🌡️ Thermal Expansion – Handy Notes

Heat makes most things grow in size and cooling makes them shrink. You’ve seen it: a tight metal lid loosens after a dip in hot water, or a 🎈 balloon swells in the sun. This size-change is called thermal expansion :contentReference[oaicite:0]{index=0}.

1. Three Ways Things Expand 🤔

  • 📏 Linear expansion (only length changes)
  • 📐 Area expansion (length & width change)
  • 📦 Volume expansion (all three dimensions change)

Key Formulas

For a small temperature rise ΔT:

  • Linear: \[ \frac{\Delta l}{l}= \alpha_l \,\Delta T \tag{10.4} \] :contentReference[oaicite:1]{index=1}
  • Area: \[ \frac{\Delta A}{A}= \alpha_A \,\Delta T \] (where \(\alpha_A \approx 2\alpha_l\) for a rectangle) :contentReference[oaicite:2]{index=2}
  • Volume: \[ \frac{\Delta V}{V}= \alpha_v \,\Delta T \tag{10.5} \] :contentReference[oaicite:3]{index=3}

Whenever a solid cube expands the same in every direction, the two coefficients lock together:

\[ \alpha_v = 3\alpha_l \tag{10.9} \]

That “3” pops up because a cube has three equal sides 📦 :contentReference[oaicite:4]{index=4}.

2. Typical Expansion Numbers 🔢

The coefficients are tiny, but metals still beat glass by a mile:

  • Linear αl (10–5 K–1): aluminium ≈ 2.5, copper ≈ 1.7, glass ≈ 0.32, lead ≈ 0.29 :contentReference[oaicite:5]{index=5}.
  • Volume αv (K–1): aluminium ≈ 7 × 10–5, mercury ≈ 18.2 × 10–5, ethanol ≈ 110 × 10–5 😲 :contentReference[oaicite:6]{index=6}.

👉 Metals expand plenty, special alloys such as invar hardly move (αv ≈ 2 × 10–6 K–1), so they’re used in precision instruments.

3. Water’s Weird Twist 💧

Between 0 °C and 4 °C, water shrinks when heated and expands when cooled – the reverse of “normal” behaviour. It is densest at 4 °C, letting lakes freeze from the top and protecting aquatic life 🐟 :contentReference[oaicite:7]{index=7}.

4. Gases – Expansion Champions 🎈

For an ideal gas at constant pressure,

\[ \alpha_v=\frac{1}{T} \quad (\text{in kelvin}) \tag{10.6} \]

At 0 °C the value shoots up to 3.7 × 10–3 K–1, orders of magnitude larger than for solids or liquids :contentReference[oaicite:8]{index=8}.

5. Holding a Solid Fixed – Thermal Stress 🏗️

If a rod is clamped at both ends and heated, it can’t stretch, so it builds compressive stress equal to

\[ \text{Stress}=Y\alpha_l\Delta T \]

For steel (Y ≈ 2 × 1011 N m–2, αl ≈ 1.2 × 10–5 K–1), just a 10 °C rise generates a whopping 2.4 × 107 N m–2! :contentReference[oaicite:9]{index=9}

6. Everyday Examples 🛠️

  • Iron tyres on wooden cart wheels are heated so they loosen, fitted quickly, then shrink tight as they cool (needs ~218 °C for a 12 mm stretch) :contentReference[oaicite:10]{index=10}.
  • Bridges and rail tracks use tiny gaps or expansion joints so they can “wiggle” instead of warp.

7. Quick Temperature Scale Link 🔗

Celsius and Kelvin sizes are the same, just shifted by 273.15:

\[ T = t_C + 273.15 \tag{10.3} \]

Absolute zero sits at −273.15 °C, where an ideal-gas pressure extrapolates to zero 🌌 :contentReference[oaicite:11]{index=11}.

🎯 High-Yield NEET Nuggets

  1. Relation αv = 3 αl for uniform solids.
  2. Water’s anomalous expansion and its ecological impact.
  3. Ideal-gas expansion coefficient αv = 1/T.
  4. Thermal stress formula Stress = Y αl ΔT.
  5. Why expansion joints are essential in rails & bridges.

Keep these points in mind, practice the formulas, and you’ll ace the questions! 🚀