In our high school mathematics textbook, I came across Newton's Law of Cooling. The textbook stated that a body placed in cooler surroundings will cool down at a rate proportional to the difference in temperature between the body its surroundings.

That is, dT/dt = -k(T-E), where T is the temperature of the object, t is time, E is the (constant) temperature of the surroundings, and k is a constant of proportionality (this same law applies to a body warming up when placed in a warmer environment).

I was wondering, why is it that warm bodies cool off proportionally to the temperature difference with the surroundings? Is there a physical explanation of this, or did Newton just find this result experimentally?

Thanks in advance.

## Newton's Law of Cooling question

### Re: Newton's Law of Cooling question

It is an empirical law and it works well for heat conduction.

Via the kinetic theory of heat, it can be related to the thermal motion of molecules, though Newton didn't know about that theory or even about molecules. Basically, hot (rapidly moving) molecules on average lose kinetic energy in collisions with cold (slow) molecules, who gain it. So kinetic energy moves down the gradient of T, causing the hot to cool and the cold to warm.

Newton probably did experiments (he was a good experimentalist) but I haven't looked it up.

When the heat loss is due to radiation, a linear law like Newtonian cooling also works approximately, but only if the temperature difference is small compared to the absolute temperature: radiation goes as T^4, but for small differences in T the difference between the two T^4 terms is approximately linear.

For convection, it doesn't work well.

Joe

PS. An interesting result of Newtonian cooling and other effects:

http://www.animations.physics.unsw.edu.au/jw/Mpemba.htm

Via the kinetic theory of heat, it can be related to the thermal motion of molecules, though Newton didn't know about that theory or even about molecules. Basically, hot (rapidly moving) molecules on average lose kinetic energy in collisions with cold (slow) molecules, who gain it. So kinetic energy moves down the gradient of T, causing the hot to cool and the cold to warm.

Newton probably did experiments (he was a good experimentalist) but I haven't looked it up.

When the heat loss is due to radiation, a linear law like Newtonian cooling also works approximately, but only if the temperature difference is small compared to the absolute temperature: radiation goes as T^4, but for small differences in T the difference between the two T^4 terms is approximately linear.

For convection, it doesn't work well.

Joe

PS. An interesting result of Newtonian cooling and other effects:

http://www.animations.physics.unsw.edu.au/jw/Mpemba.htm

### Re: Newton's Law of Cooling question

This web page also gives accurate explanation on Newton's law of cooling: http://www.ugrad.math.ubc.ca/coursedoc/ ... /cool.html

https://plus.google.com/110340643133455149294/posts