Physics · Molecular physics and thermodynamics · Grade 10

Thermal expansion

Why are gaps left between rails, and why doesn't a lake freeze to the bottom?

The situation

In summer in Uzbekistan the air rises above +40 °C, and in the sun railway tracks heat up to about +60 °C; in winter they cool to −20 °C. That is why a gap of a few millimetres is left between 25-metre rail sections. An iron switches itself off, the liquid in a thermometer rises, and in winter a lake freezes only from the surface — the fish survive at the bottom. All of this is linked to a single phenomenon — thermal expansion. Why do bodies expand when heated, and why does water not obey this rule?

Topic

Thermal expansion. Gap between railway rails and the anomaly of water at 4 °C

Key ideas

Linear and volume expansion
Bimetallic thermostat
Anomalous expansion of water

Formulas

Δ L = α L_{0} Δ T

Linear expansion: $α$ is the coefficient of linear expansion, 1/K. For steel $α \approx 1.2 \cdot 10^{- 5}$ 1/K.

Δ V = β V_{0} Δ T, β \approx 3 α

Volume expansion. For a solid ${(1 + α Δ T)}^{3} \approx 1 + 3 α Δ T$ .

σ = E α Δ T

Thermal stress in a rod that is not allowed to expand ( $E$ is Young's modulus).

\frac{1}{R} \approx \frac{3}{2} \frac{(α_{1} - α_{2}) Δ T}{h}, d = \frac{L^{2}}{2 R}

Curvature of a bimetallic strip (for layers of equal thickness and equal Young's moduli), $h$ is the total thickness, $d$ is the displacement of the end.

β = \frac{S Δ h}{V_{0} Δ T} + β_{s h}

Dilatometer: the level in the capillary rises by $Δ h$ ; the glass vessel also expands, so its $β_{s h} = 3 α_{s h}$ is added.

Theory

Why do bodies expand?

In a solid the atoms oscillate about their equilibrium positions. The potential energy of the interaction between atoms, $U (r)$ , is not symmetric: it is much harder to push atoms closer together than to pull them apart. As the oscillation energy increases, the atom climbs further up the softer side of the “well” and the mean distance $⟨ r ⟩$ grows — this is anharmonicity (see the $U (r)$ graph in the laboratory). In a harmonic (parabolic) potential there would be no thermal expansion. That is why $α$ also depends on temperature: at very low temperatures it tends to zero, like the heat capacity (Grüneisen's law), and above room temperature it grows slowly. Anharmonicity also makes phonons scatter off one another — which is why the thermal conductivity of crystals is limited.

Rails, bridges, pipes

If expansion is prevented, a large thermal stress appears in the body: $σ = E α Δ T$ . For steel $E α \approx 2.5$ MPa/K: a rise of 40 K gives 100 MPa, and in an R65 rail a compressive force of more than 800 kN. That is why a gap is left in jointed track, expansion joints (comb-like steel plates) are placed at the ends of bridges, U-shaped expansion loops are fitted to hot-water pipes, and electric wires sag more in summer. In gapless (welded) track the rails are fixed firmly to the sleepers and carry the stress themselves. If it overheats, the track buckles sideways. Model: the buckling limit is taken as about 125 MPa (≈ 50 K); in jointed track the bolt holes open the gap by at most 21 mm, after which the rail is pulled tight.

Bimetallic strip and thermostat

Strips of two different metals (for example brass and invar) are joined along their whole surface. When heated, the brass lengthens more and the strip bends towards the metal that expands less. In an iron the end of the strip lifts and breaks the contact: at the set temperature the circuit opens and the heater switches off; when it cools, the contact closes again. The knob raises and lowers the support screw — this is how the switch-off temperature is chosen. The strip heats up slightly later than the soleplate, so the temperature oscillates around the set value in a “sawtooth” pattern. Electric kettles and heaters have the same kind of thermostat. Model: the soleplate is uniform, Timoshenko's formula is used for the strip, and time is speeded up 5 times.

Volume expansion and the thermometer

If the edge of a cube lengthens by a factor $(1 + α Δ T)$ , its volume increases by a factor ${(1 + α Δ T)}^{3} \approx 1 + 3 α Δ T$ : β ≈ 3α. Liquids expand 10–100 times more than solids: for alcohol $β \approx 1.1 \cdot 10^{- 3}$ 1/K, for mercury $1.8 \cdot 10^{- 4}$ 1/K. The liquid thermometer is based on this: the liquid in the bulb expands and rises into a thin capillary; the thinner the capillary, the higher the sensitivity. The glass expands together with the liquid, so the level shows only the apparent expansion — $β - β_{s h}$ . The flask warms up before the liquid does: when heated sharply, the level may even drop slightly at first.

Lab work

Linear expansion. A 500 mm rod is heated in a tube furnace, and its end pushes a 0.01 mm dial indicator through a quartz rod. Quartz hardly expands at all ( $α \approx 0.55 \cdot 10^{- 6}$ 1/K), but in a precise measurement it has to be included in the result. For each interval calculate $α = Δ (Δ L) / (L_{0} Δ T)$ : the value at 200–300 °C turns out larger than at 20–100 °C. Volume expansion. A 50 mL flask is filled with liquid and a capillary of internal diameter 2 mm ( $S = 3.14$ mm²) is attached to it. Change the temperature of the thermostat, and once the level has settled write down $T$ and $h$ , then calculate $β = S Δ h / (V_{0} Δ T) + β_{s h}$ ; for laboratory glass $β_{s h} = 9.9 \cdot 10^{- 6}$ 1/K. The $β$ of water depends strongly on temperature: $2.1 \cdot 10^{- 4}$ at 20 °C and $4.6 \cdot 10^{- 4}$ 1/K at 50 °C.

Anomalous expansion of water

When water is heated from 0 °C to 4 °C it contracts, and it is densest at 3.98 °C (999.97 kg/m³). In autumn the lake surface cools and the water, now heavier, sinks — the whole lake mixes down to 4 °C. After that, the cooling water becomes lighter and stays at the surface, freezing at 0 °C. Ice is lighter than water (917 kg/m³): it floats at the surface and acts as a “lid” that shields the water below from the cold air and wind — the wind can no longer stir the water, heat escapes only through the ice, and this slows down as the ice thickens. Ice itself conducts heat about 4 times better than water (≈ 2.2 and 0.6 W/(m·K)); but snow on the ice is a real heat insulator. At the bottom the water stays at about +4 °C all winter — the fish live. The reason is hydrogen bonds: in ice the molecules form an open, porous structure. Model: 40 layers of a lake 4 m deep, wind mixing and convection; the snow layer and heat from the bottom are not taken into account, 1 s = 1 day.

Did you know?

The invar alloy (about 64 % iron and 36 % nickel) expands about 10 times less than steel at ordinary temperatures. For discovering it Charles Édouard Guillaume received the Nobel Prize in Physics in 1920.
When water freezes its volume increases by about 9 % — that is why pipes and glass vessels with water left in them burst in winter.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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