Whenever I see something about temperature, there’s all these little atoms or molecules in a block or jar and they’re whizzing around at different speeds. More motion, more temperature.
However, if a single atom or molecule were in the jar, could it be said to have a temperature? If so, how is that temperature being stored? Kinetic energy?
Temperature is proportional to the average kinetic energy. For one molecule the average kinetic energy is just that one molecule’s kinetic energy. But it practice temperature is really only a bulk and statistical property. That is, you are only sampling some of the molecules with your thermometer.
I should have added this is an idealized law which holds pretty well for a monatomic gas. For a molecule with more than one atom, some energy can be in the form of rotation of the molecule or vibration along the bonds, etc. The kinetic energy relation is KE = 1.5 k T where T is the absolute temperature (in kelvins) and k is the Boltzman gas constant which is Avagardo’s number divided by the ideal gas constant. And you’re averaging the KE per particle. That is, don’t compute the average speed and then use 0.5mv[sup]2[/sup]. Rather directly average 0.5mv[sup]2[/sup] or equivalently use the root-mean-square speed.
I don’t know as much about this stuff as OldGuy but it bears mentioning that temperature is not stored. Temperature is an indirect measure of heat, which is a type of kinetic energy, and as you can see from OldGuy’s second post there is a mathematical directly proportional relationship between the two. You can store energy but you don’t really store temperature. It’s like you can store a piece of paper but you can’t store 11 inches.
Strictly speaking, temperature cannot be defined for a single particle… But if you have a particle in a jar, you could define a temperature for the system consisting of the particle and the jar.
I assume that’s because you cannot define the velocity, and therefore the kinetic energy, of a single particle as that would depend on the frame of reference. So for a group of particles is temperature defined by the average kinetic energy in the frame of reference where the center of mass is stationary.
Think of it as “if a truckload of molecules all had the exact same kinetic energy as this one, the temperature of the system would be X”
Yeah, pretty much, OldGuy. Though in a system of two (or a few hundred) particles, while it might be possible to define temperature, it’s really not useful. Much better, in that case, to just enumerate the energies or velocities of all of the individual particles.
It depends to some extent on what you mean. Historically, temperature was a thermodynamic property, which means it only has meaning for macroscopic observations of (in the pristine limit) an infinite number of infinitely small and unobservable degrees of freedom. From that purist 19th century point of view, no, the temperature of a single molecule – or in fact any finite number of molecules – is not well defined.
However, with the advent of statistical mechanics, and its connection to thermodynamics, it became possible to make statistical mechanical definitions of temperature which in the limit of an infinite number of infinitely small degrees of freedom – usually referred to as the “thermodynamic” limit – becomes functionally identical to the thermodynamic temperature. That is usually the way temperature is defined and used today.
In that sense, you can certainly defined the temperature of a single molecule. It is defined as a derivative on a partition function, which is a large sum (or integral in the classical limit) over all possible states of the molecule, with different weighting factors and different restrictions depending on what kind of boundary conditions you set – which is, more or less, how you imagine this molecule interacting with the rest of the universe. If it does not at all, that leads to one type of partition function. If it can exchange energy but is confined to a fixed volume, that is another. And so on.
What you can see is that the temperature in this sense is a statistical beast – it describes the typical behaviour of the molecule (and some of the typical variations from the typical behaviour). That typical behaviour will depend on, for example, the physical nature of the molecule itself: how many internal degrees of freedom does it have? What is the spacing between internal energy levels? How big is the box in which it moves? What is its typical energy, if the energy is not fixed? And so on. These things are characteristic of the physical nature of the molecule and of the box (or whatever) in which it’s being held. And they determine the temperature.
What the temperature does not depend on, however, is the trajectory of the molecule during any particular specific period of time. That’s because this is inherently a statistical description of the molecule. It does not attempt to describe any actual portion of the molecules history, but only what one can say about the average properties of its entire trajectory, from long before the period of observation to long after. It’s essentially the difference between describing my average day at work and a specific actual day at work.
If you think about it, that can lead to some subtle disconnects. For example, if I work alternate days, my “average” day at work is 4 hours long, but there is no actual one day when I work 4 hours – it’s always 0 or 8. So there is a serious disconnect between the statistical description of my day and a direct observation of my trajectory over a short time. That can happen with molecules, too, although it is quite difficult to set up. A mechanical system, like a molecule, could be “non-ergodic,” which essentially means its mechanical behaviour is such that even averaging over very long times there will be disconnects between the behaviour of a typical stretch of trajectory and the statistical description. (For example, consider the case where the atom is set to bounce perfectly between two perfectly parallel walls. It will never visit the corners of the box, while a statistical description of the molecule would assert, absent some asymmetry in the box, it should be equally likely to visit all regions of the box.) The question of how a system becomes ergodic, or at least ergodic enough, is deep and subtle, because it connects directly to the emergence of an “arrow of time” (i.e. the Second Law of Thermodynamics) from mechanical systems which are inherently time-reversible.
But to return to your core question: the temperature of a single molecule, bearing in mind this is a statistical description, is essentially the inverse of the average increase in the number of states the molecule can be when the molecule’s energy is increased a little. There are some logs and factors of Boltzmann’s constant in there as well, and perhaps some complicated integrals, but that’s the core of the matter. If the molecule is in a situation where the number of states it might be in increases rapidly when a little energy is added, then it is at a low temperature, and if the number of states it could be in increases only slowly with energy, then it is at a high temperature.
We chatted this idea up a bit here in a thread I started wondering how small a unit of anything can be said to have the quality of “hotness,” by which I was trying to get at the idea of “absolute zero” having any meaning at the level of teeny tiny particles or groups thereof.
I remain confused, but I have attributed that to my stupidity and not the kindness of folks here trying to explain the concept to me.