An object cools fastest when it is much hotter than its surroundings, and slows as it approaches room temperature.
How to use the Newton’s Law of Cooling Calculator
- Initial temperature
- Surrounding temperature
- Cooling constant k
- Time
- Want to work backwards? Type a value into Temperature after time and Time is calculated for you.
Results update as you type, and every field accepts the units shown in its menu.
How it works
The temperature difference decays exponentially with a rate constant k that depends on the object.
Inputs explained
| Field | What to enter |
|---|---|
| Initial temperature | Units: °C, °F, K |
| Surrounding temperature | Units: °C, °F, K |
| Cooling constant k | In per minute |
| Time | Units: min, h, s |
What each result means
- Temperature after time = [Surrounding temperature] + ([Initial temperature] – [Surrounding temperature])×exp(-[Cooling constant k]×[Time])
- Time to halve the difference = ln(2)÷[Cooling constant k]
Example values
How the main result changes when Time changes and everything else stays at the example values above:
| Time (min) | Temperature after time (°C) |
|---|---|
| 10 | 63.2 |
| 15 | 54.1 |
| 20 | 47 |
| 25 | 41.5 |
| 30 | 37.2 |
| 40 | 31.2 |
Formula
T(t) = T_ambient + (T₀ − T_ambient) × e^(−k t)Worked example
Inputs — Initial temperature: 90 °C; Surrounding temperature: 22 °C; Cooling constant k: 0.05 per minute; Time: 20 min.
Results — Temperature after time = 47 °C; Time to halve the difference = 13.9 min.
Frequently asked questions
How do I find k?
Measure the temperature at two times and solve the equation for k.
Is the law exact?
It is a good approximation when the temperature difference is moderate.
What formula does the Newton’s Law of Cooling Calculator use?
T(t) = T_ambient + (T₀ − T_ambient) × e^(−k t).
Can I work backwards from the result?
Yes. Type the result you want into Temperature after time and the calculator finds the matching time.