Temperature and internal energy

A cup of water and a whole bath both read 40 °C. Why doesn't the same thermometer reading mean the same amount of energy? And what exactly does a thermometer measure after it has had time to settle?

Put a cold thermometer into warm water: its reading rises at first. The water transfers energy to it until their temperatures become equal. Net heat transfer then stops: they are in thermal equilibrium. Individual particles still exchange energy, but on average there is no transfer in one direction. That is why we wait for a steady reading. The thermometer itself gains energy and may noticeably change the temperature of a very small amount of water.

Can your hand compare water temperatures reliably? After touching cold or hot water, the same water can feel different. We need a reproducible numerical measure of thermal state: temperature. It tells us the direction of spontaneous heat transfer, from higher temperature to lower temperature. Celsius temperature is written t and measured in °C. In other problems, t can mean time: the context tells us what a symbol means.

Imagine two rulers with equal-sized divisions but different zero positions. That is how Celsius and Kelvin relate: the same warming spans the same number of divisions. Absolute temperature is written T, and its unit is the kelvin, К, without a degree sign. Zero on this scale is the lower temperature limit for ordinary thermal states. It does not mean that all internal energy has disappeared, or that a single thermometer reading can tell us that energy.

To convert a reading, allow for the different zeros: T in К = t in °C + 273,15. At 0 °C we get 273,15 К. Lowering t by one degree also lowers T by one kelvin. For a temperature change, the offset cancels: (t₂ + 273,15) − (t₁ + 273,15) = t₂ − t₁. A rise of 10 °C is therefore a rise of 10 К. If the temperature does not change, both differences are zero, not 273,15.

A cup rests on the table, but its particles keep moving and interacting. The energy of their random motion and interactions together is called internal energy U, measured in joules. Lifting the cup or carrying it across the room changes the energy of the object as a whole; that is separate from U. Temperature is linked to thermal motion, but by itself it accounts for neither the number of particles nor all their interaction energy.

Imagine combining two equal portions of water at the same temperature. Their particle numbers and internal energies add, but the temperature does not double: both portions already had the same temperature. This shows why a bath and a cup differ in internal energy. The comparison is for the same substance in the same state, with a common choice of energy zero. For arbitrary different objects, “hotter means more U” is not a valid rule.

Hot tea warms a spoon without moving it noticeably: heat transfer changes its internal energy. Now rub your palms together. They warm up even without a hotter object nearby, because you do work against friction. This is the second way to change U. When your hands later cool, energy passes to the surroundings rather than disappearing. In each experiment, look for how energy is transferred, rather than imagining temperature as a substance.

Air is warmed from 20 to 30 °C. Find its initial temperature in К and its temperature change.

First find the initial reading on the other scale: T₁ = 20 + 273,15 = 293,15 К. The added value accounts for the different zeros.

For the change, compare the final and initial readings: 30 − 20 = 10 °C. The scale intervals are equal, so ΔT = 10 К.

Check by converting both readings: 303,15 − 293,15 = 10 К. Adding 273,15 to a difference would predict a change even when there was no warming.

The initial temperature is 293,15 К and the change is 10 К. The air warmed by ten equal-sized divisions on either scale.

Equal temperatures stop net heat transfer, but do not stop the particles' thermal motion.

Temperature describes thermal state; internal energy includes the motion and interactions of all the particles.

Add 273,15 to a Celsius temperature reading, but not to a temperature difference.

Energy can be transferred to an object by heating or by doing work.

Heat can be transferred in three different ways. We will compare them in the next lesson.