18-3 Thermal Expansion#
Prompts
Write the equation for linear thermal expansion. What is \(\alpha\)? Does it apply to holes and thickness as well as length?
For an isotropic solid, how is the volume expansion coefficient \(\beta\) related to the linear coefficient \(\alpha\)?
A bimetallic strip is made of brass and steel bonded together. When heated, which side bends outward? Why?
Why do lakes freeze from the top down? What is special about water between 0°C and 4°C?
Lecture Notes#
Overview#
Most materials expand when heated—atoms gain energy and move farther apart against interatomic forces.
Linear expansion: \(\Delta L = \alpha L \Delta T\); volume expansion: \(\Delta V = \beta V \Delta T\).
For isotropic solids, \(\beta \approx 3\alpha\). Applications: bridges, bimetallic strips, thermometers.
Linear expansion#
For a rod of length \(L\) and temperature change \(\Delta T\):
\(\alpha\): coefficient of linear expansion (units: K\(^{-1}\) or °C\(^{-1}\)).
Applies to every linear dimension—length, thickness, diameter, hole size. A disk that fits snugly in a hole continues to fit if both undergo the same \(\Delta T\).
Material |
\(\alpha\) (10\(^{-6}\)/°C) |
|---|---|
Aluminum |
23 |
Brass |
19 |
Steel |
11 |
Glass (ordinary) |
9 |
Pyrex |
3.2 |
Invar (low-expansion alloy) |
0.7 |
Poll: Coefficient units (K vs °C)
The coefficient of linear expansion for a special alloy of steel is \(1.0 \times 10^{-5}\)/K. What is the coefficient in 1/(°C)?
(A) More than \(1.0 \times 10^{-5}\)/(°C)
(B) Less than \(1.0 \times 10^{-5}\)/(°C)
(C) Equal to \(1.0 \times 10^{-5}\)/(°C)
Volume expansion#
For a solid or liquid of volume \(V\):
\(\beta\): coefficient of volume expansion.
For isotropic solids: \(\beta = 3\alpha\) (each of three dimensions expands by \(\alpha\)).
Liquids
For liquids, \(\beta\) is measured directly (no simple relation to \(\alpha\)). Liquids typically expand more than solids—e.g., mercury in a glass thermometer.
Poll: Area expansion (systematic)
The coefficient of linear expansion for a steel alloy is \(1.0 \times 10^{-5}\)/K. How much must you increase the temperature to increase the area of a plate by 1%?
(A) 5 K
(B) 10 K
(C) 50 K
(D) 100 K
(E) None of these
Poll: Brass ring hole
Brass has a positive coefficient of thermal expansion \(\alpha\) (where \(\Delta L/L = \alpha \Delta T\)). A ring (annulus) of brass is heated. Does the hole in the middle get larger or smaller in area?
(A) Larger
(B) Smaller
(C) Stays the same
Bimetallic strip#
A bimetallic strip bonds two metals with different \(\alpha\) (e.g., brass and steel). When heated, the side with larger \(\alpha\) expands more → the strip bends. Used in thermostats and thermometers.
Water anomaly#
Water is unusual: between 0°C and ~4°C it contracts with increasing temperature—density is maximum near 4°C. Above 4°C it expands normally.
Consequence: As a lake cools from the surface, water below 4°C is less dense and stays on top → ice forms at the surface. If water froze from the bottom up, lakes could freeze solid and aquatic life would be threatened.
Poll: Expansion
A steel railroad track is 100 m long at 20°C. On a hot day (40°C), by how much does its length increase? (\(\alpha_{\text{steel}} = 11 \times 10^{-6}\)/°C)
(A) 0.011 m
(B) 0.022 m
(C) 0.044 m
Summary#
\(\Delta L = \alpha L \Delta T\)—linear expansion; applies to all linear dimensions.
\(\Delta V = \beta V \Delta T\)—volume expansion.
\(\beta = 3\alpha\) for isotropic solids.
Bimetallic strip: different \(\alpha\) → bending when heated.
Water: density maximum near 4°C; explains why lakes freeze from the top.