Newton's law of cooling calculator
Enter the starting temperature of the object and the temperature of its surroundings. Then give the cooling constant and a time to get the temperature, a target temperature to get the time, or one measurement to find k. It works for warming too, such as a cold drink in a warm room.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose what to find: the temperature, the time or the constant k.
- Enter the starting temperature and the temperature of the surroundings.
- Enter the other two values: k and the time, k and the target temperature, or a measured temperature and its time. Pick minutes, hours or seconds; k is per that unit.
- Temperature: T(t) = Tₐ + (T₀ − Tₐ) × e^(−k·t).
- Time: t = ln((T₀ − Tₐ) ÷ (T − Tₐ)) ÷ k.
- Constant: k = ln((T₀ − Tₐ) ÷ (T − Tₐ)) ÷ t. The gap halves every ln 2 ÷ k.
Worked example
Coffee at 90 °C sits in a 20 °C room with k = 0.05 per minute. After 10 minutes it is 20 + 70 × e^(−0.5) = 62.46 °C; 60.65 % of the starting difference is left, and the difference halves every 13.86 minutes.
When will it be 40 °C? t = ln(70 ÷ 20) ÷ 0.05 = 25.06 min.
Tips
- k depends on the setup: the object’s size, material, lid, airflow. Measure it once for your situation.
- Not for big temperature gaps with strong radiation or for phase changes like freezing, where the law no longer holds.
- Warming works the same way: a 5 °C drink in 25 °C air with k = 0.5 per hour is at 17.64 °C after 2 hours.
Frequently asked questions
What is Newton's law of cooling?
The rate at which an object cools is proportional to the difference between its temperature and its surroundings. The solution is T(t) = Tₐ + (T₀ − Tₐ) × e^(−kt), so the gap to room temperature shrinks exponentially.
How do I find the cooling constant k?
Measure the temperature once after a known time and use k = ln((T₀ − Tₐ) ÷ (T − Tₐ)) ÷ t. Coffee that drops from 90 °C to 60 °C in 10 minutes in a 20 °C room has k = ln(70 ÷ 40) ÷ 10 = 0.056 per minute.
Does it work in Fahrenheit?
Yes. Only temperature differences enter the formula, so °C, °F and K give the same times and the same k. The temperatures you get back are in the unit you chose.
How is it used to estimate time of death?
Investigators apply the same idea to body temperature, but real bodies cool less simply than the law assumes, so professional estimates use corrected models. Treat this calculator as a physics model, not a forensic tool.