The Exponential Decay of Heat
Formulated by Sir Isaac Newton in the late 17th century, Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in temperatures between the body and its surrounding environment.
Because the rate of cooling depends on the temperature difference, an incredibly hot object will cool down very rapidly at first. But as it gets closer and closer to room temperature, the rate of cooling slows down exponentially. Mathematically, it takes an infinitely long time for the object to exactly reach the ambient temperature.
Forensics and Food
- Forensic Pathology: This is the exact math used by medical examiners to estimate the Time of Death at a crime scene. A human body has a known initial temperature (roughly $37^\circ\text{C}$). By measuring the current temperature of the body and the ambient temperature of the room, pathologists can work the exponential equation backward to find exactly how long the body has been cooling.
- Coffee Cooling: If you pour a fresh cup of boiling coffee, you want to know when it will reach a drinkable $60^\circ\text{C}$. Newton's law dictates that if you add cold milk immediately, the coffee's temperature drops, meaning it will lose heat to the room much slower. If you wait 10 minutes and then add the milk, the coffee will ultimately be much colder.
- Metallurgy: When forging steel, blacksmiths use specific cooling rates (quenching in oil or water vs. letting it cool slowly in air) to lock the carbon atoms into specific crystalline structures, radically altering the hardness of the final blade.
The Formula
Example Calculation
A cup of soup is heated to $360 , \text{K}$ ($87^\circ\text{C}$) and placed on a table in a room that is $295 , \text{K}$ ($22^\circ\text{C}$). The specific cooling constant for this bowl of soup is $k = 0.005 , \text{s}^{-1}$. You want to know its temperature after $300 , \text{seconds}$ (5 minutes).
- Calculate the Temperature Difference ($T_0 - T_a$): $360 - 295 = 65 , \text{K}$.
- Calculate the Exponential ($e^{-kt}$): $e^{-(0.005 \cdot 300)} = e^{-1.5} \approx 0.223$.
- Multiply and Add Ambient: $295 + (65 \cdot 0.223) = 295 + 14.495 = 309.495 , \text{K}$.
After 5 minutes, the soup has cooled rapidly down to about $309.5 , \text{K}$ ($36.5^\circ\text{C}$), which is right around body temperature and ready to eat.