The Cup of Tea

The tea was 90°C when Trolla poured it. The room was 20°C. She set the cup on the table and began to read.

Half an hour later, she lifted the cup and tasted.

Cold.

Not lukewarm. Not tepid. Cold. The kind of cold that makes you appreciate the existence of heaters, which Trolla does not appreciate, because she built her own heater from a resistor and a fan and now the room is 28°C and the tea is 50°C and she has achieved a state of reasonable contentment.

But let us rewind. Let us speak of the tea's journey.

The tea began at 90°. It cooled quickly at first — the gap between 90 and 20 is large, and Newton's Law of Cooling says the rate of cooling is proportional to that gap. $$\frac{dT}{dt} = -k(T - T_{\text{room}})$$

The tea wanted to reach 20. The room offered 20. The tea was not willing to give up its warmth easily, but it was willing, gradually, in increments. The bigger the gap, the faster the surrender.

In the first minute, the tea dropped from 90 to about 72. A full 18 degrees. The tea fought hard here — it had just been boiled, it was still hot with memory. By minute five, it had settled to 45. The gap was closing. By minute ten, it was 32, and the tea had accepted that 20 was coming whether it liked it or not.

By minute fifteen, 25. Minute twenty, 22. Minute thirty, 21.

The tea never reached 20. Not in the time that mattered. It asymptotically approached it, as all things do. An asymptote is a polite word for "you'll never quite get there but you'll get close enough to pretend."

Trolla finished her reading. She looked at the tea. She looked at the room. She took a sip.

It was 23°. Warm enough. Not 20, not 90. Somewhere in the middle, where most things live — and where Trolla is content to let them be.

— A Story, by Trolla