What Newton’s law of cooling says
A hot object loses heat faster when it is much hotter than its surroundings, and more slowly as it gets close to room temperature. Isaac Newton described this in 1701: the rate of cooling is proportional to the temperature difference. The result is an exponential decay of that difference:
T(t) = Tₐ + (T₀ − Tₐ) × e^(−k × t)
where T₀ is the starting temperature, Tₐ the surrounding temperature and k the cooling constant, which depends on the object’s size, shape, material and how air or water moves around it.
Finding k from two readings
You rarely know k in advance, so the calculator works it out from a second measurement T₁ taken after a time t₁:
k = −ln((T₁ − Tₐ) ÷ (T₀ − Tₐ)) ÷ t₁
It then predicts the temperature at any time and the time needed to reach a target:
t = −ln((T_target − Tₐ) ÷ (T₀ − Tₐ)) ÷ k
Worked example: a cup of coffee
Coffee poured at 185 °F in a 70 °F room has cooled to 160 °F after 10 minutes:
| Result | Value |
|---|---|
| Cooling constant k | 0.02451 per minute |
| Temperature after 30 minutes | 125.1 °F |
| Time to reach 140 °F | 20.3 minutes |
| Time for the gap to room temperature to halve | 28.3 minutes |
The coffee loses 25 °F in the first 10 minutes, but only about 35 °F more over the next 20 minutes, because the difference with the room keeps shrinking.
Where it is used
- Food safety: estimating how long cooked food takes to cool through the danger zone. The FDA Food Code asks for cooling from 135 °F to 70 °F within 2 hours and to 41 °F within 6 hours in total.
- Engineering: cooling of electronics, castings or liquids in tanks.
- Forensic science: body cooling is one clue to the time of death, though experts use adjusted models such as the Henssge nomogram.
- Everyday life: when to drink your tea, or how long a drink takes to chill in the fridge.
Limits
The simple law works best when:
- heat leaves mainly by convection and conduction, not by evaporation or strong radiation;
- the surrounding temperature stays constant;
- the object’s temperature is roughly uniform inside.
A cup with a lid, a stirred drink or a large roast cooling from the outside in will deviate from the curve. Take your second reading after a reasonable gap, so measurement errors weigh less.