Newton's Law of Cooling Calculator

How it works

T(t) = Tₛ + (T₀ − Tₛ) · e^(−k t) · time to reach T: t = −ln((T − Tₛ) ÷ (T₀ − Tₛ)) ÷ k · half-life of the difference = ln 2 ÷ k · dT/dt = −k (T − Tₛ)

Newton's law of cooling says the rate at which an object loses heat is proportional to the difference between its temperature and its surroundings, so the difference decays exponentially. It holds well for convective cooling with modest temperature differences (coffee, food, water baths, forensic body-temperature estimates). The constant k depends on the object's size, material and the air flow; find it from two readings with k = −ln((T₁ − Tₛ)/(T₀ − Tₛ)) ÷ t₁. Any temperature scale works as long as all three temperatures use the same one.

Worked example

A cup of coffee at 90 °C sits in a 20 °C room with k = 0.05 per minute. After 10 min: T = 20 + (90 − 20) × e^(−0.5) = 20 + 70 × 0.6065 = 62.46 °C. To cool to 40 °C it needs t = −ln(20 ÷ 70) ÷ 0.05 = 25.06 min, and the gap to room temperature halves every 13.86 min.

Sources: LibreTexts — Newton's law of cooling · Wikipedia — Newton's law of cooling

Frequently asked questions