Newton's Law Of Cooling Calculator

A hot object cools fastest when it is hottest, because the rate follows the temperature gap. That single idea produces an exponential curve.

Starting temperatureThe temperature of the object at time zero, in degrees.Ambient temperatureThe temperature of the surroundings the object is approaching.Cooling constantThe fraction of the gap lost per unit time. Higher means a faster cool down.Time elapsedHow long the object has been cooling, in the same units as the constant.Target temperatureThe temperature you want the time for. Keep it between ambient and the start.

Temperature after time t

35.6191

Time to reach the target: 11.1923
Half difference time: 13.8629
Initial cooling rate: -3.5
Share of the gap remaining: 22.31%

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

T(t) = Ta + (T0 - Ta) x e^(-kt)

T(t): The temperature after time t
Ta: The ambient temperature the object approaches
T0: The starting temperature of the object
k: The cooling constant in units of one over time

How to check the result by hand

1
Find the starting gap
Coffee at 90 degrees in a 20 degree room starts with a gap of 70 degrees. This gap is what decays, not the temperature itself.
2
Evaluate the decay factor
e to the minus k times t is e to the minus 0.05 times 30, which is e to the minus 1.5, giving 0.22313. Just over a fifth of the gap remains.
3
Apply it to the gap
70 times 0.22313 is 15.6191, so the coffee has fallen 54.3809 degrees and only 15.6191 of the original gap is left.
4
Add the ambient back
20 plus 15.6191 is 35.6191 degrees. Forgetting this last step is the most common error, because the formula decays the gap, not the temperature.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate Newton's Law Of Cooling page.