It’s been a while since I took thermodynamics and statistical mechanics, so I’m using this post partly as a way to refresh my memory—please tell me I’m wrong! It’s also a way for me to start a conversation on definitions of temperature, some of which are abysmally bad. Because science has a misconception problem, we ought to think of a good way to progress toward more sophisticated notions of temperature throughout one’s educational timeline and to make clear when we break from the naïve notions and why.

One has only to google “definition of temperature” to realize the problem: Once one skips the dictionary definitions about temperature being a scale of hotness (which, as I will argue, is arguably better than what follows), one gets to definitions that say something about how temperature is a measure of the average translational kinetic energy of the atoms in a substance.  (I won’t link to these because I don’t want to increase their Google PageRank.)

### Bad definition 1: Temperature is a measure of the average translational kinetic energy of the atoms in a system.

Compare to NGSS draft 2: DCI PS3.A: “Temperature is a measure of the average kinetic energy of particles of matter. The relationship between temperature and the total energy of a system depends on the types, states, and amounts of matter present.”
It’s a nice picture.  It’s just not hard to break.  For instance:

• What would happen to the temperature if you threw the system?

### Bad definition 2: Temperature is a measure of the average translational kinetic energy of the atoms in the rest frame of a system.

Even if you try to correct for this naïvely by specifying that the kinetic energy be measured in the center of mass frame:

• What would happen to the temperature if you split the system in two pieces and threw them in opposite directions?

Or even worse:

• What would happen to the temperature if you spun the system?

### Bad definition 3: Temperature is a “measure of the average translational kinetic energy associated with the disordered microscopic motion of atoms and molecules”.

HyperPhysics has the best bad definition of temperature.  However, I wouldn’t go as far as they do and call it an “operational definition”.  What is “kinetic energy associated with disordered…motion”?  What counts as disordered motion?  Is a sound wave (phonon) disordered?  It’s what we might think of as a vibration (which many texts cite when they talk about temperature).  How do we measure disorder for the purpose of calculating the “associated” kinetic energy?  Why doesn’t rotational energy count?  What do we mean by “measure” (another problem with calling it an operational definition)—even if I account for all the “disordered” kinetic energy, what value do I put down for temperature?

What’s the simplest system we could envision that breaks this definition? Note that to break the measure part, there must be a non-monotonic relationship between temperature and kinetic energy.  Candidates include:

• photon gas (Many thermal systems include EM radiation.  How do we measure the temperature of light when it isn’t made of atoms?)
• low-temp quantum solids???
• diatomic gases???

### Good definition 1: Temperature is the rate of change of energy with entropy (TODO: specifics on how this is defined) (with volume and the number of particles fixed)

HyperPhysics also gets credit for this good definition.  However, the authors don’t give any examples where the behavior of temperature differs drastically from the simplistic $T\propto U$ idea.
The standard definition in thermodynamics is:

$T=\frac{Q_\textrm{\scriptsize reversible}}{\Delta S}$

However, the standard definition of temperature in statistical mechanics is:

$T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}$

Does this always work?  What does it mean? Without teaching entropy well, this might be hopeless. However, I recently read an interesting paper on a simple model to introduce the need for entropy as a thermodynamic variable:

• Abreu, Rodrigo de, and Vasco Guerra. “Introducing Thermodynamics Through Energy and Entropy.” American Journal of Physics 80.7 (2012) : 627–637. 4 Jan. 2013. <http://link.aip.org/link/?AJP/80/627/1>.

## How could we make the teaching of temperature better?

I envision a tripartite system of teleological, conceptual, and operational definitions, where we scale up the complexity of the conceptual definitions (i.e. develop new models for temperature) as a student progresses through the system. This has to be explicit, or students won’t understand either the nature of science or why their old ideas are not quite right.

### Teleological definition: Temperature is a scale that tells us the direction of heating when two systems (both in equilibrium) come into contact with each other.

Even this isn’t obvious and needs some justification. If we have three systems in equilibrium, A, B, and C, why couldn’t A heat B, B heat C, and C heat A if they were brought into pairwise contact?
It does get the point across why we care about temperature, which helps to ground our other definitions and provide continuity in the notion of temperature.

### Operational definition 2 (later elementary): Temperature is what a thermometer measures.

This should be connected to the idea of heating and reference temperatures (from special systems with understood behavior). It can later include the idea of different scales. The idea of absolute zero should come from an extrapolation of gas law data from a student experiment.

### Conceptual definition 1: Temperature is a measure of how frenetic and disordered the motion of atoms/molecules in a substance is.

Note that we’re pretty close to a bad definition but that it’s always augmented by our teleological and operational definitions. This is not worth getting to until students understand that matter is made of molecules. Students should see simulations of matter at different temperatures to get a feel for what we’re talking about. It’s simplistic and qualitative.

### Conceptual definition 2: Temperature is a measure of the distribution of energies in a substance given by $T=\left(\frac{\partial S}{\partial U}\right)_{N,V}^{-1}$.

That is, roughly, it’s the ratio of added energy to the change in entropy that results. We’re moving past our bad definitions.

### Conceptual definition 3: We should define a new “temperature” as an inverse energy scale $\tau=-\beta=-\frac{1}{k_B T}=-\frac{1}{k_B}\frac{\partial S}{\partial E}$

Regular absolute temperature fails our teleological definition at negative absolute temperatures. This new definition gets the direction of thermal transfer right when two systems come into contact. See, for instance:

• Braun, S. et al. “Negative Absolute Temperature for Motional Degrees of Freedom.” Science 339.6115 (2013) : 52–55. <http://arxiv.org/abs/1211.0545>