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Newton's Law of Cooling Calculator

Predict how an object cools (or warms) toward its surroundings with T(t) = Tₛ + (T₀ − Tₛ)·e^(−k·t) — enter the starting and ambient temperatures, the cooling constant and the time.

 

Formula

$$ T(t) = T_s + (T_0 - T_s)\,e^{-k t} $$

Worked example

A 100 °C (373.15 K) object in a 20 °C (293.15 K) room with k = 0.05/min after 10 min reaches \( 293.15 + (373.15-293.15)e^{-0.05\times10} \approx 341.7\ \text{K} = 68.5\ °\text{C} \).

How it works

Newton's law of cooling says an object's temperature approaches its surroundings at a rate proportional to the temperature difference: T(t) = Tₛ + (T₀ − Tₛ)e^(−kt). The bigger the gap, the faster it cools.

The cooling constant k (per unit time) captures how quickly heat is exchanged — larger k means faster cooling. Keep k and t in matching time units. The result is given in kelvin with the Celsius equivalent.

Frequently asked questions

What is Newton's law of cooling?

It states that the rate of temperature change of an object is proportional to the difference between its temperature and the ambient temperature, giving T(t) = Ts + (T0−Ts)e^(−kt).

What is the cooling constant k?

k is a positive rate constant (per unit time) that depends on the object's surface area, material and surroundings. A larger k means the object exchanges heat and cools faster.

Does Newton's law of cooling work for heating too?

Yes. If the object is colder than its surroundings, the same formula gives warming — the temperature rises exponentially toward the ambient value.

Why use kelvin here?

You can enter Celsius or Fahrenheit and the calculator converts, but it works internally in kelvin. Because only the temperature difference matters, the exponential decay is the same in any scale.

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