Coefficient of Linear Expansion - Formula and Examples
What is Coefficient of Linear Expansion?
To put simply, linear expansion is the phenomena of increase in the length of a specimen or material because of the rising temperature of this same material. Consequently, the coefficient of linear expansion is expressed as the per degree Celsius, or change in the length of a 1 unit long material when there is a 10 C rise in temperature.
Coefficient of Linear Expansion Formula
As per the definition, the formula is expressed as
αL1 = ∆L / ∆T, or
αL1 = dL / dT,
Where,
α define coefficient of linear expansion.
L1 is the initial length of the material.
dL indicates a unit change in length.
dT indicates a unit change in temperature.
SI unit & Dimension
The SI unit of coefficient of linear expansion can be expressed as °C-1 or °K-1. Here, C indicates Celsius and K indicates Kelvin.
The dimension of coefficient of linear expansion will be \[M^{0}L^{0}T^{0} K^{-1}\]
Solved Numericals
1. A metal rod is of length 64.576 cm at a temperature 90°C whereas the same metal rod has a length of 64.522 cm at a temperature 12°C. Calculate the coefficient of linear expansion.
Ans: Consider L2 as 64.576 cm and L1 as 64.5222; then the value of dL will be 0.054. Consider T2 as 90°C and T1 as 12°C then the value of dT will be 78. Subsequently, from the formula, α will be 1.073 x 10-5/ °C.
Functions of Linear Expansion
Every material is packed with numerous atoms and the extent to which it can expand entirely depends upon the cohesive force between these particles. In simpler words, if the cohesive force is high, then the expansion in length of the material will be low even when the temperature is increased.
From the above, it can be concluded that the coefficient of linear expansion is an inherent property of a material, and it will vary from one element to another. For instance, soft metals expand quickly as the cohesive force between their atoms is low.
Coefficient of Linear Expansion Value for a Few Compounds
Since this is the intrinsic property of a compound, each one has a different value than others, based on their properties. To define coefficient of linear expansion of solid, these have a high coefficient because of the high cohesive force between the atoms in solid. Hard solids have a higher coefficient in the range of 10-7/K, whereas organic liquids can be in the range of 10-3/K.
Read on to familiarise yourself with various accessible materials and their coefficient of linear expansion.
Application
This concept is popularly used in various industrial applications as well as day to day operations, like opening the lid of a tight bottle by dipping it in hot water. Here are a few other applications related to this same concept.
The idea is used in thermometers for measuring temperature.
Riveting.
Thermostats.
Development of firm buildings.
In the creation of metal alloys.
These are a few concepts which will help you understand the idea of the coefficient of linear expansion equation better.
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Example question on the concept of the Coefficient of Linear Expansion
Example 1:- At the temperature of 0°C a brass rod was 200 inches long in length and on top of that the coefficient of linear expansion of brass is given at 0·0000188 per degree C, what will be the length of the brass rod at 50°C?
Solution: Given
Length of the brass rod at the temperature of 0°C = 200 inches.
The Coefficient of Expansion = \[ \frac{\text { Increase/decrease in length} } {\text {Original length } } \] per 1 degree celsius of temperature rise
From the equation above we can drive the formula for the increase in length
Change in length = 0.0000188 200 50
= 0.188 inches.
Hence the length of the brass rod at the temperature of 50°C is = 200 + 0.188 inches
= 200.188 inches.
FAQs on Coefficient of Linear Expansion
1. What does the coefficient of linear expansion mean in Physics?
The coefficient of linear expansion represents the fractional change in length of a material per unit change in temperature. It is a fundamental property that indicates how much a material expands or contracts along its length when heated or cooled, under constant pressure.
2. What is the standard formula for calculating coefficient of linear expansion as per CBSE 2025–26?
The formula for the coefficient of linear expansion (α) is:
α = ΔL / (L × ΔT), where:
- ΔL = Change in length
- L = Original length
- ΔT = Change in temperature
3. What is the SI unit and dimensional formula of coefficient of linear expansion?
The SI unit of coefficient of linear expansion is K⁻¹ (per Kelvin) or °C⁻¹ (per degree Celsius). The dimensional formula is M⁰L⁰T⁰K⁻¹.
4. How do different materials compare in terms of their coefficient of linear expansion?
Each material has a unique coefficient of linear expansion:
- Metals generally have higher coefficients due to weaker interatomic bonds (e.g., aluminium: ~24 x 10⁻⁶ /K).
- Hard solids usually have lower coefficients due to strong cohesive forces.
- Organic liquids may have even higher values, ranging up to 10⁻³ /K.
5. Why is the coefficient of linear expansion important in engineering and daily life?
The coefficient of linear expansion is crucial in construction, manufacturing, and science because it helps predict how materials behave with temperature changes. This ensures structures and devices remain safe and effective, such as:
- Preventing cracks in bridges or railway tracks
- Ensuring proper fit in mechanical parts
- Designing thermostats and thermometers
6. How would you calculate the change in length of a metal rod using the coefficient of linear expansion?
To calculate the change in length (ΔL) of a rod, use:
ΔL = α × L × ΔT, where:
- α = coefficient of linear expansion
- L = initial length of the rod
- ΔT = change in temperature
7. What factors affect a material’s coefficient of linear expansion?
The coefficient of linear expansion depends on:
- The material's atomic or molecular structure
- Strength of cohesive forces between the particles
- Purity or presence of alloys/impurities
8. Can you give real-world examples of using coefficient of linear expansion?
Examples include:
- Dipping a tight bottle lid in hot water – The lid expands and loosens due to increased temperature.
- Railway tracks/gaps – Small gaps are left to allow rails to expand in summer, preventing bending or damage.
- Thermometers – Use expansion of liquids to measure temperature changes accurately.
9. What misconceptions should students avoid about linear expansion?
Common misconceptions include:
- Believing that all materials expand at the same rate — expansion varies widely across substances.
- Thinking expansion occurs in only one dimension — solids can expand in all directions, but linear expansion focuses on length only.
- Assuming expansion is always significant — for many solids, it is a very small but critical change.
10. How is the coefficient of linear expansion used to design instruments or constructions for varying climates?
Engineers select materials with suitable coefficient of linear expansion values to ensure safety and durability across temperature ranges. For example, bridges use expansion joints, and buildings may use combinations of materials that expand at similar rates to prevent structural damage when temperatures change.
