α in millionths per °C (OpenStax table). Drawings exaggerate the size change; the readout and zoom box give true sizes.
Heating is sped up: a real bridge takes hours to warm.
Thermal expansion is how most materials get bigger when they warm up and smaller when they cool. Heated particles move faster and push farther apart. The change in length is ΔL = αL₀ΔT. For steel α is about 12 × 10−6 per °C, so a 100 m steel bridge grows about 4 cm when it warms by 35 °C.
| Material | α (× 10−6 per °C) | 100 m warmed by 10 °C grows |
|---|---|---|
| Aluminum | 25 | 25 mm |
| Brass | 19 | 19 mm |
| Copper | 17 | 17 mm |
| Steel (iron) | 12 | 12 mm |
| Concrete (about) | 12 | 12 mm |
| Glass (ordinary) | 9 | 9 mm |
| Glass (Pyrex) | 3 | 3 mm |
| Invar (nickel-iron) | 0.9 | 0.9 mm |
Linear expansion. ΔL = αL₀ΔT, or L = L₀(1 + αΔT). The unit of α is 1/°C, which is the same as 1/K (K−1), because a change of 1 °C equals a change of 1 K. Worked example: a 100 m steel rail warmed by 50 °C grows 12 × 10−6 × 100 × 50 = 0.06 m = 6 cm. Values of α change a little with temperature and from table to table (aluminum is listed as 23 to 25 × 10−6 per °C); the simulator uses the OpenStax values.
Area and volume. A flat sheet grows in area by ΔA ≈ 2αA₀ΔT and a solid grows in volume by ΔV = βV₀ΔT with β ≈ 3α, because (1 + αΔT)² ≈ 1 + 2αΔT and (1 + αΔT)³ ≈ 1 + 3αΔT when αΔT is tiny. Liquids only have β: mercury 180, water 210 × 10−6 per °C (at 20 °C).
Thermal stress. If the ends are held, the blocked strain αΔT becomes a stress σ = EαΔT (E = Young’s modulus). Steel (E = 200 GPa) warmed by 100 °C: σ = 200 × 109 × 12 × 10−6 × 100 = 240 MPa, close to the yield strength of mild steel (about 250 MPa). Concrete has a smaller E (about 20 GPa), so the same warming gives one tenth of the stress. Heating a clamped bar squeezes it (compression); cooling pulls it (tension).
Bridge lab model. The joint is set at 15 °C with a 45 mm gap; deck cross-sections (0.2 m² steel or aluminum, 2 m² concrete) and the 1,000-tonne buckling limit are illustrative, chosen so the numbers are realistic in size. The bimetal strip uses Timoshenko’s formula for two 0.5 mm layers (brass on steel); its tip moves about 0.05 mm per °C.
Water’s anomaly. Water is densest at about 4 °C (999.97 kg/m³). Cooling below 4 °C, hydrogen bonds start to hold the molecules in a more open pattern, so the water expands; ice (917 kg/m³) has an open hexagonal crystal. The thermometer lab uses a real density table; it leaves out the small expansion of the glass bulb.
Standards. NGSS HS-PS3-2 (energy at the macroscopic scale as motion of particles) and HS-PS2-6 (why the molecular structure of materials matters in designed materials, e.g. invar in clocks and measuring tools).
α and E come from OpenStax tables (E for concrete: 20 GPa). Try invar (α 0.9): it hardly grows, which is why it is used in measuring tools.
Pick one answer for each question, then press Check my answers to see the explanations. New questions picks another set.
A 45 minute lesson. Start in Bridge on a projector (Full screen button): ask how much a 100 m bridge grows on a hot day, then press Heat wave. Switch the joint off and heat again to show why gaps matter. Pairs then work through the six missions; each asks for a prediction first, and a certificate prints when all six are done.
The expansion coefficients are textbook values (OpenStax Table 1.2) and the water densities come from a standard table. The joint gap, the deck cross-sections, the buckling limit and the ball-and-ring clearance are illustrative (realistic in size). Drawings exaggerate the size change (the scale is written on the picture); the readout and the zoom box show true sizes. The particle lens is a picture of the idea, not to scale.
For middle school (NGSS MS-PS1-4, MS-PS3-4) focus on the particle lens and the gaps. Go deeper has ΔL = αL₀ΔT, area and volume expansion, thermal stress σ = EαΔT and water’s anomaly for high school (HS-PS3-2, HS-PS2-6). Continue with heat transfer or states of matter.
Standards: NGSS MS-PS1-4, MS-PS3-4 · high school extension (Go deeper): HS-PS3-2, HS-PS2-6
| Time | Step | What students do | Go to |
|---|---|---|---|
| 5 min | Hook | Predict how much a 100 m bridge grows; press Heat wave | Simulator |
| 10 min | Explore | Missions 1–3: bridge grows, no joint, aluminum | Simulator |
| 10 min | Apply | Missions 4–6: ring hole, flip the strip, water thermometer | Simulator |
| 5 min | Explain | Order game: why a heated bar grows (particle story) | Order game |
| 10 min | Check | Quiz (10 questions) | Quiz |
| 5 min | Exit ticket | Worksheet question 10: explain with particles | Worksheet |
Worksheet: a print-out for students and a separate answer key page for the teacher (opens the print dialog; choose "Save as PDF" to keep a file).
Thermal expansion is the increase in size of a material when its temperature rises. The particles move faster and push slightly farther apart, so solids get longer, liquids take up more volume and gases spread out. Cooling does the opposite (thermal contraction).
Key takeaway: hotter particles move more and need more room.A bridge deck gets longer on hot days and shorter on cold nights: a 100 m steel deck changes by about 1.2 mm for every °C. Expansion joints (often interlocking metal fingers) leave room for that movement. Without them, the deck would push on its supports with huge forces and could buckle or crack.
Key takeaway: joints give the deck room to grow and shrink.Older tracks leave small gaps between rails so each rail can expand in summer (that is the “clack-clack” sound). Modern welded rail has no gaps; it is laid and anchored so it can take the stress, but in extreme heat it can still bend sideways into a “sun kink”.
Key takeaway: rails need room to grow, or they are built to hold the push.The linear coefficient α is the fraction of length gained per degree, so its unit is per degree: 1/°C, which is the same as 1/K (K−1). Values are tiny, so tables write them as × 10−6 per °C: steel 12, aluminum 25.
Key takeaway: unit = 1/°C = 1/K.Bigger. Every part of the object grows by the same fraction, so the hole is enlarged along with the metal around it, like an enlarged photo. In the ball-and-ring lab, heating the brass ring to 300 °C widens the hole from 25.45 to about 25.59 mm.
Key takeaway: heated holes grow.Two metals with different expansion coefficients (like brass, 19, and steel, 12 × 10−6 per °C) are bonded together. When heated, the brass side gets longer than the steel side, so the strip curls toward the steel. Old thermostats, toasters and dial thermometers use this movement.
Key takeaway: different expansion makes the strip curl.Near freezing, water molecules begin to line up in an open, ice-like pattern held by hydrogen bonds. Below about 4 °C this open pattern wins, so cooling water expands instead of shrinking. That is why ice floats and lakes freeze from the top down, leaving liquid water below for fish.
Key takeaway: water is densest at about 4 °C; ice floats.When boiling water hits cold glass, the inside surface warms and expands before the outside does, and the strain can crack it (thermal shock). Ordinary glass has α of about 9 × 10−6 per °C; Pyrex has about 3, so the same temperature difference strains it only a third as much. For the same reason dentists use filling materials that expand about as much as teeth.
Key takeaway: smaller α means less strain and less cracking.A small bulb of liquid is joined to a very thin glass tube. When the liquid warms it expands, and because the tube is so thin, a tiny change in volume makes a long change in the column. In the simulator a mercury column rises about 0.57 mm per °C.
Key takeaway: a thin tube turns a tiny volume change into a long column change.