Newton's Law of Cooling

Hot coffee cools quickly at first and slower as it nears room temperature. The rate depends on the temperature gap, which is how investigators estimate time of death.

How it works

Exponential decay ODEs

In one sentence

An object at temperature $T$ in surroundings at $T_a$ changes temperature at a rate proportional to the difference, $T' = -k(T - T_a)$ , so $T(t) = T_a + (T_0 - T_a)e^{-kt}$ .

General formula

{\htmlData{sym=0}{\frac{dT}{dt}}} = -{\htmlData{sym=4}{k}}\,({\htmlData{sym=1}{T}} - {\htmlData{sym=2}{T_a}}) \;\;\Rightarrow\;\; {\htmlData{sym=1}{T}}({\htmlData{sym=5}{t}}) = {\htmlData{sym=2}{T_a}} + ({\htmlData{sym=3}{T_0}} - {\htmlData{sym=2}{T_a}})\,e^{-{\htmlData{sym=4}{k}}{\htmlData{sym=5}{t}}}

The bigger the gap between the coffee and the room, the faster heat flows; the gap shrinks by the same fraction every minute.

What the symbols mean

$\frac{dT}{dt}$
Cooling rate
How fast the object's temperature is changing.
$T$
Temperature
The object's temperature right now.
$T_a$
Room temperature
The temperature of the surroundings. The object heads toward it.
$T_0$
Starting temperature
The object's temperature at t = 0.
$k$
Cooling constant
How easily heat escapes. A bigger k cools faster.
$t$
Time
Minutes since the object was put in the room.

The cooling rate is proportional to the gap $T - T_a$ .
The gap shrinks exponentially, so the temperature approaches room temperature without ever quite reaching it.
Two readings are enough to solve for $k$ and run the clock backwards.

Learn the math

Separable EquationsDiff Eq · Lesson 5, First Order Differential Equations Modeling with First Order EquationsDiff Eq · Lesson 10, First Order Differential Equations