What it shows
A hot body cools at a rate proportional to its excess temperature over the surroundings. A graph of the natural log of the excess temperature against time is a straight line.
The science
For a small temperature difference, the rate of heat loss to the surroundings is proportional to the excess temperature: dT/dt = −k (T − Ts). Solving gives T − Ts = (T0 − Ts) e^(−kt), so the curve is a decaying exponential and ln (T − Ts) against t is a straight line with slope −k. The constant k depends on the surface area, the surface and the material. Cooling is quick at first and slows as the water nears room temperature.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A steel or glass cup with 150 mL of hot water (about 80 degrees C)
- A thermometer reading up to 100 degrees C
- A stopwatch or a phone timer
- Cloth, foil and paper to try different coverings
- A second thermometer for room temperature
- Graph paper and a calculator
Safety
For this project: An adult must heat the water and pour it; hot water causes scalds. Keep the cup on a firm tray.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Note the room temperature Ts from the second thermometer, kept away from the cup.
Pour the hot water into the cup and stir gently with the thermometer until the reading steadies; start the stopwatch at about 80 degrees C.
Record the temperature every minute for the first 20 minutes and every two minutes afterwards, up to 40 minutes.
Do not stir between readings but swirl just before each reading to even out the temperature.
Plot temperature against time and draw a smooth curve.
Calculate T − Ts for every reading, take its natural log, and plot ln (T − Ts) against time.
Find the slope of the straight line; its magnitude is k per minute.
Repeat with the same cup in a cloth wrapping and compare the values of k.
What you should see
The curve falls quickly then flattens, and the ln graph is nearly straight for the first 20 to 30 minutes. The cloth-wrapped cup has a smaller k and cools more slowly.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1State Newton’s law of cooling.
The rate of loss of heat is proportional to the excess temperature over the surroundings, for small differences.
2What does the slope of the ln graph give?
The negative of the cooling constant k.
3Why does the ln graph curve at high temperatures?
Evaporation and convection grow faster than linearly when the excess is large, so the law is no longer accurate.
4Why is a hot drink in a cup covered with a lid kept warm longer?
The lid cuts evaporation and convection from the surface.
Take it further
Compare a black-painted tin and a shiny tin of the same size cooling side by side, to test the effect of surface radiation.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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