THERMAL EXPANSION

physical phenomenon where materials change their size with their temperature changes

Thermal expansion is the physical phenomenon where materials change their size when their temperature changes. As a substance is heated, the particles inside it gain energy and vibrate more intensely, which increases the average distance between them. This microscopic movement translates into a macroscopic change in dimensions. Depending on the shape and measurement of the object, this expansion can be observed as linear expansion (change in length), area expansion (change in surface area), or volume expansion (change in overall size). The extent of expansion depends on the material’s coefficient of thermal expansion, which is a property unique to each substance.

Thermal expansion is the change in size of matter in response to change in temperature.

LINEAR EXPANSION

Linear expansion is a type of thermal expansion that refers specifically to the increase in length of a solid object when its temperature rises. This phenomenon occurs because heat causes the atoms or molecules in a material to vibrate more vigorously. As they vibrate, they push slightly farther apart, which results in the material stretching or elongating. Although the change in length is usually very small, it becomes significant in engineering and construction where precision matters — like in bridges, railways, or metal pipes

FORMULA

The amount of expansion can be calculated using the formula:

ΔL=α⋅L_0⋅ΔT

Where:

ΔL= change in length

α= coefficient of linear expansion (depends on the material)

L_0= original length of the object

ΔT= change in temperature

To find the final length after expansion:

L=L_0+ΔL

This formula shows that the longer the object and the greater the temperature change, the more it will expand — assuming the material has a significant coefficient of expansion.

ILLUSTRATIVE EXAMPLE

a steel rod that is 2 meters long. Steel has a coefficient of linear expansion of about 1.2×10^(-5) " " "K" ^(-1). If the temperature increases by 50°C:

ΔL=(1.2×10^(-5))⋅2⋅50=0.0012" m"=1.2" mm"

the rod will expand by 1.2 mm — a small but important change in engineering applications.

2. a steel bridge is built in several segments, each 20 meters long. the bridge was constructed when the temperature was 20 0C. if a gap of 4 cm long is left between neighboring segments what would be the maximum temperature the bridge can manage before buckling?

Solution:

Given: L_0=20" " mΔL_"gap" =0.04" " mα=1.2×10^(-5) " " K^(-1)

Formula: ΔL=αL_0 ΔT

ΔL=ΔL_"gap" : ΔT=(ΔL_"gap" )/(αL_0 )

ΔT=0.04/((1.1×10^(-5))(20) )=ΔT=0.04/(2.2×10^(-4) )

ΔT≈180^∘ C

Tmax=20+180=200∘C

LINEAR EXPANSION COEFFICINT OF SOME SOLIDS

Material

α×10-6(0C-1)

Aluminum

23.1

Copper

16.5

Brass

19.0

Iron

11.8

Steel (Carbon)

10.8

Glass (Borosilicate)

3.3

Concrete

12

Gold

14

Diamond

1.0–1.3

Invar

1.2

AREA EXPANSION

Area expansion is the increase in surface area of a solid when its temperature rises. It occurs because both the length and width of the object expand simultaneously. This is important in materials like metal sheets or glass panels used in construction.

FORMULA

ΔA=β⋅A_0⋅ΔT

Where: ΔA= change in area, β= coefficient of area expansion, A_0= original area, and ΔT= temperature change

Final area: A=A_0+ΔA

Relationship Between α and β

β=2α

Since area is two-dimensional, both length and width expand, doubling the effect of linear expansion.

ILLUSTRATIVE EXAMPLE

A metal sheet is 100 cm × 50 cm, so A_0=5000" " 〖"cm" 〗^2. If α=1.2×10^(-5) " " 〖"°C" 〗^(-1), then β=2.4×10^(-5) " " 〖"°C" 〗^(-1). For a temperature rise of 50°C:

ΔA=(2.4×10^(-5))(5000)(50)=6" " 〖"cm" 〗^2

So the new area is 5006" " 〖"cm" 〗^2.

VOLUME EXPANSION

Volume expansion is the increase in the overall size (volume) of a solid, liquid, or gas when its temperature rises. Just like linear and area expansion, this happens because particles vibrate more and move farther apart. Since volume is three-dimensional, expansion occurs in length, width, and height simultaneously.

Peculiarity of Water Expansion

Water behaves differently from most substances. Normally, materials expand when heated and contract when cooled. But water has an anomalous expansion between 0°C and 4°C:

As water cools from 4°C to 0°C, it expands instead of contracting.

This is why ice at 0°C is less dense than liquid water at 4°C, allowing ice to float.

This anomaly is crucial for life: lakes and rivers freeze at the surface while remaining liquid below, protecting aquatic organisms during winter.

Formula

The change in volume is given by:

ΔV=γ⋅V_0⋅ΔT

Where: ΔV= change in volume, γ= coefficient of volume expansion, V_0= original volume, ΔT= temperature change

Final volume: V=V_0+ΔV

Relationship Between α and γ

Since volume involves three dimensions, the coefficient of volume expansion is approximately:

γ=3α

This means volume expansion is three times the effect of linear expansion.

ILLUSTRATIVE EXAMPLE:

a glass container holds 1 liter of ethanol at 20°C. Ethanol has γ≈1.1×10^(-3) " " K^(-1). If the temperature rises by 30°C:

ΔV=(1.1×10^(-3))(1)(30)=0.033" " L

So the ethanol expands by 33 mL, making the final volume 1.033 L. This shows why liquid-filled containers need space at the top to prevent overflow when heated.

a metal cube of volume 1000" " 〖"cm" 〗^3at 20°C. If the coefficient of linear expansion is

α=1.2×10^(-5) " " K^(-1), then: γ=3α=3.6×10^(-5) " " K^(-1). For a temperature rise of 100°C:

ΔV=(3.6×10^(-5))(1000)(100)=3.6" " 〖"cm" 〗^3

So the cube’s new volume is 1003.6" " 〖"cm" 〗^3.