the flask in a temperature‑controlled bath (e.g., water bath) and gradually raise the temperature to a higher value T₂.
5. Allow thermal equilibrium, then record the final liquid level (height h₂).
6. Compute the apparent change in volume from the change in stem height, knowing the stem’s cross‑sectional area A_stem.
7. Calculate the apparent coefficient of expansion γ_app using:

where Δh = h₂ – h₁.

Correction for Glass Expansion

The flask itself expands, contributing to the observed change in level. The volume expansion of the glass (γ_glass) is known from tables or can be measured separately. The absolute coefficient of the liquid is then obtained by:

where γ_vessel ≈ γ_glass (assuming the flask’s volume change dominates).

Example 4: Dulong and Petit Experiment with Ethanol

In a Dulong and Petit setup, the stem of the flask has a cross‑sectional area of 0.2 mm². At 20 °C, the ethanol column height is 150 mm. After heating to 60 °C, the height reads 158 mm. The glass flask has γ_glass = 8.5 × 10⁻⁶ °C⁻¹. Determine the absolute coefficient of volume expansion of ethanol.

Solution:

1. ΔT = 60 °C – 20 °C = 40 °C
2. Δh = 158 mm – 150 mm = 8 mm = 0.008 m
3. Initial height h₁ = 150 mm = 0.150 m
4. Apparent fractional change in height = Δh / h₁ = 0.008 / 0.150 = 0.05333
5. γ_app = (Δh / h₁) / ΔT = 0.05333 / 40 = 1.333 × 10⁻³ °C⁻¹
6. γ_vessel = γ_glass = 8.5 × 10⁻⁶ °C⁻¹
7. γ_abs = γ_app + γ_vessel = 1.333 × 10⁻³ + 8.5 × 10⁻⁶ ≈ 1.342 × 10⁻³ °C⁻¹

Thus, the absolute volume expansion coefficient of ethanol is approximately 1.34 × 10⁻³ °C⁻¹, which aligns with literature values (~1.1 × 10⁻³ °C⁻¹) considering experimental uncertainties.

Summary of Key Formulas

Applications and Importance

Understanding thermal expansion is vital in many fields:

• Engineering: design of bridges, railways, pipelines to accommodate expansion joints.
• Manufacturing: precision machining where temperature variations affect tolerances.
• Everyday life: operation of thermostats, bimetallic strips, and liquid‑in‑glass thermometers.
• Science: accurate measurement of material properties and calibration of instruments.

Neglecting expansion can lead to structural failure, measurement errors, or malfunction of devices. The Dulong and Petit method exemplifies how combining simple measurements with known container properties yields accurate liquid expansivity data, essential for industries such as petrochemicals and beverage production.

Review Questions

1. Derive the relationship β ≈ 2α and γ ≈ 3α starting from the definition of linear expansion for an isotropic solid.
2. A copper wire of length 10 m at 15 °C is heated to 65 °C. Calculate its change in length (α_Cu = 17 × 10⁻⁶ °C⁻¹).
3. Explain why liquids generally exhibit a larger coefficient of expansion than solids.
4. In a Dulong and Petit experiment, the observed height change of a liquid column is 5 mm in a stem of area 0.5 mm² when the temperature rises by 30 °C. If the initial height is 200 mm and the vessel’s γ_vessel = 1.0 × 10⁻⁵ °C⁻¹, find the absolute coefficient of volume expansion of the liquid.