Tuesday, June 21, 2011

Kinetic Energy and Temperature

A long-standing discussion among colleagues has concerned the understanding of temperature. Central to our discussions has been the heuristic, which is often taught, that temperature is a measure of kinetic energy or average kinetic energy (per something).

There are many touchstone examples we have discussed that have drawn our attention to critical features of not only the concepts and their connections, but what it means to understand them.

Included in these touchstone examples are
  1. An ideal gas in an external gravitational field. How do we think about the kinetic energy and temperature varying with height? Theoretically, intuitively, and empirically.
  2. The free expansion of a non-ideal gas. How do kinetic and inter-molecular potential energies change? How does temperature change? Why? How are these changes related?
  3. A gaseous systems composed of boson and fermions in near classical regime. How do kinetic energies compare?
From these discussion, we have enriched our understanding of all the ways we might think about temperature, how it is related but distinct from average kinetic energy, what it has to do with heating and entropy, how the theoretical construct relates to the ways in which we attempt to access it through measurement, and importantly when it suffices to associate temperature with energy per particle or mode.

Ralph Baierlein in "Thermal Physics" writes the following in a section called "Temperature recapitulated" in a subsection called, "Temperature is deeper than average kinetic energy":

"The misconception that introduces this subsection is propagated with the best of intentions: to make temperature easier to understand. The root of the conceptual error lies in this: a belief that the purpose of absolute temperature is to tell us about a physical system's amount of energy. That is not the purpose of the temperature notion. Rather, temperature is intended to tell us about a system's hotness, its tendency to transfer energy (by heating). All physical systems are capable of heating or cooling others. The purpose of temperature is to rank the systems with respect to their ability to heat one another."


  1. I really like how Tom Moore in "6 ideas" discusses this. His treatment has me saying "avg energy per mode" a lot. I think you need to add the equipartition thm to that to enter into the "ability to heat one another" conversation.

  2. My question is: when should we open this package for students? I feel like I often complicate models too early for my high school sophomores (please, someone gag me before I talk about friction so much). I know you guys are talking about older students, but does a freshman coming in with "temperature is a macroscopic measure of average kinetic energy of atoms" an issue? That said, students who take my second year (calculus-based) physics course come away with a more nuance (or maybe just more confused?) view of temperature, since we do statistical thermo and define temperature in terms of entropy. But that's only a about 20% of our school that gets that far.

  3. When I think about what I assume my students in, say, a calc-based intro course, know about this, I guess it's just the typical societal "hot means hot". I really like showing them this for the 0th law to get them to realize it's all pretty complicated: http://en.wikipedia.org/wiki/Checker_shadow_illusion

  4. In someways, I think the package is easier to make sense of-- temperature is a measurement that is used to rank systems for their capacity to heat other systems they make contact with. In that sense temperature is much more closely related to the phenomena of heating, rather than to abstract notions.

    The relationship between temperature to energy in microscopic world is complex. Sure, dumping energy into a system will often make that system more able to heat other systems, thus making it move "up" in the Temp ranks. But there are other ways to change the temperature of a gas without changing its energy (i.e., free expansion). It's also the case that two systems can have equal temperatures but very different average kinetic energies.

    If all of this isn't confusing (it is to me), try this definition of temperature: "a numerical scale for a concept of hotness which exists on a one-dimensional manifold" Thanks wikipedia, that helped. ;)