Thermal expansion, linear, areal and volume expansion, explanation and relation between them, practice problems, FQAs

If you have ever seen the railway tracks carefully, there is a gap between them as shown in the figure. Ever wondered why engineers would give such provisions. Well to know the reason behind it one should be aware about thermal expansion.

Table of contents

Thermal Expansion
Thermal expansion at molecular level
Thermal expansion in solids
- Isotropic solids and anisotropic solids
- Linear expansion
- Areal expansion
- Volume expansion
Variation of density due to thermal expansion
Apparent Expansion of Liquids
Expansion of cavity
Time Period of Simple Pendulum under Temperature Variation
Real life consequences of thermal expansion:
Practice problems
FAQs

Thermal Expansion

Thermal expansion is the property of matter to change in shape, volume, area, and configuration in response to a change in temperature.
It can be thought of as a photographic enlargement of the object, where the distance between any two points increases/decreases in accordance with the temperature.

The graphical representation of an object at two different temperatures T and T0 (T >T0) is shown in the figure. It can be observed that the photographic enlargement is not only restricted to an increase in length, breadth, and area, but also increases the distance between any two points chosen at random.

Thermal expansion at molecular level

We can understand thermal expansion qualitatively on the molecular level. The interatomic forces in a solid can be considered as springs, as in figure below. There is an analogous relationship between spring forces and interatomic forces. Each atom vibrates about its equilibrium position. As spring is easier to stretch as compared to compress, the interatomic spring behaves in a similar manner and these are not symmetrical about equilibrium position.

With the rise in temperature, the amplitude of vibrations at atomic level increases inside the matter. This results in the increase of energies and average distances between the atoms. Therefore, the matter as a whole expands.

Point to Remember

With the rise in temperature, the amplitude of vibrations at atomic level increases inside the matter, which results in the increase of energies and average distances between the atoms. Therefore, the matter as a whole expands.

Thermal expansion in solids

Solids can expand in one, two, or three dimensions. These are known as linear expansion, areal expansion, and volume expansion, respectively.

Isotropic Solids

The solids whose physical properties are independent of the orientation of the system are known as isotropic solids. In other words, substances that expand linearly at the same rate in every direction are known as isotropic substances.

The coefficient of linear expansion (𝛼) of an isotropic solid is the same in x, y, and z directions.

Anisotropic Solids

The solids whose physical properties change in different orientations of the system are known as anisotropic or non-isotropic solids.

For an anisotropic solid, let 𝛼1, 𝛼2, and 𝛼3 be the coefficients of linear expansion in x, y, and z directions. Then, the coefficients of areal and volume expansion will be as follows:

𝛽= 𝛼1 + 𝛼2

𝛾= 𝛼1 + 𝛼2 + 𝛼3

Linear expansion

It refers to the change in the length of the body due to thermal expansion.

Consider a rod calibrated at length L0 initially at temperature T1. Heat has been supplied to the rod for some time. The temperature of the rod rises up to T2 and its length increases by amount ΔL.

According to the principle of thermal expansion, the change in length (ΔL)depends on the following two factors.

Introducing a proportionality constant,

Where is coefficient of linear expansion and it is property of a material.

Units of is or

For small increments in length,

Even for temperature differences in the range of 102, the value of expression αΔT is of order 10−3.

Hence, for practical purposes, one can expand the exponential series.

Linear expansion coefficients for some material

Areal expansion

Areal or superficial expansion refers to the expansion in the area of an object due to the increase in temperature.

The expression for areal thermal expansion is similar to that of linear expansion.

coefficient of real expansion

Initial area

and represents change in area and temperature respectively.

Final area

Consider a case of square sheets of area A0 for simplicity. It undergoes expansion due to a rise in temperature.

L0 and L1 are the lengths of the sides of the sheet before and after the expansion takes place. The final value of the area of the sheet is A.

Therefore,

Since

Thus the coefficient of areal expansion is twice as much as linear expansion.

Volume expansion

Volume expansion, also known as cubical expansion, is the increase in the volume of a solid due to the rise in temperature.

Similar to linear and areal expansions, the expression for volume expansion can be deduced as follows:

Thus,

Consider a cubical solid of side L0 initially at temperature T. The temperature of this solid has been raised by an amount ΔT. It goes under volume expansion and as a result, the length of its sides increases up to L.

From the expression of linear expansion, we can write it as:

Using binomial expansion as ,

Thus the coefficient of volume expansion is thrice as much as linear expansion.

It can be concluded that for solids,

Variation of density

Consider a body being heated by an external source. The temperature of the body rises by an amount ΔT. Consequently, the volume of the body also increases according to the volume expansion. Since the mass of the body remains unchanged, the density of the body decreases.

Density before heating,

Density after heating,

Therefore,

For solids, value of is generally small.

Using binomial expansion,

Apparent Expansion of Liquids

Consider a solid container filled with a liquid upto height h such that the excess volume can be drained into a small beaker as shown in the figure. Let the coefficient of volume expansion of the solid container and liquid be 𝛾C and𝛾L, respectively. The container is provided with heat from an external source, causing the temperature of both the container and liquid to rise by an amount ΔT.

Initially,

Volume of the container (VC) = Volume of the liquid (VL)

Volumes after heating,

Therefore the overflow volume of the liquid relative to the container is as follows:

is known as the apparent coefficient of volume expansion and this phenomena is known as the apparent expansion of liquid.

If or ,then the liquid’s level appears to have risen.
If or ,then the liquid’s level appears to have fallen.

Expansion of cavity

Since thermal expansion leads to the enlargement of the dimensions of an object, it does the same for the cavities. Thus if there is a cavity in a body or a hole in a lamina, the volume of the cavity or the area of the hole will increase with thermal expansion. Also the size and shape of the cavity(or hole) does not have any effect on the expansion of the body or plate.

For example: two spheres of same material, radius and at equal temperature, if heated to same temperature, will be having same outer radius

Time Period of Simple Pendulum under Temperature Variation

Consider a simple pendulum of natural length L0. A temperature change of ΔT has been introduced in the medium it is oscillating in, and therefore, the length of the pendulum will change due to thermal expansion/contraction.

Initially,at temperature time period,

Time period at temperature ,

Ratio final time period and initial time period,

Using binomial expansion of and neglecting higher order terms, we get

Fractional change in time period

Hence,

As temperature decreases (T < T0), time period (t) decreases (t < t0) and the clock speeds up. In other words, the clock gains time.
As temperature increases (T > T0), time period (t) increases (t > t0) and the clock slows down. In this case, the clock loses time.

Time gained/lost by the clock over interval of time

Real life consequences of thermal expansion:

The following points are some examples of thermal expansions in everyday life.

Rail tracks have small gaps at intervals to provide space for undesirable change in the length of rails, which otherwise would result in the bending of the rails at places.
The change in the length of measuring scales/instruments due to temperature difference may contribute to error in measurement.
An iron rim with slightly less diameter than a wooden wheel is heated up to a certain temperature, such that it barely can sit on the wooden wheel. As it cools off, the rim gets attached firmly with the wheel due to its tendency to contract.

Practice problems

Q. If the temperature of an iron rod of length changes by , what will be the percentage change in its length? ()

A. Given,

Percentage change in length

Q. Find the ratio of the lengths of an iron rod and an aluminium rod for which the difference in the lengths is independent of temperature. ( and.

Temperatures of both the rods were raised from T0 to T, which results in the increment of their lengths.

Assuming that the lengths of aluminium and iron rods at temperature T0 are l1 and l2, respectively. In response to the rise in temperature, their lengths are increased to l1’ and l2’, respectively.

It is also given that the difference in the lengths of both the rods (l1 − l2) is independent of temperature.

Therefore,

[Since is same for both the rods]

Hence, the ratio of the lengths of iron and aluminium rods is

Q. A glass flask is filled upto a mark with of mercury at . How much mercury will be above the mark, if the flask and contents are heated to ?

For glass is and coefficient of volumetric expansion of mercury is

Volume of mercury at ,

Similarly

Change in volume of flask upto mark due to change in temperature from to ;

Volume of mercury at above the mark

Q. A thin brass sheet which is at a temperature of have same surface area as that of a thin steel sheet at . Find the common temperature at which both would have the same area.

Coefficient of areal expansion for brass and steel are and respectively.

A. We know ,

Final area

where is the initial area.

Given:

is the same for both brass and steel sheet at and respectively.

Let at temperature , both have the same area .

FAQs

Q. Two rods having the same material and initial temperature but different lengths are heated to the same temperature. For which rod change in length will be greater?
A. Change in length will be more for longer rod as compared to shorter rod having same material and same initial temperature.

Q. For the same rise in temperature of a body, how much will be the areal expansion as compared to linear expansion?
A. Percentage change in area will be twice as much as percentage change in length since .

Q. If two metallic strips, one of steel and another of brass having the same length are joined together, what will happen if the temperature for the whole combination is raised? ()

A. Since the coefficient of expansion of brass is more than steel. Therefore, the brass

expands more and the steel expands less. This uneven expansion of the same strip results in the bending of the strip as shown in figure.