Thermal expansion

What is thermal expansion?

The increase in the dimensions of a body due to the increase in its temperature is called thermal expansion.

NCERT Physics Part II (Class 11), Section 10.5, page 205

In plain words: most things get bigger when heated and smaller when cooled. A tight metal lid loosens in hot water; mercury climbs a thermometer; a half-blown balloon fills out in warm water (NCERT, pages 204–205). This proportionality is a local or small-change approximation; the coefficient depends on the material and temperature.

Linear expansion

For a long rod, the fractional increase in length is proportional to the temperature rise:

Some values from NCERT Table 10.1 (page 205), for 0 °C to 100 °C, in units of 10^⁻⁵ K^⁻¹: aluminium 2.5, brass 1.8, iron 1.2, copper 1.7, silver 1.9, gold 1.4, glass (pyrex) 0.32. Metals expand more than glass: copper expands about five times as much as pyrex for the same rise in temperature (NCERT, page 205).

NCERT’s Table 10.1 also lists lead at 0.29 × 10^⁻⁵ K^⁻¹. That value is ten times too small: the accepted coefficient for lead is about 2.9 × 10^⁻⁵ K^⁻¹, larger than for most common metals. (At 0.29, lead would expand less than pyrex, which contradicts NCERT’s own remark on the same page that metals normally expand more.)

Area and volume expansion

For volume, NCERT defines the coefficient of volume expansion by

It depends on material, state and temperature. NCERT’s copper graph shows a nearly constant high-temperature region; that is not a universal rule for every material. Some values from NCERT Table 10.2 (page 206), in 10^⁻⁵ K^⁻¹: iron 3.55, ordinary glass 2.5, pyrex 1, invar (an iron-nickel alloy) 0.2, mercury 18.2, water 20.7, ethanol 110. Solids and liquids expand only slightly; alcohol expands more than mercury.

Why area goes as 2α and volume as 3α. Heat a rectangular sheet of sides a and b. Each side grows by the factor (1 + α_lΔT), so the area grows from ab to ab(1 + α_lΔT)^² = ab(1 + 2α_lΔT + α_l²ΔT^²). Since α_lΔT is tiny (about 10^⁻⁵ × ΔT), its square can be dropped, giving ΔA/A ≈ 2α_lΔT (NCERT Example 10.1, page 207). The same argument for a cube gives ΔV/V ≈ 3α_lΔT, so α_V = 3α_l (NCERT Eqs. 10.7–10.9). A hole in a uniformly heated, unconstrained isotropic plate expands as though it contained the same material (Example 3). The relations between differential coefficients, β = 2α and γ = 3α, are exact for isotropic expansion; the finite-change formulas neglect higher powers of αΔT.

Density falls when volume increases. The mass is fixed while the volume grows, so the density becomes ρ′ = ρ/(1 + α_VΔT) ≈ ρ(1 − α_VΔT).

Gases. Gases expand far more than solids and liquids, and their coefficient depends strongly on temperature. For an ideal gas (μ moles, R the gas constant) at constant pressure, the ideal-gas equation PV = μRT gives PΔV = μRΔT. Dividing by PV = μRT gives ΔV/V = ΔT/T, so, with T in kelvin,

At 0 °C this is 3.7 × 10^⁻³ K^⁻¹, and at room temperature about 3.3 × 10^⁻³ K^⁻¹ (NCERT, page 206), larger than for most liquids (page 207); at room temperature it is about 18 times mercury’s value but only about three times the tabulated ethanol value.

Thermal stress

For a uniformly heated rod with fixed ends, initially unstressed and remaining linearly elastic without buckling, the supports compress it by the amount it would have expanded. The compressive strain is α_lΔT, and by Hooke’s law the stress set up, called thermal stress, is

where Y is the Young’s modulus of the material (NCERT, page 207). It can be very large (Example 2), which explains gaps in traditional jointed rails and expansion joints in bridges. Continuous welded rail is largely restrained against expansion and develops thermal stress; its stability depends on rail anchorage and the stress-free installation temperature.

Anomalous expansion of water

Water exhibits an anomalous behaviour; it contracts on heating between 0 °C and 4 °C.

NCERT Physics Part II (Class 11), Section 10.5, page 206

As water cools from room temperature its volume falls, as for other liquids, until 4 °C; below 4 °C it expands as it cools further. So:

This means that water has the maximum density at 4 °C.

NCERT Physics Part II (Class 11), Section 10.5, page 206

This matters for life in lakes and ponds. As a lake cools towards 4 °C, the surface water becomes denser and sinks, and warmer water rises. Once the surface water is below 4 °C it is less dense and stays on top, where it freezes. So lakes freeze from the top down; otherwise, NCERT notes, they would freeze from the bottom up, destroying much of their animal and plant life (page 206). In a sufficiently deep freshwater lake, floating ice insulates water below, which can remain near 4 °C. Shallow ponds can freeze through; depth, mixing and salinity affect the outcome.

Solved examples

Common mistakes to avoid

Quick revision summary

• Most substances expand on heating: Δl/l = α_lΔT, ΔV/V = α_VΔT.
• For an isotropic solid, area expansion is 2α_lΔT and α_V = 3α_l; many common metals expand more than pyrex, though this is not universal.
• For an ideal gas at constant pressure, α_V = 1/T, far larger than for solids and liquids.
• A rod kept from expanding develops a thermal stress Yα_lΔT.
• Water contracts on heating from 0 °C to 4 °C and is densest at 4 °C, so lakes freeze from the top.
• In short: α_V = 3α_l.

Questions students ask

What is the relation between α, β and γ in thermal expansion?

For an isotropic solid the coefficients of linear (α), area (β) and volume (γ) expansion are related by β = 2α and γ = 3α, so α : β : γ = 1 : 2 : 3. NCERT writes α_l for α and α_V for γ. For instantaneous differential coefficients these relations are exact under isotropic expansion. Using them in linear finite-change formulas assumes αΔT is small, so higher powers can be neglected. For a crystal that expands by different amounts along three perpendicular directions, the volume coefficient is the sum α_x + α_y + α_z.

What is the SI unit and dimensional formula of the coefficient of linear expansion?

The coefficient of linear expansion has the SI unit K^⁻¹ (per kelvin, which is the same as per °C) and the dimensional formula [K^⁻¹], because α_l = Δl/(lΔT) is a fractional change in length, which has no dimensions, divided by a change in temperature. Its value depends on the material and slightly on the temperature, but not on the length or shape of the body: a 1 m and a 10 m iron rod have the same α_l, about 1.2 × 10^⁻⁵ K^⁻¹, though the longer rod grows ten times as much. The coefficients of area and volume expansion have the same unit and dimensions.

Why does water expand when it is cooled below 4 °C?

Water expands when cooled below 4 °C because of hydrogen bonding between its molecules: as it nears freezing, more and more molecules join into open, ice-like groups that take up more room than the closely packed liquid. Between 0 °C and 4 °C this opening-up outweighs the normal contraction on cooling, so the volume grows as the temperature falls; above 4 °C ordinary thermal contraction wins, which is why water is densest at 4 °C. On freezing, the whole structure becomes the open hexagonal lattice of ice, about 9% larger in volume than the water it came from, which is why ice floats and frozen water pipes can burst.

What are some examples of thermal expansion in daily life?

Everyday examples of thermal expansion include the small gaps left between rail lengths and the expansion joints in bridges and concrete roads, which give them room to grow on hot days, and overhead power lines, which sag more in summer than in winter. A thick glass tumbler can crack when boiling water is poured into it, because its inner surface expands before the outer one; heat-resistant glassware is made of borosilicate glass (pyrex), which expands less than half as much as ordinary glass. Thermostats in electric irons and water heaters use a bimetallic strip, two metals that expand by different amounts and so bend when heated, to switch the current off and on.

Does a metal scale read too high or too low on a hot day?

A metal scale calibrated at a lower temperature reads too low on a hot day, because its marks spread apart, so each ‘centimetre’ on it is slightly longer than a true centimetre. The true length of the object, at the temperature of measurement, is about the reading × (1 + α_lΔT), where α_l is for the scale’s metal and ΔT is the rise above the calibration temperature. For a steel scale calibrated at 20 °C and used at 45 °C (α_l = 1.2 × 10^⁻⁵ K^⁻¹), a reading of 80.00 cm means 80.00 × (1 + 1.2 × 10^⁻⁵ × 25) ≈ 80.024 cm.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A steel bridge is 100 m long. Over a year its temperature ranges from −10 °C to 40 °C. By how much does its length change? Take α_l = 1.2 × 10^⁻⁵ K^⁻¹.

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   Δl = 1.2 × 10^⁻⁵ × 100 × 50 = 0.06 m, that is 6 cm, which is why bridges have expansion joints.

2. Q2. Why does a lake freeze at the top first, while the water at the bottom stays liquid?

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   Below 4 °C water expands as it cools, so the coldest water is the least dense and stays at the surface, where it freezes. Ice is less dense still and floats, and the water at the bottom stays near 4 °C.

3. Q3. Using NCERT Table 10.1, estimate the coefficient of volume expansion of aluminium and compare it with Table 10.2’s value of 7 × 10^⁻⁵ K^⁻¹.

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   α_V = 3α_l = 3 × 2.5 × 10^⁻⁵ = 7.5 × 10^⁻⁵ K^⁻¹, close to the table’s 7 × 10^⁻⁵ K^⁻¹.