Coefficient of Linear Expansion

Coefficient of linear expansion is a constant that measures the change in a material's length due to the temperature change. It is denoted by α. Every material has a unique coefficient of linear expansion. The unit of coefficient of linear expansion is K^-1.

In this article, we are going to learn about the Coefficient of Linear Expansion in detail, including its formula, applications, and a few solved examples based on it.

What is Coefficient of Linear Expansion?

Linear expansion is the phenomenon of an increase in the length of a substance due to a temperature change. The coefficient of linear expansion (denoted as α) is a physical property that shows how much a material expands or contracts in length due to a change in temperature. It is an intrinsic property of a material and therefore, it is different for different materials.

• Coefficient of linear expansion measures how much a material expands per unit length.
• It is a constant value unique to each material.
• It is denoted by the symbol α.
• Different materials have different coefficients, depending on their molecular structure.

Coefficient of Linear Expansion Definition

Coefficient of Linear Expansion is defined as the rate of change in unit length per unit change in degree temperature is defined as the coefficient of linear expansion. 

Coefficient of Linear Expansion Formula

The rate of change of one unit length for every one-degree rise in temperature is known as the coefficient of linear expansion. Numerically, it can be expressed as:

α_L = ΔL/LΔT

where,

• L= initial length
• α_L = coefficient of linear expansion.
• ΔL= change in length
• ΔT= change in temperature

Dimension and S.I. units

The coefficient of linear expansion has dimensions of reciprocal temperature, typically in Kelvin. Its SI unit is Kelvin inverse (K^-1). This unit represents the change in length per unit length per degree Celsius.

• The linear expansion coefficient will have a dimension formula of [M⁰ L⁰ T⁰ K^-1].
• The S.I. unit of coefficient of linear expansion is °C^-1 or °K^-1 .

Factors Influencing Coefficient of Linear Expansion

Factors influencing the coefficient of linear expansion are:

• Material Composition: Different materials have varying coefficients of linear expansion due to their atomic structures. Materials with stronger interatomic bonds tend to have lower coefficients of linear expansion.

• Temperature: Coefficients of linear expansion generally increase with rising temperature. Higher temperatures cause atoms or molecules to vibrate more vigorously, leading to greater expansion.

• Crystal Structure: Crystal structures of materials influence their coefficients of linear expansion. Materials with a more ordered crystal lattice often exhibit lower coefficients of expansion.

• Pressure: Pressure can affect coefficients of linear expansion, especially in gases. Increased pressure can reduce the spaces between atoms or molecules, reducing expansion.

• Presence of Impurities: Impurities in materials can alter their coefficients of linear expansion. Impurities may disrupt the regular arrangement of atoms or molecules, affecting expansion behavior.

• Mechanical Stress: Mechanical stress can influence coefficients of linear expansion, particularly in solids. Applied stress can change the spacing between atoms, impacting expansion properties.

Coefficient of Linear Expansion of Materials

The coefficient of linear expansion varies for different materials. This coefficient determines how much a material expands or contracts when heated or cooled. Here is a list of the coefficients of linear expansion for various materials:

Material	Coefficient of Linear Expansion (α) at 20°C (10-6 K-1)
Aluminum	23.1
Brass	19
Copper	17
Glass	8
Iron	11.8
Lead	29
Steel	11
Concrete	12
Water	69
Air	3.4
Nylon	75
PVC	70
Silver	18
Silicon	2.56
Platinum	9

These values represent the change in length per unit length per degree Celsius. They change depending on the material's molecular structure and composition.

Relation between Coefficient of Linear Expansion and Volume Expansion

The coefficient of linear expansion (α) and the coefficient of volume expansion (?) are related in the following way:

? = 3α

This relationship arises from the fact that volume expansion involves changes in three dimensions (length, width, and height), each governed by the coefficient of linear expansion as volume is proportional to the cube of the length. 

Applications of Coefficient of Linear Expansion

The coefficient of linear expansion has various practical applications in everyday life and engineering. It is important for designing structures and systems that can handle temperature changes without damage. Here are some common applications:

• Bridge Construction: It helps design bridges that can expand and contract with temperature changes.

• Railway Tracks: Railway tracks use expansion joints for expansion and contraction of metal tracks due to temperature changes. Without these joints, the tracks could buckle, leading to accidents.

• Pipelines: It ensures pipelines can accommodate temperature variations without rupturing.

• Thermometers: It is used in designing thermometers to accurately measure temperature changes.

• Automobile Engines: Engine parts are designed considering thermal expansion to maintain performance.

Read More,

• Thermal Expansion
• Coefficient of Thermal Conductivity
• Thermal Properties of Matter

Solved Examples on Coefficient of Linear Expansion

Example 1:A steel rod has a length of 1.5 meters at 20°C. If its coefficient of linear expansion is 11 ×10^-6 K^-1, find its length at 100°C.

Solution:

Given, α = 11 × 10^-6 K^-1,

Δ? = 100°? − 20°? = 80°? and L=1.5 m.

Using the formula Δ? = αLΔT, we find

Δ?= (11 × 10^-6 )(1.5)(80) = 0.0132 m.

Thus, the length at 100°C is 1.5 + 0.0132 = 1.5132 m.

Example 2: A copper wire has an original length of 2 meters. If its length increases by 0.05 cm when heated from 20°C to 100°C, find its coefficient of linear expansion.

Solution:

Given, ΔL= 0.05 cm,

Δ? = 100°?−20°? = 80°? and L = 2 m.

Converting ΔL to meters (0.05 cm = 0.0005 m), we use the formula

? =Δ?/?Δ?

? = 0.0005/ 2 × 80 = 3.125 × 10^-6 K^-1.

Example 3:A brass rod has a length of 50 cm at 10°C. If its length increases by 0.1 cm when heated to 100°C, find its coefficient of linear expansion.

Solution:

Given, ΔL= 0.1 cm = 0.001 m,

Δ? = 100°?−10°? = 90°?, and L = 50 cm = 0.5 m.

Using α = ΔL/LΔT​, we find ? = 0.001/ 0.5×90 = 2.22×10^-5 K^-1.