Rate of Cooling

Question

I was given the following problem:

Two objects have a temperature difference of 20 degrees C at time 0. Two minutes later, they have a temperature difference of 18 degrees. How long does it take for them to have a difference of 10 degrees.

Initially, I tried to approach the problem with Newton's Law of Cooling. However, I believe my solution is flawed:

$$\frac{dT}{dt} = -kT$$

letting $T$ be the temperature difference between the two objects. The solution to the above equation is $T(t) = ce^{-kt}$. Inserting the provided values, I found that $c = T(0) = 20$ and $T(2) = 18 = 20e^{-2k}$, so $k = 0.05268$. Setting $20e^{-0.05268t} = 10$, I found that $t = 13.2$ minutes.

Is Newton's Law of Cooling applicable if I simply let $T$ be the difference in temperatures between two objects? Also, the problem implies that both objects are changing temperature. What do I need to do to accommodate for both objects changing temperatures?

Answer 1

You needn't do anything. To a reasonable approximation, Newton's law of cooling can apply to two idealized objects in thermal contact. Your solution seems fine to me.

Newton's law of cooling describes cooling by convection, which is what happens when you leave a hot drink sitting out in a cool room and it steadily gets cooler because the air near it heats up and rises, carrying away heat. Since the air moves away and has thermal contact with a whole lot of other things, it's assumed that the temperature of the air does not change. Now, your problem seems to be describing two objects in thermal contact, which would be conduction. While real conduction is a bit more complicated than Newton's law describes, the end result is the same if we assume that the objects have zero temperature gradient (that is, the temperature is the same throughout the entire object). This is what it means when it's said that Newton's law is a discrete approximation of Fourier's law, which describes gradient, or gradual, conduction.

Now, since you asked about temperature difference as opposed to just temperature: Newton's Law is actually about temperature differences, and doesn't treat absolute temperatures. Look at what the equation you posted says: the rate at which $T$ changes is proportional by $-k$ to $T$. The object will stop changing temperature when $dT$ is zero, which will be when $T$ is zero. Would it make sense for something left out in a room to get colder until it reached absolute zero? Not at all! An object left out to the air cools (or warms up) until it's the same temperature as the air around it. So really, the $T$ in your equation is temperature difference between the two objects, not the temperature of just one object. While it's true that both objects will change temperature, the heat exchange between them will always be proportional to their temperature difference, and that difference will approach zero just as you calculated. We might not know what the final temperature will be, but we still know how fast they'll get closer to being the same temperature. Your solution is sound.