Heat and Temperature: The Bath and the Cup

The bath and the cup of tea

Fill a bath with 80 litres of water at 40 °C and pour a 250 ml cup of tea at 40 °C. Every particle statistic that temperature reports is the same in both: the average kinetic energy of the molecules is identical, and a thermometer put into either reads 40. On the temperature question there is nothing to choose between them.

Now ask how much energy it took to get each of them there from a cold tap at 20 °C. The equation is Q = mcΔT — mass, times specific heat capacity, times temperature change. For the bath: 80 × 4,200 × 20 = 6,720,000 J, which is 6.72 megajoules. For the cup: 0.25 × 4,200 × 20 = 21,000 J, or 21 kilojoules.

The bath took 320 times as much energy as the cup, and the two are at the same temperature. That ratio is the entire difference between the two quantities, in one calculation. Temperature says how energetic the particles are on average. Energy content says how many of them there are doing it.

Heat is a transfer, not a stock

This is the part that textbooks often blur and that repays getting right. An object does not contain heat. It contains internal energy — the total kinetic and potential energy of all its particles. Heat is the name for energy moving from one place to another because of a temperature difference, and once it has arrived it is no longer heat. It is just internal energy again.

Heat also has a direction built into its definition. Energy transfers from the hotter body to the colder one, never the reverse, until the two temperatures are equal. There is no such thing as transferring cold. A fridge does not put cold into food; it removes energy and dumps it into the kitchen, which is why the back of a fridge is warm and why leaving the door open heats a room.

Put plainly: temperature is a property of an object, and heat is something that happens between two of them.

Which instrument reads which, and in what units

A thermometer reads temperature, in degrees Celsius or kelvin. The scales are offset by 273 but their divisions are the same size, so a rise of 1 K is a rise of 1 °C — which is why a temperature change can be quoted in either without conversion, while a temperature itself cannot.

Nothing in a school laboratory reads heat directly. Energy transferred has to be calculated — either from Q = mcΔT using a mass, a specific heat capacity and a measured temperature change, or from the electrical input using power multiplied by time, or a joulemeter. It is measured in joules, and joules are a unit of energy, not a unit of hotness.

Specific heat capacity is the bridge between the two quantities: the energy needed to raise one kilogram of a substance by one degree. Water's is unusually high, about 4,186 J per kilogram per degree, which is why the sea moderates coastal climates and why a hot water bottle stays warm for hours.

Two things at the same temperature that feel nothing alike

A sparkler burns at a temperature far above that of boiling water, and children hold them at arm's length without injury. Boiling water at 100 °C causes serious burns. The difference is quantity: each glowing spark has a tiny mass and therefore carries almost no energy, and it cools to nothing in the instant it touches skin. Temperature was never the thing doing the damage.

A second, separate reason is worth keeping distinct from the first. On a cold morning a metal railing feels far colder than the wooden bench beside it, though both sat in the same air all night at the same temperature. Metal conducts energy away from your hand much faster, so your skin cools quickly and reports cold. What you feel there is a rate of transfer, not a temperature.

The proof: energy going in, temperature standing still

The cleanest demonstration that heat and temperature are different quantities is a heating curve. Take crushed ice at −10 °C and warm it steadily with a constant supply of energy, plotting temperature against time.

The temperature climbs to 0 °C and then stops. Energy is still going in at exactly the same rate, and for several minutes the reading does not move at all. It resumes climbing only when the last of the ice has melted, rises to 100 °C, and flattens out again for much longer while the water boils.

During those plateaus the energy works against the forces holding the particles together rather than making them move faster — breaking the lattice at 0 °C, separating molecules at 100 °C. This is latent heat, and it is why steam at 100 °C scalds far worse than water at 100 °C: it releases all of that energy again as it condenses on skin. If heat and temperature were one quantity, a heating curve could not have a flat section.