Newton's Law of Cooling Calculator
This calculator applies Newton's Law of Cooling to predict how an object's temperature changes over time as it exchanges heat with its surroundings. Whether you want to know how quickly a hot cup of coffee cools down, how long a metal ingot takes to reach room temperature, or how fast a human body cools in a cold environment — this tool gives you the answer instantly.
Thermal Conduction and Convection
There are three main mechanisms of heat exchange: thermal conduction, convection, and radiation. Newton's Law of Cooling applies best when conduction and convection dominate — for example, the cooling of a cup of tea or a hot metal object in still air. In these cases, the primary heat exchange happens at the surface between the object and the surrounding medium. Warm liquid evaporates and convection carries heat away from the surface, gradually lowering the object's temperature.
How fast things cool depends on two factors:
- The temperature difference between the object and its surroundings — the larger the difference, the faster the cooling.
- The cooling coefficient k — which depends on the material, shape, surface area, and heat transfer mechanism.
Newton's Law of Cooling Formula
The formula is:
T(t) = Tenv + (T0 − Tenv) × e−k·t
Where:
- T(t) — temperature of the object at time t
- Tenv — ambient (environment) temperature
- T0 — initial temperature of the object
- k — cooling coefficient [min−1] (rate of heat loss)
- t — elapsed time
- e — Euler's number ≈ 2.71828
The cooling coefficient k can be expressed as:
k = hA / C
Where:
- h — heat transfer coefficient [W/(m²·K)]
- A — surface area of the object [m²]
- C — heat capacity of the object [J/K]
Typical Cooling Coefficients
| Object | Cooling coefficient k (per minute) | Condition |
|---|---|---|
| Cup of hot coffee/tea (ceramic, 250 ml) | ≈ 0.017 | Room temperature ~20°C, no lid |
| Cup of hot coffee (with lid) | ≈ 0.010 | Room temperature ~20°C, insulated lid |
| Small metal object in still air | ≈ 0.03–0.08 | Forced convection increases k |
| Human body (in water at 15°C) | ≈ 0.02–0.04 | Rapid cooling — hypothermia risk |
| Large metal casting | ≈ 0.001–0.005 | Low surface-to-mass ratio |
How Long Does a Cup of Tea Take to Cool?
Starting at 90°C with an ambient temperature of 20°C and a typical ceramic mug cooling coefficient of k = 0.017/min:
- After 10 minutes: T ≈ 79°C
- After 20 minutes: T ≈ 69°C
- After 30 minutes: T ≈ 60°C (comfortably warm)
- After 60 minutes: T ≈ 43°C (still warm)
- After 120 minutes: T ≈ 29°C (nearly room temperature)
Key Derived Values
-
Half-cooling time — time for the object to lose half the temperature difference to the environment:
t½ = ln(2) / k ≈ 0.693 / k -
99% equilibrium time — time to reach 99% of ambient temperature:
t99 = ln(100) / k ≈ 4.605 / k
Units Supported
This calculator supports:
- Temperature: Celsius (°C) — metric system; Fahrenheit (°F) — US/imperial; Kelvin (K) — SI base unit
- Time: seconds, minutes, hours
Note: Newton's Law uses temperature differences, which are numerically identical in °C and K. In °F, a 1°F difference equals 5/9°C. The cooling coefficient k must always match the time unit used in the calculation.
FAQs
Is Newton's Law of Cooling accurate?
It is an excellent approximation when the temperature difference between the object and environment is not too large (less than ~50°C), and when forced convection is small. For large temperature differences or highly dynamic conditions, more complex heat transfer equations are needed.
How do I find the cooling coefficient k experimentally?
Measure the temperature of your object at two time points t1 and t2, then use:
k = −ln[(T2 − Tenv) / (T1 − Tenv)] / (t2 − t1)
Does the formula work for heating too?
Yes! If the object is colder than the environment, the same formula describes heating. The temperature difference becomes negative, and the object warms exponentially toward Tenv.