# What is the physical explanation for Newton's laws of cooling?

In my calc 1 class we covered these laws and (rate of change proportional to difference in temperature) I would like to know why they are true, or at least why they work as close approximations. Is it because the greater kinetic energy of warmer particles allows for more frequent and significant trades of energy with the particles in the surrounding environment?

• Regarding Gert comments below, see the links I provided. In applying the law the temperature of the surroundings, a.k.a the ambient temperature, is assumed constant as I stated in my answer. – Bob D Jun 6 at 21:19
• In applying the law the temperature of the surroundings, a.k.a the ambient temperature, is assumed constant as I stated in my answer. This is simply incorrect. – Gert Jun 6 at 21:44

First you need to know that heat is energy transfer due solely to a temperature difference between things. Heat naturally transfers from a warmer body to a cooler body. Heat transfers kinetic energy from the molecules of the higher temperature surroundings to the lower temperature object.

Normally the heat transfer results in the temperature of the warmer body to decrease and the temperature of the cooler body to increase. But newton’s Law of cooling assumes the lower temperature surroundings is a thermal reservoir so that its temperature stays constant when heat transfers from the higher temperature object.

The law then gives the object’s temperature as a function of time with the object’s temperature eventually equaling the lower temperature of the surroundings. The equation is:

$$T(t)=T(s)+[T(o)-T(s)]e^{-kt}$$

Where $$T(t)$$ is the object’s temperature as a function of time, $$T(s)$$ is the temperature of the surroundings (a constant), $$T(o)$$ is the initial temperature of the object, $$k$$ is a cooling constant depending on the object in 1/s. $$t$$ is time in seconds and all temperatures are in degrees Kelvin.

From the equation, at $$t=0$$ $$T(t)=T_o$$ and at $$t=∞$$ $$T(t)=T_{S}$$. In between the difference in temperature between the object and the surroundings decreases exponentially to zero.

Hope this helps

• Comments are not for extended discussion; this conversation has been moved to chat. Also, please try to keep things cordial. – tpg2114 Jun 7 at 1:44