The analysis of a single sentence
Copyright 2001 Kevin T. Kilty, All Rights Reserved.

Recently my friend and colleage, HD, wrote to ask a question or two. He had read a sentence in a college chemistry textbook that made less and less sense the more he read it. He asked for my thoughts on the matter.

"Subject: Temperature & KE

I ran into this in a textbook,

"heat energy transfer CO2(s,-78 C) <--> CO2(g, -78 C) gas molecules have higher kinetic energy."

If they are at the same temperature, how does the gas have a higher kinetic energy? If it has a higher kinetic energy then wouldn't it be able to transfer some kinetic energy to a third body (a thermometer)?

I wonder, how do solid molecules communicate temperature to a thermometer to the same extent as a gas?

For an Ideal gas we say there are no attractive forces between molecules and the molecules have no potential energy. The total energy E of an ideal gas is therefore entirely kinetic. As the average kinetic energy per molecule is 3/2kT, we have...

E = 3/2NkT or 3/2nRT

The energy of an ideal gas is proportional to the absolute temperature and depends on temperature only, independent of the pressure and volume. This of course is true for a very restricted system, rigid, "billard-ball" molecules which exert no forces on one another, which do not rotate, vibrate or dissociate.

So, if the opening quote is true, what am I missing?


As he always does, HD has asked some very probing questions regarding this sentence, and for good reason. This single sentence covers a lot of physics and chemistry, and, depending on how you interpret it, contains a lot of error. Rather than answer off the top of my head, or off the cuff, and say something stupid that I would have to retract later, I ended up thinking about this during my wakeful period from 3 am to 5am that night. Here is the analysis I did.

First, I began to worry about that two headed arrow in the sentence. This one...'<-->'. A fellow shouldn't just leave one of these hanging around without explanation. A chemist might interpret such a construction as saying something like "This chemical equation is reversible under the correct circumstances," and therefore to say nothing more than solid CO2 can become a vapor, and vice versa. I don't know for sure that this is what a chemist would think because I'm only an amateur chemist, and I really don't think like a real one. A physicist, however, definitely sees that double arrow as saying something more specific; that there is equilibrium between the solid and the vapor. In fact, even a casual reading shows that the author(s) placed a temperature of -78C in the equation as though they meant something. So I grabbed my copy of Lange's Handbook and found, sure enough, at 1 atmosphere of pressure -78.52C denotes the melting temperature of solid CO2.

By now that double headed arrow really bothered me. You see, it conflicts with the statement at the beginning of the sentence about heat transfer. This is just plain wrong. There is no input of heat anywhere in the sentence. And if there is equilibrium implied in the chemical equation, with both sides at the same temperature, then nothing in the tiny universe of that sentence evolves. For every infinitesimal dn of solid molecules that change phase from solid to vapor taking on whatever new properties the vapor phase requires, there is a dn of vapor becoming solid and returning those properties in whole. The author(s) claim that there is heat transfer somewhere, but the tiny universe is isothermal and nothing else apparently changes. If the authors had really meant a heat transfer they needed to write the equation as...

CO2(s)--> CO2(g)

in which the CO2 vapor is allowed to escape, and the CO2 solid cools by sublimation. Since the cooling CO2 solid is obviously loosing heat, the vapor must be taking heat away with it. This is decidedly non-equilibrium, though, and a <--> simply won't do anywhere in its description.

It dawned on me; just then. This textbook sentence violates one of the laws of thermodynamics. Most likely, I thought at 3 am, it is the second law, or maybe the zeroth, but whichever law it violates, I realized I was pondering a perpetual motion machine. What a wonder. I had never in my life seen such an abbreviated, and unadorned description. Usually the design of a perpetual motion machine is complicated and difficult to understand, but this one is just a naked, skrawny, stripped to its absolutely essential elements, perpetual motion machine.

So, I have concluded that there is something wrong with the textbook statement, and maybe I can fix it up by just dumping its opening phrase. Is this revised one any better, I wondered?

" CO2(s,-78 C) <--> CO2(g, -78 C) gas molecules have higher kinetic energy."

Now, here, I have a problem with the comparison of kinetic energy of the molecules in the vapor with those, if I can call them molecules, in the solid. HD, has already noted some of the difficulty in that by using the term "kinetic energy" in describing the molecules, the author(s) are bring in a mechanical concept. Thermodynamics contains no such concept, and the authors have implicitly brought kinetic theory into the problem. As HD points out, kinetic energy (average kinetic energy actually) and temperature are one and the same in the ideal gas, so if the gas has "higher" kinetic energy, then how can it and the solid be at the same temperature?

Or as HD asks, if the gas molecules really do have more kinetic energy, then how is it they don't convey some of this to a third body that is at the same temperature as the solid? What HD has noticed is that if kinetic energy determines temperature, then why doesn't the higher kinetic energy of the gas molecules "diffuse" to anything with less kinetic energy. Here perhaps the author(s) have confused the idea of heat content in thermodynamics with kinetic energy. After all, it is possible for two materials to be in thermal equilibrium, and yet for one of the two to have greater heat content. For example, a mole of water vapor and a mole of liquid water are at thermal equilibrium at 100C and one atmosphere pressure, but the mole of vapor has obviously more heat content. Maybe this is what they meant? It is confusing to take concepts from a simple kinetic theory, like that of the ideal gas, and just apply them carelessly to the real world. Heat content doesn't diffuse, but kinetic energy does. In the kinetic theory of conduction electrons, for example, diffusion of their kinetic energy from the heated end of a metal bar, to take a homely example, carries heat to the cold end, and we credit this diffusion for the good thermal conductivity of metals.

The most fundamental problem becomes apparent only after filling in lots of details. If the author(s) are using kinetic energy as a measure of heat content, then the author(s) mean to say that the vapor or gas is ideal. An ideal gas is one in which the individual atoms (particles we really should say), have only translational kinetic energy, no internal rotation or vibration, and no potential energy of interaction. The only way a person can suppose that such a mixture manages to achieve internal equilibrium is that its atoms can interact through elastic collisions. But here is an interesting issue. If there is no potential energy of interaction between atoms, then there is no way to bind the ideal gas atoms into a solid. A strict ideal gas model can never be in equilibrium with its solid the way the equation...

CO2(s,-78 C) <--> CO2(g, -78 C)

suggests because there are no attractive forces with which to bind the solid. Therefore an ideal gas in equilibrium with its solid is an internally inconsistent idea.. I have emphasized the previous sentence because it is the most important one that I'll make in all this ranting. Theories, hypotheses, statements of fact, even measurments, all have to be internally consistent for us to make any sense of them or to draw valid conclusions from them. There is all too little attention paid to this simple fact; and my drawing attention to it at this juncture gives me a self-serving opportunity to plug another web page of mine that is solely concerned with drawing valid conclusions.

At this point I see the textbook statement as pretty self-contradictory thoughout. Yet, HD brought up several related issues with which I can prolong the beating of this dead horse.

HD asks how does a solid convey its temperature to a thermometer in a way comparable to a gas conveying it through collisions? Here we enter the weird topic of how to define temperature in a microscopic system. Temperature is a bulk property. It only has microscopic meaning when applied to a distribution of molecular "states." Within a small enough collection of molecules temperature fluctuates constantly. Because of this there are routine violations of the second law in the microscopic realm. When I say that temperature has meaning only in a statistical sense, I mean to say, only when the system is large enough for statistics to imply a reasonably consistent and constant temperature. In the case of the gas, I can view temperature as a parameter in its Maxwell distribution of molecular speed. I find the value of this parameter that best "fits" or describes the current situation, and this becomes the temperature of that system. Note, however, that a distribution of speeds means that some molecules have very little kinetic energy while others have very much, and only the average is 1/2kT per degree of freedom.

In a solid material there is also a distribution of energy among available states. In a solid insulator, like carbon dioxide, there are no conduction electrons. The molecules, if that is how to describe them, cannot float around. There is, however, a new class of particles which move, and these are the lattice vibrations, or phonons. Here too temperature is a parameter of their distribution of occupied states, but it is likely not the only parameter. The equivalent of kinetic energy for phonons is their energy, E=hw. There is a distribution, the Bose-Einstein Distribution, of energy of lattice vibrations or phonons, and temperature is a parameter. When a solid and gas are placed in contact with one another they can exchange energy back and forth (heat flow or fluctuations) but at equilibrium the distribution of energy in either does not change with time. It is in this sense that phonons interact with a thermometer to convey a measurement of temperature.

When I first wrote back to HD, I had the idea that no one had come up with a satisfactory model of phonon distributions in a solid. I have found lately that I was wrong. I knew of the Einstein model, in which all phonons have the same frequency, and the Debye model in which they vary according to some law up to a maximum frequency. I knew that both of these models are able to explain some features of heat capacity of solids and so forth. It turns out there are more sophisticated models as well. But the pertinent issue in this case is that a person cannot compare kinetic energy of ideal gas molecules to kinetic energy in phonons, because phonons have more than just kinetic energy. If phonons were classical particles, then their energy would be split on average between kinetic energy and potential energy. While ideal gas molecules have kinetic energy, phonons have mechanical energy.

In summary. The opening phrase of the textbook statement conflicts with the implied equilibrium in its chemical equation to embody some sort of perpetual motion machine. While the statement of equilibrium implies that the vapor and solid are the same material, only an ideal gas has heat content depending on kinetic energy alone. Unfortunately an ideal gas has no corresponding ideal solid. Finally, it is erroneous to compare kinetic energy of molecules in a vapor, real or ideal, with kinetic energy of lattice vibrations because lattice vibrations actually involve mechanical energy. There are three "wrongs" in the textbook sentence, and using the well known principle that two wrongs do not make a right, anyone can use induction to prove that three don't either.