Chemistry Versus Physics: Why does “Heat Rise”? (Part 2 of 2)

Part 1 presented a basic conundrum: particles that are hotter (or lighter) than their peers tend to rise; for example steam will rise off boiling water to condense on a glass lid. The “explanation” given in physics textbooks makes no sense because the hotter (or lighter) particles are not bound together in a separate volume. One cannot treat hot steam rising in the same way as one treats the gas in a hot air balloon. The steam particles can freely intermix with their cooler peers and there is no surface for the cooler air to push up against to buoy the steam upwards.

Thus, the rising of hot (or light) particles appears to directly contradict basic principles students learn in chemistry and physics courses. They are taught the Kinetic Molecular Theory, which claims the behavior of a gas is determined by chaotic, random collisions among its particles. They are also taught that the downward acceleration of gravity for each particle is the same. So how can some particles be favored to move upward if gravity treats them all the same and the chaotic motion is random?
Here is the 3-sentence answer:

The collisions between particles are random but on average they tend to work against gravity. Lighter particles are affected more by collisions than heavier particles, and faster particles undergo more collisions than slower particles. Thus, lighter and faster particles gain more upward momentum than their (heavier or slower) peers.

Two notes regarding the above explanation bear discussion. The first is simple. Lighter particles are almost always moving faster than their heavier peers because temperature is a measure of kinetic energy, which depends on both mass and speed. Thus, if a gas is a mixture of two or more kinds of particles (in thermodynamic equilibrium) the lighter particle must be moving faster on average to make up for having less mass. Thus, lighter particles are often buoyed by both principles.

The second note is more involved. In my books I emphasize contradictions taught by science textbooks. For example, a chemistry textbook will state that temperature is a measure of the average kinetic energy possessed by the microscopic particles of a substance. The same book will later claim that particles in a gas move much faster than particles in a liquid. But how can these statements both be true if water and steam can exist at the same temperature and pressure? If they have the same temperature, their particles have the same average kinetic energy, and if their particles have the same average kinetic energy (and the same mass), how can the particles of steam be moving much faster than those of water?

Hence, when I give the 3-sentence answer above, I immediately hope a student will ask, “But how can collisions work to push particles upward if all collisions conserve momentum? In any given collision, whatever upward momentum one particle gains is lost by the other particle.”

It would be great if students were taught to have such skeptical, discriminating minds. Unfortunately, most students are conditioned to accept whatever is told to them, in science class no less!

Regarding this hypothetical student’s question, the first thing that must be noted is that if collisions produced no net lift, all the particles would be lying on one another on the floor, and we would not be having this conversation because there would be no oxygen to breathe. The second point is that not all collisions conserve momenta among the particles. Consider a single particle striking the floor and rebounding. It gains upward momentum and is the only particle of interest involved.

These two points go together, of course. Consider a particle that has a mass of 1. If the negative acceleration of gravity is g, then gravity is reducing this particle’s momentum by 1*g every second. Assuming this particle is at equilibrium and is not changing its height much over time, the particle must be gaining 1*g momentum from some other source. The indirect source is the floor, which presses up on some particles, but the direct source is the individual collisions, which on average supply the particle with 1*g upward momentum.

These observations give us some confidence in the explanation, but one might still be puzzled as to how collisions have an average positive effect if only collisions between particles and the floor can have a net upward change in momentum, and most particles never hit the floor. The answer lies in a mathematical subtlety: the collisions undergone by each individual particle can (on average) increase momentum even if all the individual collisions between particles conserve momentum.

As an example, let’s say we looked at the collisions in a certain window of time and found:

• Particle A rebounded off the floor, which increased its (upward) momentum by 5.
• Then particle A, now moving upward, hit particle B, losing 4 units of momentum (which particle B gained).
• This sent particle B moving upward, where it hit particle C, losing 3 units of momentum, which particle C gained.
• Similarly, particle C hits particle D, transferring 2 units of momentum, and particle D hits particle E, transferring 1 unit of momentum. Particle E is at the top of the “cloud” of particles, so it does not hit another particle until it begins falling back down.

In this example, each individual collision conserved momentum, yet every particle enjoyed a net gain of (upward) momentum because it gained slightly more momentum in one collision than it lost in the other. This was, of course, a toy example, but it is easy to see that it must be this way because the interaction of the particles must counteract the force of gravity.