Thermal Expansion

Formulas

Linear expansion (1D):

ΔL = α · L₀ · ΔT

where:

• ΔL = change in length (m).
• α = linear expansion coefficient (1/K or 1/°C).
• L₀ = original length.
• ΔT = temperature change.

Volumetric expansion (3D):

ΔV = β · V₀ · ΔT

where β ≈ 3α for isotropic solids (cube of α).

Common expansion coefficients

Material	α (10⁻⁶ /K)	Notes
Quartz (fused)	0.5	Used in precision optics, telescopes
Invar (alloy)	1.2	"Invariable" — used in clocks, calipers
Pyrex (borosilicate)	3	Lab/oven glass
Diamond	1.0	—
Tungsten	4.5	Lamp filaments
Concrete	10	Construction
Steel	12	Common alloy
Iron	12	—
Glass (window)	9	—
Copper	17	—
Brass	19	Lamps, hardware
Aluminum	23	Aerospace, kitchen
Lead	29	—
Polyethylene	200	Plastic; very high
Liquid water (20°C)	207	β value
Air (gas at 0°C)	3,667	= 1/T_K (for ideal gas)

Real-world consequences

System	Implication
Bridge expansion joints	Allow ~10 cm movement per 100 m bridge over seasonal range
Railroad sun kinks	Tracks buckle in extreme heat; managed by stress and gaps
Bimetal thermostats	Two metals bend → close/open electrical contacts
Mercury thermometers	Mercury column height ∝ T (β ≈ 180 × 10⁻⁶/K)
Pyrex vs regular glassware	Pyrex resists thermal shock 3× better
Aircraft fuselages	Aluminum panels expand in flight; rivets allow movement
Engine pistons	Tighter fit when cold, looser when hot — designed for hot operation
Dental fillings	Material α should match tooth (~7 × 10⁻⁶) to prevent cracking
Concrete sidewalks	Expansion gaps every 4-5 m to prevent cracking

JavaScript — thermal expansion calculations

// Linear expansion
function linearExpansion(L0, alpha, deltaT) {
  return L0 * alpha * deltaT;
}

// Volume expansion (β ≈ 3α for solid)
function volumeExpansion(V0, alpha, deltaT) {
  const beta = 3 * alpha;
  return V0 * beta * deltaT;
}

// Steel bridge: 100 m, 25°C swing
console.log(`100m steel bridge, 25°C: ΔL = ${linearExpansion(100, 12e-6, 25) * 1000} mm`);
// 30 mm = 3 cm

// 100 m railway, 50°C summer-to-winter
console.log(`Railway 50°C: ΔL = ${linearExpansion(100, 12e-6, 50) * 1000} mm`);
// 60 mm = 6 cm — must accommodate or buckle

// Mercury thermometer: how much rise for 1°C?
function thermometerRise(V_bulb, alpha_mercury, alpha_glass, deltaT, capillary_area) {
  // Net expansion = (β_Hg - β_glass) · V · ΔT, divided by capillary area
  const beta_Hg = 180e-6;  // /K
  const beta_glass = 25e-6;
  const net_volume = V_bulb * (beta_Hg - beta_glass) * deltaT;
  return net_volume / capillary_area;
}

// V_bulb = 0.5 cm³, capillary = 0.2 mm² → 0.000002 m²
console.log(`Mercury thermometer ΔL: ${(thermometerRise(0.5e-6, 0, 0, 1, 2e-7) * 1000).toFixed(1)} mm/°C`);

// Bimetal strip: how much does it bend?
function bimetalBend(L, alpha_top, alpha_bottom, t, deltaT) {
  // Curvature 1/r = (α_top - α_bottom) · ΔT / t
  // Tip deflection ≈ L²/(2r) = (α_top - α_bottom) · ΔT · L² / (2t)
  return (alpha_top - alpha_bottom) * deltaT * L * L / (2 * t);
}

// Brass + Steel bimetal, 100 mm long, 1 mm thick, 50°C rise
const tipDeflection = bimetalBend(0.1, 19e-6, 12e-6, 0.001, 50);
console.log(`Bimetal tip deflection: ${(tipDeflection * 1000).toFixed(2)} mm`);
// ~1.75 mm — significant for switching contacts

// Stress generated if expansion is constrained
function thermalStress(alpha, deltaT, E) {
  // σ = E · ε = E · α · ΔT (constrained ε = α·ΔT)
  return E * alpha * deltaT;
}

// Steel constrained, 50°C rise (E = 200 GPa)
console.log(`Constrained steel stress: ${(thermalStress(12e-6, 50, 200e9) / 1e6).toFixed(0)} MPa`);
// 120 MPa — significant fraction of yield (~250 MPa for mild steel)

// Volume change of water (anomaly!)
function waterDensity(T_C) {
  // Approximate water density 0-30°C
  // Maximum at 4°C
  if (T_C < 0) return 999.84 + (T_C - 4) * (-0.4);  // rough below 0
  if (T_C > 30) return 999.84 + (T_C - 4) * (-0.5);
  // Quadratic-ish around 4°C
  const dT = T_C - 4;
  return 999.972 - 0.0085 * dT * dT;
}

console.log(`Water at 0°C: ${waterDensity(0).toFixed(3)} kg/m³`);   // ~999.83
console.log(`Water at 4°C: ${waterDensity(4).toFixed(3)} kg/m³`);   // ~999.97 (max)
console.log(`Water at 20°C: ${waterDensity(20).toFixed(3)} kg/m³`); // ~998.20

Where thermal expansion shows up

• Civil engineering. Bridges, roads, railways, buildings — all have expansion joints to accommodate temperature swings.
• Mechanical engineering. Engine clearances designed for operating temperatures (cold engine = tight, hot = looser).
• Electronics. Solder joints subjected to thermal cycling; mismatched α between chip and substrate causes failures.
• Optical instruments. Telescopes, microscopes use low-α materials (Invar, Zerodur) for stability.
• Plumbing. Hot/cold water pipe expansion (especially copper) — flexible joints and movement allowance.
• Manufacturing. Shrink-fit assembly (heat outer part, fit on inner, cool — locks together by α).
• Geology. Rock expansion/contraction with daily/seasonal T cycles drives weathering, crack propagation.

Common mistakes

• Forgetting volume expansion is ~3× linear. β = 3α for isotropic solids. Don't use α for volume problems.
• Confusing α units. Always 1/K (or equivalently 1/°C) for ΔT calculations. Coefficient values often given as 10⁻⁶/K.
• Treating water normally. Water has anomalous expansion below 4°C — actually expands when cooled below this. Important for limnology, ice formation.
• Underestimating expansion in long structures. 100 m × 25°C × 12 ppm = 30 mm. Sounds small but matters for bridges and rails.
• Ignoring constrained-expansion stress. If expansion is prevented, large stresses develop (σ = E·α·ΔT). Can cause buckling, cracking. Why expansion joints exist.
• Forgetting α can vary with T. Coefficients listed in tables are at room T. At very high or low T, materials behave differently.