Newton's Law of Cooling Calculator
Predict how an object's temperature changes as it cools or warms toward its surroundings, using Newton's law of cooling, T(t) = ambient + (initial - ambient) times e^(-k t). Enter the ambient and starting temperatures, a cooling constant and a time.
How to use Newton's law of cooling
- Enter the ambient (surrounding) temperature in degrees Celsius.
- Enter the object's initial temperature in degrees Celsius.
- Enter the cooling constant k in per minute.
- Enter the elapsed time in minutes and read the temperature.
Examples
Coffee cooling
Ta = 20 C, T0 = 90 C, k = 0.1, t = 10 min
T = 45.75 C
At time zero
Ta = 20 C, T0 = 90 C, k = 0.1, t = 0 min
T = 90 C (starting temperature)
Frequently asked questions
What is Newton's law of cooling formula?
T(t) = Ta + (T0 - Ta) * e^(-k * t), where Ta is the ambient temperature, T0 is the initial temperature, k is the cooling constant and t is the elapsed time.
What is the cooling constant k?
The cooling constant sets how fast the object approaches ambient. A larger k means faster cooling. It depends on the object, its surface area and the surroundings, and has units of inverse time.
Can this model warming too?
Yes. If the initial temperature is below ambient, the same formula gives a rising temperature that approaches ambient from below, so it works for both cooling and warming.
What temperature does the object reach after a long time?
As time grows, the exponential term goes to zero and the temperature approaches the ambient temperature. The object can get very close to ambient but never crosses it.
What units should I use?
Use a consistent set. This tool uses degrees Celsius for temperatures, minutes for time and per minute for k. The temperature scale only needs to be consistent for the difference.
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