How Fast Does Coffee Cool? Newton's Law of Cooling

A mug of coffee on a desk is one of the cleanest experiments in physics. You pour it, you wait, and the temperature falls along a curve you can predict from two measurements. A school problem that many students meet is exactly this: measure the temperature drop of a cup of hot coffee and discover Newton’s law¹. Below you get the law, the steps to apply it to your own cup, real data from three experiments, and the places where the simple model drifts from the truth. It is a calculus topic, so the daily life section and the world section both fit.

Newton’s law of cooling in plain words

The rate of heat loss of a body is directly proportional to the difference in temperature between the body and its environment⁵. In symbols, with T the coffee’s temperature, Ts the room’s and k a constant for the cup, the rate dT/dt equals −k times (T − Ts)⁴¹. Because the rate depends on the gap, and the gap keeps shrinking, the solution is an exponential decrease of the temperature difference over time⁵:

T(t) = Ts + (T0 − Ts) × e^(−kt)¹

Here T0 is the starting temperature. At t = 0 the formula returns T0, and as t grows it settles at the room temperature Ts¹. The constant k carries everything about the cup: it encompasses conduction, convection and radiation¹, and it depends on the cup’s heat transfer properties, so an experiment must use the same cup in the same orientation¹. A bigger k means faster cooling.

The idea is old. Isaac Newton published his work on cooling anonymously in 1701⁵. It was the first heat transfer formulation⁵. That is why a single page of calculus, the one on what a derivative is, is enough to follow the whole argument.

How to calculate how fast your coffee cools

You need two temperature readings and the room temperature. Then three steps give you everything else.

1. Measure. Record the temperature now (T0) and again after a known number of minutes. Record the room temperature (Ts)⁴.
2. Find k. Divide the later gap by the starting gap and take the natural logarithm: k = −ln[(T − Ts) / (T0 − Ts)] / t⁴.
3. Predict. Put k back into the formula to get the temperature at any time, or solve it for the time to reach a target temperature: t = ln[(T0 − Ts) / (T − Ts)] / k⁴.

A textbook version of this has a pizza, but the arithmetic is the same as for coffee: a pizza comes out of the oven at 200 °F in a 75 °F kitchen and is 190 °F after one minute⁴. The aim is to wait until it reaches 150 °F⁴. The textbook answer is an additional 6.12 minutes⁴.

Worked example (our arithmetic): the textbook pizza, with the formula above⁴

The result matches the textbook’s 6.12 to rounding⁴. Swap the pizza for a mug and only the numbers change.

A measured cup: the Florida data

A University of Florida write-up dated December 2017 measured water and coffee in a cup with a thermocouple¹. For a cup of water heated to about 190 °F, the fit used k = 0.0453 with a starting gap of 186 − 74 = 112¹. Warm drinkable coffee typically sits between 110 °F and 140 °F¹. Those two facts are enough to ask when the cup becomes drinkable.

Worked example (our arithmetic): coffee poured at 190 °F in the 74 °F room of that experiment, k = 0.0453 per minute¹

The curve makes the first-minutes behavior plain. The cup loses about 42 °F in the first 10 minutes and about 27 °F in the next 10 (our arithmetic), then slows further, even though each interval has the same length¹. The percentage decay is constant. The number of degrees lost per minute is not.

The same write-up added cream, about 10% of the coffee volume, and found that black coffee and coffee with cream followed nearly the same exponential rate¹. It reads that as evidence that radiation plays only a minor role in the cooling¹, which fits its statement that radiation is of negligible importance at sub-boiling temperatures¹.

Why a larger volume cools more slowly: three beakers

A University of Georgia experiment used three beakers of water, 100 ml, 300 ml and 800 ml, all starting at 100 °C in a room at 23 °C². The smaller beakers cooled more quickly than the larger ones². Fitting Newton’s law to each gave a different constant:

Worked example (our arithmetic): half-lives and model predictions from the fitted constants²

The constant for 800 ml was about half the constant for 100 ml². That is the geometry of a cup: more water holds more heat for the same surface to lose.

Where the simple law fails

The table above shows the model’s flaw: it predicts a cooler beaker at minute 30 than the thermometer reads. In every beaker the measured temperature fell faster than the model early and more slowly than the model late². The author points to evaporation, which is high near boiling and removes more heat than convective exchange with the room does, and to the countertop, which pulled heat out at first². The conclusion is still that the law is a reasonable approximation for an object cooling in a constant ambient temperature². The larger sample followed the model more closely than the smaller ones².

A 2024 Physics Education paper on soup bowls and coffee cups reaches the same split verdict: Newton’s law was verified as applicable, the contributions of different heat transfer mechanisms were quantified, and the demonstrations show the important role of evaporative cooling in many cases, versus convection and radiation⁶. The general limits are written into the law itself: radiative heat transfer does not obey it⁵, and it holds well for forced air and pumped liquid cooling⁵. It is more accurate when the working temperature variations are small⁵. A hot drink in still air is on the edge of that range, so treat the numbers on this page as a good shape and a rough forecast.

Does adding milk early keep coffee hotter?

A popular belief says cream added at the start keeps the mixture warmer for longer. A Memorial University worked problem asks the question precisely: does white coffee cool faster if the milk goes in immediately or if the milk is added later³? Its model assumes instant complete mixing³. The result depends on the milk’s temperature. If the milk is at room temperature, the time at which it is added has no effect on the temperature of the mixture afterward³. If the milk is colder than the room, delaying its addition cools the coffee down faster, and if the milk is warmer than the room, adding it immediately cools fastest³.

The Florida write-up reaches the same ordering from a different angle. The curve after cream is added stays below the black-coffee curve, and the mixture follows the same curve no matter when the cream goes in¹. Its conclusion is about insulation: to keep coffee as hot as possible, good insulation such as a thermos bottle is required¹.

Worked example (our arithmetic): 200 ml of coffee at 90 °C, 20 ml of milk at 5 °C, room 20 °C, k = 0.0453 per minute¹³

Delaying the cold milk gives a mixture about 0.2 °C cooler at minute 30, which is the direction the model predicts³ and a difference far below what a kitchen thermometer resolves. The volumes, temperatures and k here are our assumptions, chosen to show the size of the effect.

The calculus behind it

The law is a differential equation, an equation linking a quantity to its own rate of change. Solving it is the standard way to introduce separable equations in a calculus course, and Newton’s law of cooling is one of the two applications that the textbook uses to illustrate them⁴. The steps are short: separate the variables, integrate both sides, exponentiate, and use the starting temperature to fix the constant⁴. The rate you see on the dashboard of this problem is a derivative, and the formulas that produce it are on the derivative rules cheat sheet. For the same idea set inside a whole morning, with a speedometer and a filling tub, read calculus in everyday life.

Mistakes with the cooling formula

• Mixing temperature scales. The ratio of gaps works in Fahrenheit or Celsius, since only differences enter. A starting gap of 112 degF is the same physics as a gap of about 62 degC (our arithmetic)¹.
• Dropping the room temperature. The gap is measured from the room, not from zero. Using the raw temperature in the logarithm gives a nonsense k⁴.
• Reusing one cup’s k for another cup. The constant depends on the cup’s heat transfer properties¹, and in the beaker data it differed by a factor of two between 100 ml and 800 ml².
• Trusting the model near boiling. Evaporation is high there, and the measured curve departs from the model².
• Expecting precision. Even a careful fit has errors of several degrees, as the beaker table shows².