Big picture
A metal spoon warms in tea, a railway rail expands in summer and melting ice absorbs energy without an immediate temperature rise. These are different thermal properties of matter. Heat transfer concerns energy moving because of temperature difference; temperature itself describes thermal state rather than a quantity of stored heat.
Separate three questions: does temperature change, does phase change, or does energy flow between locations? A single situation can involve all three. Clear stage-by-stage reasoning prevents errors such as using a specific-heat formula during melting or treating Celsius ratios as absolute-temperature ratios in radiation.
Learning goals
Convert temperature scales and distinguish heat from temperature.
Calculate linear, area and volume expansion.
Explain water's anomalous expansion.
Solve calorimetry and phase-change problems.
Compare conduction, convection and radiation quantitatively where appropriate.
Core ideas
1. Temperature, equilibrium and thermometers
Two bodies in thermal equilibrium have the same temperature and exchange no net heat when placed in thermal contact. Thermometers rely on a reproducible temperature-dependent property, such as liquid volume, electrical resistance or gas pressure. The thermometer must reach equilibrium with the body without significantly changing its temperature.
Celsius and Fahrenheit scales relate by F = 9C/5 + 32. Absolute temperature is T = C + 273.15. For 35°C, F = 63 + 32 = 95°F and T = 308.15 K. A change of 1°C equals a change of 1 K, but an absolute value of 0°C is not zero kelvin. Temperature differences therefore need no 273.15 offset.
2. Heat and heat capacity
Heat is energy transferred because of temperature difference, measured in joules. Heat capacity, C = Q/ΔT, refers to a particular body. Specific heat capacity, c = Q/(mΔT), is heat capacity per unit mass, in J kg⁻¹ K⁻¹. Molar heat capacity is per mole, in J mol⁻¹ K⁻¹.
For a 0.40 kg metal block with c = 500 J kg⁻¹ K⁻¹ warmed from 20°C to 70°C, Q = mcΔT = 0.40 × 500 × 50 = 10,000 J. Its heat capacity is mc = 200 J K⁻¹. The same material's specific heat does not double when its mass doubles, although the body's heat capacity does. Use Q = mcΔT when c is approximately constant over the interval and no phase transition occurs.
3. Thermal expansion
For small temperature changes, linear expansion is ΔL = αL₀ΔT, where α has unit K⁻¹. Area expansion is ΔA = βA₀ΔT and volume expansion ΔV = γV₀ΔT. For an isotropic solid with small expansion, β ≈ 2α and γ ≈ 3α; these approximations need not apply directly to anisotropic crystals.
A 2.0 m rod with α = 1.2 × 10⁻⁵ K⁻¹ heated by 80 K expands by 1.2 × 10⁻⁵ × 2 × 80 = 0.00192 m = 1.92 mm. A hole in a uniformly heated plate expands as if it were filled with the same material; it does not shrink because surrounding metal expands inward. Rails and bridges use expansion arrangements to avoid excessive stress. A bimetallic strip bends because bonded metals would expand by different amounts if free.
4. Liquids, apparent expansion and water
A liquid is measured inside an expanding container, so observed or apparent expansion differs from real liquid expansion. For small changes, γapparent ≈ γliquid − γcontainer. If γliquid = 9.0 × 10⁻⁴ K⁻¹ and γcontainer = 3.0 × 10⁻⁵ K⁻¹, γapparent = 8.7 × 10⁻⁴ K⁻¹. Expansion coefficients can vary with temperature, so a constant-coefficient approximation should be stated.
Water contracts when warmed from 0°C to about 4°C, then expands above that region. Its density is greatest near 4°C. This anomalous expansion helps explain why lakes can freeze from the top while water below remains liquid. Cooling surface water sinks until it approaches maximum density; still colder water can remain above and freeze. Floating ice also insulates the water beneath. Do not generalise the usual expansion rule across the entire 0–4°C interval.
5. Calorimetry and thermal balance
In an isolated ideal calorimetry system, total heat lost by hotter components equals total heat gained by cooler components, with the calorimeter included when necessary. The final equilibrium temperature must lie between initial temperatures if no phase change or other energy source is involved.
Mix 0.20 kg water at 80°C with 0.30 kg water at 20°C in an insulated negligible-capacity vessel. Since both have the same c, 0.20(80 − T) = 0.30(T − 20). Thus 16 − 0.20T = 0.30T − 6; 22 = 0.50T, giving T = 44°C. A weighted average is valid here because specific heats match and there is no phase transition. It is not a universal shortcut for mixtures of different substances.
6. Phase changes and latent heat
During a first-order phase transition at fixed pressure, energy may change phase while temperature remains approximately constant. Specific latent heat, L, satisfies Q = mL, in J kg⁻¹. Fusion changes solid to liquid; vaporisation changes liquid to gas. A heating curve should show sloping single-phase regions and plateaus during ideal melting or boiling.
To melt 0.050 kg ice initially at 0°C and warm the resulting water to 20°C, use two stages. With Lf = 3.34 × 10⁵ J kg⁻¹, melting requires 0.050 × 334,000 = 16,700 J. With cw = 4,200 J kg⁻¹ K⁻¹, warming needs 0.050 × 4,200 × 20 = 4,200 J. Total is 20,900 J. Ice initially below 0°C would require an additional warming stage before melting.
Evaporation occurs from a liquid surface below the boiling point, with faster molecules escaping and cooling the remainder. Boiling occurs throughout the liquid when vapour pressure equals external pressure. Lower external pressure lowers boiling temperature, which helps explain cooking difficulties at high altitude.
7. Conduction
Conduction transfers energy through matter without bulk material transport. In a uniform slab in steady state, H = Q/t = kA(T₁ − T₂)/L, where thermal conductivity k has unit W m⁻¹ K⁻¹. The temperature gradient drives flow from hotter toward cooler regions.
For k = 0.80 W m⁻¹ K⁻¹, area 0.50 m², thickness 0.020 m and temperature difference 15 K, H = 0.80 × 0.50 × 15/0.020 = 300 W. Thermal resistance is Rth = L/(kA), with H = ΔT/Rth. Resistances add for slabs in series under suitable steady one-dimensional conditions. For parallel paths, heat rates add. A thick insulating layer reduces heat transfer by increasing resistance; it does not create heat.
8. Convection
Convection transfers energy through bulk fluid motion. Natural convection arises because heating changes density and buoyancy; forced convection uses fans or pumps. In a pot heated from below, warmer less-dense liquid rises while cooler liquid moves downward, producing circulation.
Convection requires mobile material, so it does not occur through a rigid solid in the same way. Land and sea breezes involve differential heating and resulting air circulation, although the atmosphere's complete behaviour also involves pressure gradients and other effects. Do not say that heat itself rises: buoyant warm fluid often rises, while heat can conduct downward or radiate in any direction.
9. Radiation and cooling
Thermal radiation transfers energy by electromagnetic waves and can occur through a vacuum. A blackbody is an ideal perfect absorber and emitter. For a real surface, Stefan–Boltzmann law gives emitted power P = εσAT⁴, where emissivity ε lies between 0 and 1. Net exchange with large surroundings is εσA(T⁴ − Ts⁴) under the usual idealised conditions.
If absolute temperature doubles with area and emissivity unchanged, emitted power becomes 16 times larger. Wien's displacement law is λmaxT = b; heating shifts the peak toward shorter wavelengths. With b = 2.90 × 10⁻³ m K and T = 2,900 K, λmax = 1.00 × 10⁻⁶ m, or 1.00 µm.
For modest temperature differences under suitable conditions, Newton's law of cooling approximates cooling rate as proportional to excess temperature. If one object has excess 20 K and an otherwise identical object excess 10 K in the same surroundings, its initial cooling-rate magnitude is approximately twice as large. This is not a universal replacement for radiation and convection physics at every temperature.
10. Experimental interpretation
To estimate specific heat by mixing, measure masses and initial temperatures, minimise heat losses and include the vessel's heat capacity. Stir gently to establish a uniform final temperature. A measured value can be biased if the hot sample cools during transfer.
A cooling experiment records temperature against time at regular intervals. Compare slopes at similar excess temperatures rather than assuming a constant temperature drop per minute. For expansion demonstrations, use clear fixed reference marks and allow uniform heating; unequal heating can cause bending rather than simple uniform length change.
A reliable approach to questions
Decide whether energy changes temperature, phase or both.
Divide multi-stage processes at every phase transition.
Apply energy conservation and include all participating bodies.
Use kelvin for absolute-temperature radiation laws.
Check units, signs and whether the final temperature is physically possible.
Connections that matter
Kinetic theory explains temperature and microscopic energy transfer.
Thermodynamics develops heat, work and internal-energy accounting.
Elasticity explains stresses when expansion is constrained.
Common mistakes
Calling heat stored in a body ignores that heat describes transfer.
Adding 273.15 to a temperature difference corrupts calorimetry.
Using mcΔT during a melting plateau omits latent heat.
Using Celsius in T⁴ radiation ratios gives incorrect scaling.
Revision checklist
I can convert scales and distinguish thermal quantities.
I can calculate expansion and explain water's anomaly.
I can solve mixing and multi-stage phase-change problems.
I can compare conduction, convection and radiation.
I can interpret cooling curves and their assumptions.