A hot object left alone does not cool at a steady rate. It sheds heat quickly while it is far above its surroundings and more slowly as it gets close, so a plot of temperature against time is a curve that flattens out rather than a straight line running down to ambient. Newton's law of cooling is the statement that makes this precise: the rate of temperature change is proportional to the difference between the object and its surroundings. Integrate that and you get an exponential decay of the gap, never of the temperature itself.
This Newton's law of cooling calculator from Arb Digital works the way the measurement is usually actually made. Rather than asking for a cooling constant that almost nobody knows, it asks for one observed temperature at one known time and derives the constant from it. Everything else follows: the temperature at any later moment, the time to reach a target, and the time constant that characterises the object and its surroundings together.
What This Cooling Calculator Does
Four inputs define the problem. The ambient temperature is where the object is heading. The starting temperature is where it begins. The observed temperature and the time it was taken supply the one piece of empirical information the model needs, because the rate of cooling depends on the object's size, shape, surface and surroundings in ways no formula on a web page could guess. From those, the calculator solves for the cooling constant k and then uses it for every subsequent answer.
The reciprocal of k is the time constant, and it is the more intuitive number of the two. It is the time taken for the gap to ambient to fall to about 37 per cent of its starting value, and after five time constants the gap has closed to under one per cent, which is usually where a measurement stops being able to tell the difference. The half-gap time, the cooling equivalent of a half-life, is the time constant multiplied by the natural logarithm of two.
The same model runs backwards without modification. If the object starts colder than its surroundings, the gap is negative, the exponential still decays, and the object warms towards ambient on precisely the same curve. Newton's law is symmetric in that respect, so a drink warming in a room and a casting cooling in a shed are the same calculation with a sign change.
How to Use It
- Measure the ambient temperature properly. It is the largest single source of error in the whole model, because everything is referenced to it. A thermometer sitting on the same hot surface as the object is not measuring ambient.
- Take the calibration reading at a sensible time. Too soon and the temperature change is within measurement noise; too late and the gap has almost closed, which is equally uninformative. Somewhere around one time constant is ideal.
- Keep the time unit consistent. The cooling constant is reported per unit of whatever you selected, and the observation time, query time and answers all use the same unit.
- Set a target between the start and ambient. An object approaching ambient never reaches it, and it certainly never passes it, so a target outside that range has no finite answer and the tool says so.
- Read the bar as a progress measure. It shows how much of the original gap has closed by the query time, which is a more honest picture of how far the process has gone than the temperature alone.
The Formula and a Worked Example
The differential statement is dT/dt = −k(T − Tₐ), whose solution is T(t) = Tₐ + (T₀ − Tₐ)e⁻ᵏᵗ. Given one observation T₁ at time t₁, rearranging gives k = −ln((T₁ − Tₐ)/(T₀ − Tₐ)) / t₁, and the time to reach any target T is t = −ln((T − Tₐ)/(T₀ − Tₐ)) / k. The derivation and a worked coffee example appear in the OpenStax Calculus section on exponential growth and decay.
Work the default, which is that coffee. It starts at 200°F in a 70°F room, so the initial gap is 130 degrees. Two minutes later it reads 180°F, a gap of 110 degrees. The ratio 110/130 is 0.8462, so k = −ln(0.8462)/2 = 0.08353 per minute, and the time constant is 11.97 minutes. At the query time of ten minutes the gap has fallen to 130 × e⁻⁰·⁸₃₅ = 56.4 degrees, putting the coffee at 126.4°F. Working the other way, 175°F is a gap of 105 degrees, reached after −ln(105/130)/0.08353 = 2.56 minutes. Half the gap closes in 8.30 minutes.
Why the Cooling Constant Is the Same in Celsius and Fahrenheit
This surprises people, and the reason is worth understanding because it tells you what kind of quantity k is. The law contains only temperature differences, never absolute temperatures, and the conversion between Celsius and Fahrenheit is affine: multiply by nine fifths and add thirty-two. The additive part cancels when you subtract two temperatures, and the multiplicative part appears in both the numerator and the denominator of the ratio (T₁ − Tₐ)/(T₀ − Tₐ), so it cancels too.
What survives is a pure number, and its logarithm divided by a time is therefore also independent of the temperature scale. A cooling constant of 0.0835 per minute is 0.0835 per minute whether you measured in Celsius, Fahrenheit or kelvin. It is not independent of the time unit, though: the same object has a k of 0.0014 per second and 5.01 per hour. That asymmetry is a good check on whether you have understood the quantity, and it is the reason this page prints the time unit alongside every constant.
Where the Model Breaks Down
Newton's law is an approximation, and it is worth knowing which of its assumptions is failing when a measurement drifts from the curve. The first assumption is that the object has a single temperature. Real objects have internal gradients, and if heat cannot move through the object as fast as it leaves the surface, the surface cools first and the interior lags. The standard test is the Biot number, comparing internal conduction resistance to surface resistance; when it is below roughly 0.1 the lumped model this page uses is sound, and above that a large joint of meat or a thick casting needs a full conduction analysis.
The second assumption is that heat loss is proportional to the temperature difference. That is a fair description of convection over a modest range, but radiation obeys a fourth-power law, with heat flux proportional to the difference of the fourth powers of the absolute temperatures and to the Stefan-Boltzmann constant published by NIST. At small temperature differences the fourth-power expression is very nearly linear and Newton's law absorbs radiation into an effective k without trouble. At a few hundred degrees above ambient it does not, and a single constant will overestimate the later cooling.
Third, the surroundings must stay at a constant temperature and the mechanism must not change. A pie cooling in a closed cupboard warms its own surroundings and cools more slowly than predicted. A cup with a lid removed halfway through switches from one k to another. And anything crossing a phase change — water freezing, fat solidifying, a metal passing through its solidus — releases latent heat at constant temperature, producing a flat plateau that no exponential can reproduce. The latent heat calculator covers that energy separately.
Forensic and Food-Safety Uses, and Their Limits
Two applications drive most searches for this calculation. The first is estimating an elapsed time from a measured temperature, working the model backwards. The arithmetic is straightforward, but the sensitivity is punishing: because the gap decays exponentially, a small error in the measured temperature becomes a large error in the inferred time once the object has come close to ambient. The same measurement precision that gives a confident answer early gives an almost useless one later, which is why a single temperature reading is treated as an indication rather than a conclusion in any serious context.
The second is food cooling, where regulators specify staged limits — a drop through one temperature band within a set number of hours, then through a second band. The exponential model shows why the second stage is the hard one: the cooling rate falls as the food approaches refrigeration temperature, so the last part of the journey takes disproportionately long. That is the physical reason the practical guidance is to divide the batch, increase the surface-to-volume ratio, or use an ice bath, all of which change k rather than fighting the curve. This page models temperature against time; it sets no safety threshold and issues no food-safety guidance, which belongs to the food authority with jurisdiction.
How This Differs From the Adjacent Arb Digital Tools
This page models one object's temperature as a function of time as it approaches its surroundings, calibrated from a measurement. The specific heat calculator answers how much energy a temperature change requires, with no time in it at all. The heat transfer calculator gives a rate of heat flow for a stated mechanism rather than a temperature history. The heat loss calculator works on a building envelope in steady state. The half-life calculator shares the exponential mathematics but applies it to decaying quantities rather than to a closing temperature gap, and the temperature converter only rescales values between units.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Expecting the object to reach ambient — the gap decays exponentially and never reaches zero. Ask for a temperature a degree or two above ambient instead, or read the time constant.
- Guessing at the ambient temperature — every term is referenced to it, so an error there distorts the constant and every prediction that follows.
- Changing the time unit mid-problem — the cooling constant is per unit time. A k measured per minute is sixty times the value per second, and mixing the two moves an answer by that factor.
- Applying one constant across a phase change — latent heat holds the temperature steady while it is released, and no exponential fits a plateau.
- Extrapolating from a reading taken too late — once the gap has nearly closed, small measurement errors imply huge time errors, so calibrate early rather than near ambient.
Related Free Tools From Arb Digital
For the energy behind a temperature change use the specific heat calculator and the latent heat calculator, and for rates of flow the heat transfer calculator or the thermal conductivity converter. Buildings are covered by the heat loss calculator, exponential decay in general by the half-life calculator, and unit changes by the temperature converter. For how cold air feels rather than how objects cool, see the wind chill calculator. The full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
The statement that an object's rate of temperature change is proportional to the difference between its temperature and that of its surroundings. Integrating it shows that the temperature gap decays exponentially, so cooling is rapid at first and slows as the object approaches ambient.
The proportionality factor in the law, with units of reciprocal time. It bundles together the object's surface area, its heat capacity and the heat transfer coefficient of its surroundings, which is why it has to be measured for a particular object in a particular setting rather than looked up.
Because the law uses only temperature differences and the constant is derived from a ratio of two differences. The offset between the scales cancels in the subtraction and the scale factor cancels in the ratio, leaving a pure number that has no temperature unit at all.
Strictly it never does, because the gap approaches zero without reaching it. In practice the gap falls to about 37 per cent of its start after one time constant and under one per cent after five, at which point the difference is usually below what a thermometer can resolve.
Yes, without any change. If the object starts colder than its surroundings the initial gap is negative, the exponential decays exactly as before, and the object approaches ambient from below on the same shape of curve.
When the object has significant internal temperature gradients, when radiation dominates at large temperature differences because it follows a fourth-power law, when the surroundings themselves warm up, or when a phase change releases latent heat and holds the temperature at a plateau.
Because k depends on the specific object and its surroundings, not on the material alone. Supplying one temperature at one known time lets the calculator solve for the constant that actually applies, which is far more reliable than adopting a figure from elsewhere.
This tool is provided for educational and study use. It applies an idealised lumped-capacitance model and issues no food-safety, medical or forensic determination; those judgements belong to the qualified authority in each field.
