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.
To get a better understanding, Vedantu app can help you get a comprehensive picture of this concept while helping you solve various tricky questions. Download our app to get access to carefully curated study material by proficient teachers.
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 is the coefficient of linear expansion of aluminium?
The value of αL for aluminium at 20°C (10-6/K) is 23.1. This leads to a fractional change in length of aluminium at a temperature 20°C when there is a per unit rise in temperature.
The average coefficient of linear expansion for the pure aluminium for the commercial usage is equal to 24 × 10 -6/K (13 × 10 -6/°F). Nearly double the amount of what carried by the ferrous metals. The expansion of aluminium can be reduced by the addition of a few traces of silicon and copper.
2. What is the coefficient of linear expansion of steel?
The value of αL for steel at 20°C (10-6/K) is between 11.0 and 13.0. This leads to a minor change in length of Steel at a temperature 20°C when there is a unit increase in temperature.
The value of the coefficient of linear expansion of the steel can vary a lot, if the concentration of the compounds present in it changes. Hence, there is no exact value for the coefficient of linear expansion of steel.
3. Which metal has the highest coefficient of expansion?
Ti-Nb alloys are considered to have the highest coefficient of expansion. In these alloys, the value of αL will be around + 163.9×10-6 to −95.1×10-6 °C-1.
But in the case of pure metal, not allies, the metal carrying the highest coefficient of expansion will be the Aluminium (Al), with a value of the coefficient of linear expansion equal to 24 × 10 -6/K (13 × 10 -6/°F). Hence Ti - Nb is the alloy and Al is the metal with the highest coefficient of expansion.
4. What are the Applications of the Coefficient of Linear Expansion?
Science and Technology are growing at such a rapid pace that it is becoming more and more difficult to match with this process. The better use of materials is another major factor contributing to such growth levels and choosing the right material is crucial too like for constructing a building that can stand tall for the years to come and should be able to handle the major changes in the weather on the place where it was built, or For building a satellite, one has to consider how to protect the spacecraft from the dangerous solar wind and other phenomenon that can be harmful for the satellite. Such an issue requires a deep knowledge of the material, so one can choose wisely which material to use. That is where concepts such as the concept of Coefficient of Linear Expansion of a material do help, as it provides a deep understanding regarding a material’s thermal properties.
Leaving the world of science this concept is also used in day to day life, one example of this concept in use in everyday life will be - In order to open a tight lid of the bottle, one may first dip the lid in the hot water.
One more important use for this concept is the measurement of the temperature via a thermometer.
5. Mention some of the Important notes/points to remember in the Coefficient of Linear Expansion?
Listed down below are the most important things to remember in the concept of the Coefficient of Linear Expansion. These are -
If the given variation of the length of the given material is in a single dimension only, then it is called Linear Expansion. The measurement for those variations in length can be given with the help of the Coefficient of Linear Expansion.
The Coefficient of Linear Expansion is the measurement of variation of length of a material on a per unit change in the temperature.
The formula to find the coefficient of linear expansion (α L1) is given by
α L1 = ∆L / ∆T
The SI unit of the Coefficient of Linear Expansion °C-1 or °K-1.
The rate of expansion of a material in accordance with the change in temperature solely depends on the Bonding between the molecule or the atoms of that material.
Real life application of this will be - Dipping the lid of a tightly closed bottle in order to loosen and open it.
