How cool is your cup of tea?

Some holiday Physics fun.

Problem: Two mugs of tea are poured from a boiling urn. One of them contains milk, and the other has the milk added after 5 minutes. Which one ends up cooler after 20 minutes?

Let’s make some basic (and slightly dodgy) assumptions.

  1. Let’s state that the milk reduces the temperature of the tea by exactly 20 degrees C, no matter when added.
  2. Let’s assume that the mug/cup has negligible heat capacity, and that conditions are such that the rate of heat loss is directly proportional to the temperature difference between the tea and the room.
  3. Let’s say the room temperature is 20 degrees C.
  4. Let’s guess that the liquid cools at a rate of 0.95 x temperature difference, every minute.

Right, let’s crack out our trusty spreadsheet. Taking my assumptions above it should be set up as:

Formulae for use in Excel

This should produce the following numbers:

And finally, the fun part, here’s the graph!

So adding after 5 minutes gets you the cooler cup! Leaving the liquid hotter for the first 5 minutes means it loses a greater quantity of heat.

Of course, there are so many other factors affecting this, you might find something a little different in practice. Plus my assumptions about milk’s cooling effect, and the effect of the mug are highly suspect. Anyone care to offer an improvement? Grab a brew and get to work!