Thermal Properties of Matter— Notes, Formulas & Revision

Most solids EXPAND when heated. Linear expansion: ΔL = α·L₀·ΔT, where α is the coefficient of linear expansion (K⁻¹).

Typical α values: steel ~12×10⁻⁶/K, aluminum ~23×10⁻⁶/K, copper ~17×10⁻⁶/K, glass ~9×10⁻⁶/K, invar ~1×10⁻⁶/K.

Larger α ⇒ more expansion per unit temperature rise.

Bimetallic strips: two metals with different α bonded together bend on heating — used in thermostats.

Railway tracks have gaps between rails to accommodate expansion in summer heat.

Bridges have expansion joints — accommodate thermal length changes.

Invar (Fe-Ni alloy) has very low α ⇒ used in precision instruments where dimensional stability matters.

Thermal expansion arises from anharmonic potential at atomic level — atoms vibrate more on heating and move slightly apart.

Area expansion: ΔA = β·A₀·ΔT, where β = 2α is the area expansion coefficient.

Holds for isotropic materials (same α in all directions).

Twice the linear coefficient because A ∝ L².

Useful for thin plates, sheet metals, surface areas of objects.

Hole in a plate: increases in area same as solid plate (β·A·ΔT).

Differential expansion can warp thin plates if heated asymmetrically.

Most metals: β ~ 20-50 × 10⁻⁶/K. Glass: ~18 × 10⁻⁶/K.

Bimetallic plates curl into spherical caps when heated uniformly.

Volume expansion: ΔV = γ·V₀·ΔT, where γ = 3α is the volume expansion coefficient (for isotropic solids).

Liquids and gases have their OWN γ — usually much larger than solids.

γ (water) at 20°C: ~2×10⁻⁴/K. γ (mercury): ~1.8×10⁻⁴/K. γ (air at constant P): 1/273 = 3.66×10⁻³/K.

Gases follow ideal gas law: V/T = const at constant P ⇒ γ_gas = 1/T ≈ 3.66×10⁻³/K at 273 K.

Liquid expansion is used in mercury and alcohol thermometers.

Solid container with liquid: APPARENT expansion of liquid = γ_liquid − γ_container (the difference shows up).

Anomalous behavior of water: density MAX at 4°C; expands when COOLING from 4°C to 0°C.

Thermal expansion of pavement and concrete is why expansion joints exist in roads and bridges.

Water has ANOMALOUS thermal expansion: density INCREASES as it COOLS from 4°C down to 0°C (volume increases).

Density MAXIMUM at 4°C: ρ = 1000.0 kg/m³.

Below 4°C: water EXPANDS as it cools further (negative γ).

When water freezes at 0°C: volume EXPANDS by ~9% (ice less dense than water).

Why ice FLOATS on water: ice (ρ ≈ 917 kg/m³) < water (ρ = 1000 kg/m³).

Lakes freeze TOP-DOWN: dense 4°C water sinks; cold (less dense) water and ice float on top.

Result: aquatic life survives under ice — bottom water stays at 4°C through winter.

Origin: hydrogen bonding in water creates open hexagonal structure in ice, denser packing in liquid.

HEAT (Q) and TEMPERATURE (T) are different physical quantities.

TEMPERATURE: average kinetic energy of molecules (a measure, like 'velocity').

HEAT: energy transferred due to temperature difference (a flow, like 'distance').

Heat flows SPONTANEOUSLY from hot to cold body — equilibrating temperatures.

Units: Q in joules (J) or calories (1 cal = 4.184 J). T in Kelvin or Celsius.

Specific heat capacity c: heat needed per kg per K. Q = m·c·ΔT.

Water has very high c (~4186 J/kg·K) — that's why oceans moderate climate.

Heat capacity C = m·c — heat needed to raise the WHOLE body by 1 K.

Body temperature depends on how heat is shared among DEGREES OF FREEDOM (equipartition theorem).

Calorimetry: study of heat transfer between bodies. Used to measure specific heat capacities.

Principle: when two bodies at different T are placed in thermal contact, heat lost by hot = heat gained by cold (no losses).

Heat balance: m₁·c₁·(T_hot − T_f) = m₂·c₂·(T_f − T_cold). Solve for T_f or unknown c.

Calorimeter: container designed to minimize heat losses (insulated double-walled, or vacuum-jacketed).

Water is commonly used in calorimeters because of its high and well-known c.

Latent heat: heat required for phase change at constant T (e.g., ice → water).

Method of mixtures: classic experiment using known reference substance and water.

Bomb calorimeter measures heat of combustion at constant volume.

Conduction: heat transfer through a material WITHOUT bulk motion of matter — vibrating molecules and electrons share energy.

Dominant mechanism in solids, especially metals (free electrons).

Fourier's law: heat current dQ/dt = −kA·(dT/dx), where k = thermal conductivity (W/m·K).

Steady-state through a slab: dQ/dt = kA·ΔT/L, where L is thickness.

Good conductors: metals (Cu: ~400 W/m·K, Ag: ~430, Al: ~240). Poor conductors (insulators): wood, plastic, air.

Thermal resistance R = L/(kA) — analogous to electrical resistance. Series and parallel rules apply.

Free electrons in metals carry heat efficiently (Wiedemann-Franz law: k/σ ∝ T).

Used in: heat sinks, thermos flasks (insulation), thermal grease (improve conduction).

Thermal conductivity k measures how well a material conducts heat. Units: W/m·K.

k_metals: Cu ≈ 400, Ag ≈ 430, Al ≈ 240, Fe ≈ 80, Steel ≈ 50.

k_insulators: glass ≈ 0.8, wood ≈ 0.15, concrete ≈ 1, brick ≈ 0.7.

k_fluids: water ≈ 0.6, air ≈ 0.025.

k_air is very small — that's why fluff insulation works (trapped air pockets).

Wiedemann-Franz law: for metals, k/σ = (π²/3)(k_B/e)²·T — linear relation to electrical conductivity.

Materials with low k are used as insulators (foam, fiberglass, wood, plastic).

Materials with high k are used in heat sinks (copper, aluminum).

Convection: heat transfer by bulk motion of a FLUID (liquid or gas).

Natural convection: driven by density differences (hot fluid rises, cold fluid sinks).

Forced convection: pump or fan moves the fluid.

Doesn't occur in solids (no bulk fluid motion) or vacuum.

Examples: boiling water, atmosphere circulation (winds), ocean currents, room heating with radiator.

Newton's law of cooling: rate of heat loss ∝ (T_body − T_surroundings) — assumes convective cooling.

Convection coefficient h depends on fluid speed, geometry, fluid properties. Typical: natural air h ~ 5 W/m²·K; forced air h ~ 50; water h ~ 500.

Rayleigh number determines whether convection occurs vs pure conduction in a fluid layer.

Radiation: heat transfer by ELECTROMAGNETIC WAVES (mostly infrared at terrestrial T). Doesn't need a medium.

Stefan-Boltzmann law: power radiated per unit area P/A = εσT⁴, with σ = 5.67×10⁻⁸ W/m²·K⁴ and ε = emissivity (0 ≤ ε ≤ 1).

Black body: ε = 1, perfect absorber and emitter. Sun is approximately a black body at ~5800 K.

Net radiative exchange: P_net = εσA(T_body⁴ − T_surroundings⁴).

Wien's law: λ_max·T = 2.898 × 10⁻³ m·K. Peak wavelength of emitted radiation.

Sun's peak: λ_max ≈ 500 nm (visible). Human body (310 K): peak ~ 9.4 μm (infrared).

Radiation is how Earth gets energy from Sun (no medium between). Also how Earth loses heat to space.

Greenhouse effect: atmosphere absorbs outgoing IR, traps heat ⇒ Earth's surface stays warmer.

Newton's law of cooling: rate of cooling of a hot body in air is proportional to its excess temperature over surroundings.

dT/dt = −k·(T − T_s), where T_s = surroundings, k = cooling constant.

Solution: T(t) = T_s + (T₀ − T_s)·e^(−kt) — exponential approach to T_s.

Valid for SMALL temperature differences (in linear regime).

For large ΔT, convective/radiative cooling is more strongly nonlinear (radiation ∝ T⁴).

Cooling constant k depends on size, shape, and surface — larger surfaces cool faster.

Used in: cooling curves of liquids, forensic temperature analysis (post-mortem time of death).

Hot soup cools faster in a wide bowl (more surface area) than in a tall cup.

Phase change (transition) occurs at a FIXED temperature for a pure substance (e.g., water melts at 0°C, boils at 100°C at 1 atm).

Latent heat L: energy absorbed/released per kg during phase change. Q = m·L.

Two main types: LATENT HEAT OF FUSION (L_f) for melting/freezing; LATENT HEAT OF VAPORIZATION (L_v) for boiling/condensing.

Water: L_f = 334 kJ/kg, L_v = 2260 kJ/kg.

During phase change, temperature stays CONSTANT even as heat is added — all energy goes into breaking molecular bonds.

Sublimation: solid → gas (skipping liquid), e.g., dry ice. Deposition: gas → solid.

Pressure changes phase-transition temperatures (P-T phase diagram).

Sweating cools you because evaporation of sweat absorbs L_v from your skin.