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Temperature, Heat & Thermal Expansion

Source lecture(s): SC133 Lec 26

Intuition

Touch a metal bench and a wooden one on a cold morning: the metal feels colder, yet both sit at the same temperature. Your hand measures heat flow, not temperature. Untangling the two is the start of thermodynamics: temperature is a state — microscopically, the average kinetic energy of molecular jiggling — while heat is energy in transit from hot to cold. Nothing "contains heat"; things contain internal energy and exchange heat.

Temperature and its scales

Two bodies in contact stop exchanging net energy when they reach the same temperature — thermal equilibrium. (The zeroth law: if A ≡ C and B ≡ C, then A ≡ B — which is what makes thermometers meaningful.)

\[T_K = T_C + 273.15 \qquad T_F = \tfrac95 T_C + 32\]

Physics happens in kelvin: only there is temperature proportional to molecular kinetic energy, with a true zero (no jiggling left to remove). Room temperature ≈ 293 K.

Heat, heat capacity, and phase changes

Heat \(Q\) (joules) raises temperature according to the specific heat \(c\):

\[Q = mc\,\Delta T\]

Water's \(c = 4186\,\text{J/(kg·K)}\) is enormous — why coastal climates are mild, why water cools engines, and why a pie's filling burns when its crust doesn't.

During a phase change, heat flows with no temperature change — it pays for molecular rearrangement instead (latent heat \(Q = mL\)): for water, \(L_f = 334\,\text{kJ/kg}\) (melting), \(L_v = 2256\,\text{kJ/kg}\) (boiling — huge, which is why steam burns worse than boiling water and why sweating works).

Heat travels three ways: conduction (molecular hand-offs — metals excel: that cold bench), convection (bulk fluid motion — see fluid instabilities), and radiation (light — the only way through vacuum, \(P \propto T^4\)).

Thermal expansion

Almost everything grows when heated — the anharmonic jiggle pushes neighbors apart:

\[\Delta L = \alpha L\,\Delta T \qquad\qquad \Delta V = \beta V\,\Delta T,\ \ \beta \approx 3\alpha\]

Steel: \(\alpha \approx 12\times10^{-6}\,\text{K}^{-1}\) — a 1 km bridge breathes ~0.4 m across seasons, hence expansion joints. Famous exception: water below 4 °C expands on cooling, so lakes freeze from the top and fish survive winter.

Worked example: icing a drink

How much 0 °C ice melts to cool 300 g of water from 25 °C to 0 °C?

Heat available: \(Q = mc\Delta T = 0.3\times4186\times25 \approx 31.4\,\text{kJ}\). Ice melted: \(m = Q/L_f = 31\,400/334\,000 \approx 94\,\text{g}\) — about four cubes, and the melting (not the ice's coldness) does almost all the work.

Common mistakes

  • "This object has a lot of heat." Objects have internal energy; heat is the transfer. (Like "rain" vs "water in the lake".)
  • Feeling = temperature. Touch senses heat flux — conductivity × temperature difference — hence cold metal, "hot" metal in the same sun-baked car.
  • Forgetting latent heat plateaus when tracking \(Q\) through a phase change: the temperature graph flatlines while \(Q\) keeps flowing.
  • Expansion holes shrink? No — a heated plate expands photographically: holes grow too (why heating a stuck jar lid works).

Knowledge graph position

Prerequisites: Conservation of energy. Leads to: First law, Ideal gas, Second law.

Quiz

Q1 (computational). A 2 kW kettle heats 1 L of water from 20 °C to 100 °C. Minimum time?

Answer

\(Q = mc\Delta T = 1\times4186\times80 \approx 335\,\text{kJ}\); \(t = Q/P = 335\,000/2000 \approx 167\,\text{s}\) — nearly 3 minutes, all thanks to water's mighty \(c\).

Q2 (conceptual). Why do bridges have toothed expansion joints while railway tracks once buckled in heat waves?

Answer

Constrained expansion becomes stress: \(\sigma = E\alpha\Delta T\) (elasticity) — tens of MPa for steel across a hot day. Joints give the length somewhere to go; continuously welded rail instead relies on strong anchoring and pre-tensioning.

Q3 (multiple choice). Equal masses of water (c = 4186) and iron (c = 450) receive equal heat. Which ends hotter? (a) water (b) iron (c) equal

Answer

(b). \(\Delta T = Q/mc\) — iron's small specific heat means a ~9× larger temperature rise.