Newton's Laws of Cooling

If a vessel of hot water is left on a table, it gradually cools. Newton was the first to systematically study how the rate at which a body loses heat to its surroundings depends on its temperature, in what is now called Newton's law of cooling: for SMALL temperature differences, the rate of loss of heat (equivalently, the rate of fall of temperature) $dT/dt$ of a body is directly proportional to the difference between the body's temperature $T$ and that of its surroundings $T_0$:

where $C$ is a constant of proportionality (the minus sign reflects that temperature is FALLING while $T > T_0$).

This is verified experimentally with a calorimeter filled roughly two-thirds with boiling water, covered, with a thermometer through the lid so its bulb stays fully immersed. The calorimeter is left in surroundings whose own temperature stays roughly constant (open room air, or a constant-temperature enclosure), and the water's temperature is recorded once every minute as it falls by about 25 °C in total. Plotting temperature $T$ against time $t$ gives a cooling curve (Fig. 7.13(a)) -- steep at first, gradually flattening as the water approaches room temperature (a hotter body loses heat faster, exactly as the law predicts). Drawing a tangent to this curve at several points and reading off each tangent's slope $dT/dt$ (the instantaneous rate of fall of temperature at that point), then plotting these rates against the corresponding temperature EXCESS over the surroundings, $(T-T_0)$ (Fig. 7.13(b)), produces a STRAIGHT LINE through the origin -- direct experimental confirmation that the rate of cooling is indeed proportional to the temperature excess, as Newton's law states. …