What this calculator does
Newton's law of cooling describes how an object's temperature approaches the temperature of its surroundings over time: T(t) = T_env + (T0 − T_env) × e^(−kt). T0 is the starting temperature, T_env is the ambient or surrounding temperature, k is a cooling rate constant, and t is elapsed time. The gap between the object and its surroundings shrinks exponentially rather than in a straight line.
The cooling constant k is not a universal number: it depends on the object's surface area, material, shape and how much airflow or contact it has with its surroundings, so it has to be measured or estimated for the specific situation rather than assumed. This calculator leaves k as a free input for that reason, alongside the starting temperature, ambient temperature and elapsed time.
The formula
The temperature gap above (or below) ambient shrinks by a constant proportion for every unit of time, which is what makes the decay exponential rather than linear. Multiplying the initial gap (T0 − T_env) by e^(−kt) gives the remaining gap at time t, and adding that back to the ambient temperature gives the object’s actual temperature at that moment.
| Term | Meaning |
|---|---|
| T0 | Initial temperature of the object at time zero. |
| T_env | Ambient temperature: the constant temperature of the surroundings the object is cooling (or warming) towards. |
| k | Cooling rate constant, in units of 1/time. A bigger k means faster cooling; it depends on the object and its environment, not a fixed physical constant. |
| t | Elapsed time since the object was at temperature T0. |
The inputs explained
| Field | What to enter |
|---|---|
| Initial temperature (°C) | The temperature the object started at, at time zero. |
| Ambient (surrounding) temperature (°C) | The constant temperature of the surrounding environment the object is cooling or warming towards. |
| Cooling rate constant (k) (1/min) | The cooling rate constant for this specific object and setting. It must be measured or estimated; there is no single correct value. |
| Elapsed time (min) | How much time has elapsed since the object was at the starting temperature. |
When to use it
Predicting how a hot drink or dish cools
Given a starting temperature, a room temperature and a cooling constant estimated from past measurements, this works out the temperature at any later time without waiting to check.
Estimating time of death in forensic contexts
A simplified form of Newton’s law of cooling is the classical model behind body-temperature-based time-of-death estimates, though real forensic use adds corrections this basic formula does not include.
Checking a cooling rate constant against measured data
If a real measurement of temperature at a known time is available, comparing it against this calculator’s prediction for a trial value of k shows whether that k is a reasonable fit.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How temperature falls over time at a fixed cooling rate
The same starting and ambient temperatures, with elapsed time varied.
| Elapsed time | Temperature at time t | Remaining gap above ambient at time t |
|---|---|---|
| 0 min | 90.00 | 70.00 |
| 5 min | 74.52 | 54.52 |
| 10 min | 62.46 | 42.46 |
| 20 min | 45.75 | 25.75 |
| 30 min | 35.62 | 15.62 |
| 60 min | 23.49 | 3.49 |
How the cooling rate constant changes the outcome at a fixed time
The same 20-minute mark, with the cooling rate constant k varied.
| Cooling constant k | Temperature at time t | Remaining gap above ambient at time t |
|---|---|---|
| 0.01 /min | 77.31 | 57.31 |
| 0.03 /min | 58.42 | 38.42 |
| 0.05 /min | 45.75 | 25.75 |
| 0.08 /min | 34.13 | 14.13 |
| 0.1 /min | 29.47 | 9.47 |
| 0.15 /min | 23.49 | 3.49 |
Questions
Where does the cooling rate constant k come from?
It is not a fixed physical constant like the ones used for unit conversions; it depends on the object’s surface area, shape, material and how much airflow or contact it has with its surroundings. In practice, k is fitted from at least one measured temperature at a known time, or taken from a similar past case.
Does this formula work for warming as well as cooling?
Yes. The same equation applies whether the object starts below or above the ambient temperature; the temperature simply approaches ambient from whichever side it started on.
What happens if the ambient temperature changes during the process?
This formula assumes a constant ambient temperature throughout. If the surroundings are also changing temperature, such as a room warming up over the day, the basic formula no longer applies exactly and a more advanced model is needed.
How is this different from the drink chilling time calculator?
The drink chilling time calculator solves the same equation the other way around: given a target temperature, it finds the time needed to reach it. This calculator instead takes a fixed time and reports the temperature reached by then.
To solve the same equation the other way, for the time needed to reach a target temperature, see the drink chilling time calculator.