About the Newton's Law of Cooling Calculator
This Newton's law of cooling calculator models how an object's temperature approaches the temperature of its surroundings over time. The difference between the object and its surroundings shrinks exponentially, at a rate set by the cooling constant k. You can find the temperature after a given time, the time needed to reach a target temperature, or work out k from one measurement.
Typical uses include predicting when a cup of coffee is cool enough to drink, how long a drink takes to chill in the fridge, how quickly cooked food passes through temperature ranges, and calculus or differential-equations homework. The same law also describes warming: a cold object in a warm room heats up along the same kind of curve. The chart plots the full temperature curve.
Temperatures can be in °C or °F — just use the same scale for all of them — and time in whatever unit k is given in (minutes by default). The model assumes the surroundings stay at a constant temperature and the object's temperature is roughly uniform inside; it works best for modest temperature differences with convection, and is only an approximation when radiation or evaporation dominates. It is not a substitute for a food thermometer when food safety matters.
How to use the newton's law of cooling calculator
- 1Choose what you want to find: temperature, time or the constant k.
- 2Enter the starting temperature and the surrounding temperature on the same scale.
- 3Enter k (or a measured temperature to calculate it) and the time or target temperature.
- 4Read the answer and the halving time.
- 5Use the chart to see the whole cooling or warming curve.
Formula and method
Newton's law of cooling says the rate of temperature change is proportional to the difference between the object and its surroundings: dT/dt = −k(T − Tₐ). Solving this differential equation gives an exponential decay of the temperature difference, T(t) = Tₐ + (T₀ − Tₐ)e^(−kt). Taking natural logarithms rearranges it to find the time to reach any temperature, or the constant k from one measured point.
The constant k depends on the object’s surface area, mass, heat capacity and how well heat leaves it (for example, a lid or wind changes k). The time for the difference to halve is ln 2 ÷ k, and the time constant 1/k is when 63% of the change has happened. The model assumes constant surroundings and uniform internal temperature.
- T(t)
- Object temperature at time t
- T₀
- Initial temperature
- Tₐ
- Surrounding (ambient) temperature
- k
- Cooling constant (per unit time)
- t
- Elapsed time
Worked examples
Coffee at 90 °C after 10 minutes in a 20 °C room
T = 20 + (90 − 20) × e^(−0.03 × 10) = 20 + 70 × 0.7408 ≈ 71.9 °C. With k = 0.03 per minute the gap to room temperature halves about every 23.1 minutes.
How long until the coffee reaches 60 °C?
t = −ln((60 − 20) ÷ (90 − 20)) ÷ 0.03 = ln(1.75) ÷ 0.03 ≈ 18.65 minutes.
Finding k from a reading of 70 °C after 10 minutes
k = −ln((70 − 20) ÷ (90 − 20)) ÷ 10 = ln(1.4) ÷ 10 ≈ 0.0336 per minute. Use this k to predict later temperatures.
A 4 °C drink warming in a 22 °C room for an hour
T = 22 + (4 − 22) × e^(−0.02 × 60) = 22 − 18 × 0.3012 ≈ 16.6 °C. The same law works for warming — the temperature rises toward the room’s.
Frequently asked questions
What is Newton's law of cooling?+
Newton's law of cooling states that the rate at which an object's temperature changes is proportional to the difference between its temperature and that of its surroundings. The result is an exponential approach to the ambient temperature.
How do I find the cooling constant k?+
Measure the temperature at the start and once more after a known time, then use k = −ln((T − Tₐ)/(T₀ − Tₐ)) ÷ t. This calculator does that in the “cooling constant” mode.
Does an object ever reach room temperature?+
Mathematically the curve only approaches room temperature and never quite reaches it. In practice, after about five time constants (5/k) the difference is under 1% of the starting gap and is effectively zero.
Can Newton’s law of cooling be used for warming?+
Yes. If the object starts colder than its surroundings, the same equation predicts it warming up, with the temperature difference shrinking exponentially at the same rate k.
When is Newton’s law of cooling inaccurate?+
It is an approximation. It becomes less accurate with very large temperature differences where radiation dominates, when evaporation is significant (like an open cup of hot liquid), or when the surroundings warm up or the object heats unevenly inside.