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HEAT · B1 · 013 · Newton's law of cooling

Your coffee cools fast, then slow.

Newton's law of cooling: heat flows faster when the temperature gap is bigger. Cooling speed = k × (T − T_room), so T(t) = T_room + (T_start − T_room) × e^(−k × t).

Try itA cup of coffee on a thermal camera

Pick a cup and watch the clock run. The steep part of the curve is the big gap at the start. As the coffee gets closer to room temperature, the gap shrinks and so does the cooling. Drag the drawing to scrub through time.

Coffee at the start 80 °C

Room temperature 20 °C

Cup cooling constant k 0.03 /min

Time 0 min

Coffee now–

Gap to the room–

Cooling right now–

Gap halves every–

NumbersHow big is it?

The k values for each cup are example values that give a realistic curve; your own mug will be a bit different. Real coffee also loses heat by steam, so the very first minutes drop a little faster than the formula.

60 vs 20 °C

The gap for 80 °C coffee and for 40 °C coffee in a 20 °C room. Three times smaller, so it cools three times slower. Our own maths.

68 to 70 °C

Below this, a study found a sizable share of coffee drinkers start calling black coffee too cold.

≈ 23 min

How long the gap takes to halve in our open mug (k = 0.03 per minute). 60, then 30, then 15 °C: it never quite reaches zero. Our own maths.

HowWhat happens, step by step

Heat flows from hot to cold.
Fresh coffee at about 80 °C sits in a 20 °C room. That's a 60 °C gap, and heat leaves the cup quickly.
The bigger the gap, the faster it flows.
Cooling speed is proportional to the gap. Halve the gap and the coffee cools half as fast.
So the curve is steep, then flat.
The first minutes drop it fast. The last few degrees take ages, and it never quite reaches room temperature.

Same curve everywhere: a cooling engine, a baked cake on the counter, a cup of tea. Anything losing heat to its surroundings follows this steep-then-flat shape.

Check3 quick questions

One more thing

Want it hot later? Add the milk straight away. A cooler cup has a smaller gap to the room, so it loses heat slower while it waits.