Coefficient of Linear Expansion: Definition, Formula & Applications

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Linear expansion refers to the change in the length along one dimension over volume. This phenomenon occurs due to a rise in the temperature of a material. It is expressed as per degree Celsius. In this article, we will explore the coefficient of linear expansion, its working and application.

Table of Content

Coefficient of Linear Expansion
Formula
SI Unit and Dimension
Working
Applications
Value for some combinations
Things to Remember
Sample Questions

Coefficient of Linear Expansion

The term expansion can be explained as modification or increase in length. If the variation in length is alongside one dimension over the volume, it is hence called Linear Expansion. At this point, the reason behind the expansion is the temperature change. Therefore, it is understood that the temperature change will replicate the rate of expansion. The concept is used to explain how much a material can endure and maintain its original shape and size under the effect of heat radiation.

Check out pdf-notes for Thermal Properties of Matter

Coefficient of Linear Expansion Formula

According to the definition of linear expansion, its formula can be expressed as

α_L= ΔL / ΔT, or

α_L = dL/dT,

Where,

α_L, refers to the coefficient of linear expansion.
dL specifies a unit change in length.
dT specifies a unit change in temperature.

SI unit and Dimension of Linear Expansion

The SI unit of coefficient of linear expansion can be stated as °C^-1 or °K^-1. Here, C indicates Celsius and K indicates Kelvin.

The dimension for the coefficient of linear expansion will be M⁰L⁰T⁰K⁻¹.

Working of Linear Expansion

The linear expansion coefficient is a fundamental property of each material. Henceforth, it differs from one material to another. The rate at which a material can expand depends only on the cohesive force between the atoms. Cohesive force is the force that attaches two or more atoms.

The cohesive force is responsible for overcoming the separation between the atoms. The superior the cohesive force, the lower the expansion will be for a specified increase in temperature. Soft metals like, Lead have a low melting point and can be compressed effortlessly. On heating, lead will expand quicker with a unit growth in temperature.

Applications of Coefficient of Linear Expansion

The result of advancements in science and technology are enormous. To match with this rapid progress in industrialization and construction, one needs to be sure about the use of the material palette. From constructing a building to building a satellite, the material used acts as a backbone of the product formed.

The dissimilar variation of materials is readily available around us. Every one of them has diverse thermal properties. Comparing the expanding capability with a rise in temperature for several materials is critical for their use in appropriate situations. Normally, the material with a higher linear expansion coefficient is strong and can be used in building firm arrangements. This property can be improved further to match the required need by mixing the materials. Therefore, currently, metal alloys are getting popular. The various applications of coefficient of linear expansion are:

The lid of a tight bottle is opened by dipping the same in hot water.
Measuring temperature using a thermometer.
Thermostats
Making of firm buildings
Riveting

and many more.

Coefficient of Linear Expansion value for some combinations

Based on their properties, each compound or material has a different coefficient of linear expansion. Solids have a high coefficient of linear expansion because of the high cohesive force amongst the atoms in the solid. Hard solids have a higher value of their coefficient which is in the range of 10^-7/K. Though the value for organic liquids can be in the range of 10^-3/K.

The various values of coefficient of linear expansion for various materials are as follows:

Metals	Coefficient of linear expansion α_L at 20^oC (10⁻⁶K^-1)
Aluminum	23.1
Benzocyclobutene	42
Brass	19
Carbon steel	10.8
Concrete	12
Copper	17
Diamond	1
Ethanol	250
Gallium(III) Arsenide	5.8
Gasoline	317
Gold	14
Ice	51
Iron	11.8
Lead	29
Magnesium	26
Mercury	61
Nickel	13
Platinum	9
Water	69
Silicon	2.56
Silver	18

Things to Remember based on Coefficient of Linear Expansion

If the variation in length for a material is alongside one dimension over the volume, it is called linear expansion.
The reason behind this expansion is a temperature change.
Formula: αL = dL/dT

Where α_L refers to the coefficient of linear expansion.

SI Unit: °C^-1 or °K^-1
The rate at which a material can expand only depends on the cohesive force between the atoms.
Comparing the expanding capability with a rise in temperature for several materials is critical for their use in appropriate situations.
Application: The lid of a tight bottle is opened by dipping the same in hot water.

Sample Questions based on Coefficient of Linear Expansion

Ques. Which metal has the uppermost coefficient of expansion? 1 mark

Ans. Ti-Nb alloys are measured to have the maximum coefficient of expansion. In these alloys, the value of α_L will be around + 163.9×10^-6 to −95.1×10^-6 °C^-1.

Ques. Which metal has the uppermost coefficient of expansion? 1 mark

Ans. The worth of αL for steel at 20°C /10^-6/K is between 11.0 and 13.0. This accounts for a minor variation in measurement of Steel at a temperature of 20°C when there is a unit rise in temperature.

Ques. A metal rod is of length 64.576 cm at a temperature of 90°C whereas a similar metal rod has a length of 64.522 cm at a temperature 12°C. Analyze the coefficient of linear expansion. 3 marks

Ans. Take L₂ as 64.576 cm and L₁ as 64.52 cm; then the worth of dL will be 0.054.

Consider T₂ as 90°C and T₁ as 12°C then dT will be 78.

Consequently, putting the values in the formula, α shall be 1.073 x 10^-5/ °C.

Ques. Mention few applications of coefficient of linear expansion. 3 marks

Ans. Few applications of coefficient of linear expansion are:

The lid of a tight bottle is opened by dipping the same in hot water.
Measuring temperature using a thermometer.
Thermostats
Making of firm buildings
Riveting

Ques. A metal bar of 60 cm has a temperature of 10oC. Find the length of the bar at 110^oC. αL is given as 1.5 x 10⁵/^oC. 3 marks

Ans. l₂ = l₁(1+α (t₂ - t₁))

= 60(1 + 1.5 x 10⁵ (110-10)

= 60 x 1.0015

= 60.09 cm

Hence, at 110^oC, the material will be of 60.09 cm.

Ques. A material of 300.36 cm has a temperature of 100^oC. At 150^oC, the material is 300.54 cm. Find the length of the material and coefficient of linear expansion of iron at 0^oC. 5 marks

Ans. l₁ = l₀(1+αt¹)

300.36 = l_o(1+100α) --- 1

l₂ = l₀(1+αt₂)

300.54 = l_o(1+150α) --- 2

Dividing equation 2 by 1

300.54/300.36 = l_o(1+150α)/lo(1+100α)

1.0006 = (1+150α)/(1+100α)

1.0006(1+100α) = (1+150α)

1.0006 - 1 = 150α - 100.06α

0.0006 = 49.94α

α = 1.2 x 10^-5/^oC

From equation 1

300.36 = l_o(1+100α)

300.36 = l_o(1+100 x 1.2 x 10^-5/oC)

300.36 = l_o(1+ 1.2 x 10^-3)

300.36 = l_o(1.0012)

l_o = 300.36/1.0012

= 300 cm

Hence, at 0^oC, the length of the rod is 300 cm and the coefficient of linear expansion is 1.2 x 10-5/^oC.

CBSE CLASS XII Related Questions

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Coefficient of Linear Expansion: Definition, Formula & Applications

Coefficient of Linear Expansion

Coefficient of Linear Expansion Formula

SI unit and Dimension of Linear Expansion

Working of Linear Expansion

Applications of Coefficient of Linear Expansion

Coefficient of Linear Expansion value for some combinations

Things to Remember based on Coefficient of Linear Expansion

Sample Questions based on Coefficient of Linear Expansion

CBSE CLASS XII Related Questions

2.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

3.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

4.
Draw a circuit diagram of a full-wave rectifier using p-n junction diodes. Explain its working and show the input-output waveforms.

5.
Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

6.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

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Coefficient of Linear Expansion: Definition, Formula & Applications

Coefficient of Linear Expansion

Coefficient of Linear Expansion Formula

SI unit and Dimension of Linear Expansion

Working of Linear Expansion

Applications of Coefficient of Linear Expansion

Coefficient of Linear Expansion value for some combinations

Things to Remember based on Coefficient of Linear Expansion

Sample Questions based on Coefficient of Linear Expansion

CBSE CLASS XII Related Questions

2.What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

3.Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

4.Draw a circuit diagram of a full-wave rectifier using p-n junction diodes. Explain its working and show the input-output waveforms.

5. Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

6.The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be:

Comments

SUBSCRIBE TO OUR NEWS LETTER

2.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

3.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

4.
Draw a circuit diagram of a full-wave rectifier using p-n junction diodes. Explain its working and show the input-output waveforms.

5.
Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

6.
The figure shows three point charges kept at the vertices of triangle ABC. The net electric field, due to this system of charges, at the midpoint M of base BC will be: