Newton's Law of Cooling

Newton's law of cooling: rate of cooling of a hot body in air is proportional to its excess temperature over surroundings.

dT/dt = −k·(T − T_s), where T_s = surroundings, k = cooling constant.

Solution: T(t) = T_s + (T₀ − T_s)·e^(−kt) — exponential approach to T_s.

Valid for SMALL temperature differences (in linear regime).

For large ΔT, convective/radiative cooling is more strongly nonlinear (radiation ∝ T⁴).

Cooling constant k depends on size, shape, and surface — larger surfaces cool faster.

Used in: cooling curves of liquids, forensic temperature analysis (post-mortem time of death).

Hot soup cools faster in a wide bowl (more surface area) than in a tall cup.