Newton's law of cooling

The equation \(T'=-k(T-T_a)\) rewrites as \(T'+kT=kT_a\), linear with constant coefficients. With the integrating factor \(e^{kt}\) we get \((e^{kt}T)'=kT_ae^{kt}\), and integrating gives \(T(t)=T_a+(T_0-T_a)e^{-kt}\). Equivalently, the difference \(T-T_a\) obeys \(y'=-ky\): it decays exponentially.

The sign of \(T-T_a\) rules: if the body is hotter than its surroundings, \(T'<0\) and it cools; if colder, \(T'>0\) and it warms. In both cases \(T\) approaches \(T_a\) monotonically, neither crossing nor reaching it in finite time; \(T=T_a\) is the only equilibrium.

The constant \(k\) depends on the object and the medium, and is found from a second measurement. Since the difference is multiplied by the same factor over equal intervals, the time to halve it is \(\ln2/k\) and the characteristic time is \(1/k\).

Worked example: a body at \(100\,^\circ\)C in a room at \(20\,^\circ\)C that reads \(60\,^\circ\)C at \(5\) min has \(k=\ln2/5\) and reads \(40\,^\circ\)C at \(10\). A common mistake is to assume the temperature drops a fixed amount per minute; what stays constant is the proportion of the difference. The model assumes constant ambient temperature and a body at uniform temperature.