Thermal Physics · 21.1
Temperature & Internal Energy
A metal bench and a wooden one sit at the same temperature on a cold morning, yet the metal feels colder. Temperature, heat, and internal energy are three different things — and thermal physics starts by telling them apart.
Build the model
Connect the measurement to the mechanism.
Temperature determines the direction of spontaneous heat transfer and whether systems are in thermal equilibrium. In the classical ideal-gas model it is proportional to mean translational kinetic energy per molecule. Heat is energy transferred because of a temperature difference.
Internal energy includes microscopic kinetic and potential energies. The thermodynamic temperature scale is not defined by the response of any particular thermometer material.
- Simple definition
- Temperature describes thermal state; internal energy is the total microscopic kinetic plus potential energy; heat is energy transferred between systems because their temperatures differ.
- Example
- A sparkler spark at 1000 °C carries far less energy than a bathtub of warm water — the spark has a high temperature but only a few molecules.
One kelvin and one Celsius degree are equal temperature intervals. Zero kelvin corresponds exactly to −273.15 °C. The offset cancels in a temperature difference: a rise of 1 °C is a rise of 1 K.
Exact offset; 273 is a rounded approximation. No degree symbol on kelvin.
Heating a substance can speed its molecules up (more KE) or pull them further apart against their bonds (more PE) — internal energy counts both.
Random molecular KE + intermolecular PE
Touching objects trade energy until their temperatures equalise; after that the net flow is zero. This is what a thermometer relies on.
Net flow is always hot → cold
Solids, liquids, gases
In a solid the molecules only vibrate about fixed positions. In a liquid some bonds are broken and molecules move about within a fixed volume. In a gas most bonds are broken and the molecules fly freely. Melting and boiling are bond-breaking steps between these arrangements.
An ideal-gas extrapolation, not a real-gas cooling experiment
For a fixed amount and volume of ideal gas, p is proportional to thermodynamic temperature T. Extrapolating this model gives p = 0 at 0 K, exactly −273.15 °C. Real gases deviate and may condense before that limit, so this is not a claim that every actual gas follows a straight line all the way to zero pressure. Thermodynamic temperature is independent of the substance used to measure it.
What a thermometer reads
A thermometer reaches thermal equilibrium with the object. It measures a calibrated property, such as liquid-column length, resistance or thermocouple e.m.f., to infer the temperature. It does not directly measure all the object's internal energy; energy can change during a phase transition without a temperature change.
Heating makes things bigger
Almost everything expands when heated. Give the particles more energy and they vibrate or fly about more vigorously, so on average they sit a little further apart and the same mass takes up more volume. How much further depends on what holds them: a gas expands most, because nothing but the container restrains it; a liquid expands far less, its particles still touching; a solid least of all, every atom locked into the lattice and able only to vibrate more widely about the same site. Gases most, liquids next, solids least — and the reason is the strength of the bonds, not the size of the temperature rise.
Designing for expansion
Engineers leave room for it rather than fight it. Railway lines are laid with gaps between sections, a bridge deck ends in a toothed expansion joint and sits on rollers, and overhead cables are strung with a deliberate sag that takes up the slack in winter. A bimetallic strip puts the effect to work: two metals that expand by different amounts are bonded together, so heating bends the strip towards the metal that expands less, and that bend can open or close a circuit in a thermostat or a fire alarm.
Change one variable at a time
Make the relationship visible.
Steel c = 450, aluminium c = 900 J kg⁻¹ K⁻¹; the aluminium block is fixed at 1.0 kg. Heat flows hot → cold until both blocks share one temperature.
Equilibrium temperature55 °C
Heat transferred31.5 kJ
Directionsteel → aluminium
Catch the common trap
Explain before calculating.
What is 27 °C on the kelvin scale, to the nearest kelvin?
Choose an answer to test the model.
Worked examples
State the rule, substitute, then check units.
EasyConvert 25 °C to kelvin, and 100 K to Celsius. Round each answer to the nearest whole unit.
- T/K = 25 + 273.15 = 298.15, which rounds to 298 K.
- θ/°C = 100 − 273.15 = −173.15, which rounds to −173 °C.
Answer298 K; −173 °C
Medium100 g of water at 90 °C is mixed with 200 g of water at 30 °C in an insulated cup. Find the equilibrium temperature.
- Same substance, so the specific heat cancels: heat lost = heat gained.
- 0.100 × (90 − T) = 0.200 × (T − 30).
- 9 − 0.1T = 0.2T − 6 → 0.3T = 15 → T = 50 °C.
Answer50 °C
HardThe average molecular kinetic energy of a gas is proportional to its kelvin temperature. At what Celsius temperature is the average KE double its value at 27 °C?
- 27 °C corresponds to 300.15 K. In this ideal-gas model mean translational KE is proportional to T.
- Double the mean translational KE needs 600.30 K.
- 600.30 − 273.15 = 327.15 °C, about 327 °C — not 54 °C. Doubling must be done in kelvin.
Answer327 °C (600 K)
ChallengingA 2.0 kg steel block (c = 450 J kg⁻¹ K⁻¹) at 100 °C is clamped to a 1.0 kg aluminium block (c = 900 J kg⁻¹ K⁻¹) at 10 °C, insulated from everything else. Find the equilibrium temperature and the heat that flows.
- Heat lost by steel = heat gained by aluminium: 2.0 × 450 × (100 − T) = 1.0 × 900 × (T − 10).
- 900(100 − T) = 900(T − 10) → 100 − T = T − 10 → T = 55 °C.
- Q = 900 × (100 − 55) = 40 500 J ≈ 41 kJ flows from steel to aluminium — and the same 40 500 J arrives, confirming the bookkeeping.
Answer55 °C; about 41 kJ transferred