Newton's law of cooling
/ NOO-tunz /
Leave a hot coffee on the table and it cools fast at first, then slower and slower as it nears room temperature, sidling up to the room's warmth but never overshooting it. Newton noticed that the speed of cooling is governed by one thing: how big the temperature gap is between the object and its surroundings. A scalding cup loses heat quickly; a lukewarm one barely changes.
Written as a differential equation, dT/dt = -k(T - T_s), where T(t) is the object's temperature, T_s is the surrounding (ambient) temperature, and k > 0 is a constant set by the object and how well it trades heat with its surroundings. The minus sign and the gap (T - T_s) together say: if the object is hotter than its surroundings the temperature falls, if cooler it rises, and either way it heads toward T_s. The solution is T(t) = T_s + (T0 - T_s)e^(-kt) — the gap decays exponentially, so the temperature glides smoothly to ambient. (Heating works the same way; the law is symmetric.)
This is really exponential decay in disguise, applied not to the temperature itself but to the difference T - T_s. It is used in forensic estimates of time of death, in cooking, and in any heat-exchange estimate. It assumes the surroundings stay at a fixed temperature and the object has one uniform temperature throughout — both approximations that fail for a large, slowly-conducting body whose inside lags its surface.
Coffee at 90°C in a 20°C room cools to 60°C in 10 minutes. From T(t) = 20 + 70 e^(-kt), at t = 10 we get 40 = 70 e^(-10k), so k = (ln(70/40))/10 ≈ 0.056 per minute, and we can now predict any later temperature.
Exponential decay of the temperature gap T - T_s, gliding toward ambient.
The law assumes a single, uniform body temperature; a thick roast or a large lake has internal gradients, so its center cools quite differently from this simple formula.