# How do we know that the Law of Cooling is true?

According to James Stewart's Calculus book (exercise 14, page 609),

Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large.

In general, I've read about this law in the context of differential equations where little information is given (because the focus is the math and not the physics). So, I'd like to know:

What sort of experiments was done to establish this law as a scientific truth? How can we justify that this law is valid?

Thanks.

(If this question is not appropriate for the site, let me know and I will delete it.)

Newton's law of cooling is an empirical observation not a fundamental law. The cooling of a body in air is a formidably complicated process because the cooling is dominated by the air flow, and the air flow is complicated to model. However we find from experiment that over a limited range of temperatures the cooling rate is proportional to the temperature difference, though the constant of proportionality varies from system to system.

Response to comment:

Newton's law gives us the rate of cooling as:

$$\frac{d\Delta T}{dt} = -k\Delta T$$

where $\Delta T$ is the temperature difference and $k$ is some constant that depends on the geometry, specific heat of the body, conductivity of the body and probably lots of other parameters. Solving the differential equation we get:

$$\Delta T = \Delta T_0e^{-kt}$$

where $\Delta T_0$ is the temperature difference at time $t = 0$. Taking the log of both sides we get:

$$\ln \Delta T = -kt + \ln \Delta T_0$$

So if Newton's law is correct graphing $\ln\Delta T$ against time should give a straight line. This is how you test Newton's law of cooling. Take your system, let it cool, measure the temperature as a function of time and draw the graph.

• What should be such experiment? – Pedro Apr 7 '15 at 17:18
• Isn't the simple law often called "cooling in a draft" -- i.e., assuming enough motion so that the ambient medium has essentially infinite thermal conductivity, and the rate limiting step is the object-medium interface? – user10851 Apr 7 '15 at 17:18
• @ChrisWhite: from a careful experiment involving blowing on my fingers after I scalded them it appears the cooling rate is highly sensitive to the air flow velocity. – John Rennie Apr 7 '15 at 17:22
• Although, to be fair, evaporative cooling could have been involved in the experiment :-) – John Rennie Apr 7 '15 at 17:22
• @JohnRennie Ok, I understood how to test the Newton's law. But, before to test it, the law is just a guess. What is the motivation for this guess? What kind of data we need to "discover" the law? – Pedro Apr 7 '15 at 18:58

It is an empirical observation, with an assumption that there is enough air flow over the temperature gradient (wind) to allow more rapid change in temp.

With no wind, we end up with the law of natural cooling, which is proportional to $(\delta T)^{5/4}$, instead of $\delta T$. This law also has separate constants from newtons law of cooling! this is a photo from "Mathematical Modeling: a Differential Equations approach by Barnes and Fulford 2nd edition"click