1.12.2: Kinetic Theory of Heat

If you were to ask a group of people what the "Einstein relation" is, most of them would be likely to mention E = mc² or something associated with relativity. In fact, the Einstein relation is based on his much earlier work studying the motion of the microscopic particles that make up a liquid. This relationship describes how the properties of the molecular motion of particles in a material, specifically the diffusion coefficient, is proportional to it's temperature. Covering this in detail goes well beyond the scope of this class, but a brief overview of it will help us to connect our understanding of macroscopic particles to the concepts of heat and temperature. With that in mind, let's discuss what heat is and how it relates to what we have already covered earlier in this text.

Heat is energy. Energy makes things move. To be hot is to move fast, on average.

The last words there are an important caveat that must be kept in mind when we are talking about heat. A shot glass filled with water has about 100,000,000,000,000,000,000,000 (10²⁴ or so) molecules in it. In light of this, we will almost always restrict ourselves to discussing average properties of those many billions and billions of molecules. Many of the tools of thermodynamics and statistical mechanics have been developed specifically so that we can obtain meaningful (and accurate) insights into the behavior of these particles without the need to treat them each individually. Later in this chapter we will begin seeing these shortcuts that let us describe the bulk behavior. As we build our intuition though, we will retreat to the safety of individual particles. As we consider the molecular motion in side a glass of water, we will find that we can explain most behavior using familiar ideas like elastic collisions, kinetic energy, rotations and springs.

In Joule's experiment, a paddle spinning in a liquid increased the temperature. From the perspective of a molecule in the liquid, the paddle acts in much the same was as any other collision. A molecule of water encountering the paddle will behave in much the same way as a ping pong ball being struck by a bat. The collision transfers momentum and kinetic energy into the molecule, and sends it flying on its way. The chief difference between these two scenarios is that the molecule cannot travel very far before running into another molecule. This is not a problem for us though, as we can simply treat this as a new collision. If we have enough patience, pens and paper, there is nothing to stop us from treating each of the collisions (between paddle and liquid, and between molecules in the liquid) manually. In fact, this is similar to how modern molecular dynamics simulations work.

If we consider how heat effects molecules in a solid, we will find that the system behaves similar to a large network of springs. The molecules in a solid sit in an equilibrium position as if tethered there with a spring. If they move to close to a neighbor they will be repelled and pushed back to their origin. Giving this system more energy simply lets the molecules move a little bit further before being repelled, just as giving more kinetic energy to a mass on a spring lets it oscillate to a greater amplitude, and achieve a higher speed when it returns to the equilibrium position. 

When we rolled a disc down a ramp, we saw how we needed some kinetic energy to make its center of mass move ( \( \frac{1}{2} m v^2 \) ) and also needed kinetic energy to make it rotate ( \( \frac{1}{2} I \omega^2 \) ). We discussed how these types of motion may be related in certain cases, but can vary independently if, for example, a wheel were to slip and over rotate or start to slide without rolling. Similarly, the molecules in a liquid can not only move (translational motion) but can spin, twist and tumble (rotational motion). Some molecules might even be able to bend and wiggle as if their atoms were connected by springs. All of these potential types of motion require kinetic energy and have their own inertia associated with them. As we heat a material, energy is divided between these types of motion. The energy will be divided according to something called the equipartition theorem. The various ways that energy can be stored will in turn determine how much energy we need to add to something in order to increase its temperature, giving us something called the heat capacity of the material.

As we explore the subject further, keep in mind that we will be seeing very little new information. Instead, we are simply taking what we learned in the context of a small number of macroscopic objects, and applying it to a much larger number of much smaller objects. It's still all just a bunch of springs and simple masses.