Translational kinetic in solids

[Lujz_br]

I'm a bit confuse. In chapter 15, of Conceptual physics by Paul Hewitt:

"Temperature is related to the random motion of atoms and molecules in a sub-
stance. (...) More specifically, temperature is proportional to the average “trans-
lational” kinetic energy of random molecular motion (motion that carries the
molecule from one place to another). Molecules may also rotate or vibrate, with
associated rotational or vibrational kinetic energy—but these motions are not
translational and don’t define temperature."

In solid with only vibracion, we have absolute zero? I think most solids do not have translational kinetic energy... Can you help me understand this?

 

[jambaugh]

Yep, that's just plain wrong as written. In general, temperature relates to how changes in energy for a system and changes in its entropy relate. Any energetic mode of the system contributes to the definition of temperature. While the author may have been thinking about how systems couple, you still have the fact that internally as molecules interact there will be energy exchanged between all those degrees of freedom. So even if two systems only interact with each other via translational motion energy from their other modes is still exchanged randomly through those translational modes.

Actually the reciprocal temperature is the better quantity to start with. If you increase the energy of a system it will usually increase the ambiguity in its state because typically there are more ways to distribute a larger amount of energy among the many,many modes within the system. The rate at which this happens defines reciprocal temperature. Since entropy is proportional to the logarithm of the measure of the size of the range of available states, its rate of change with respect to mode count is the reciprocal of that count. So you can also, roughly speaking, think of the temperature as the proportionate increase in system size per unit increase in energy (units being resolved by the Boltzmann constant.)

To clarify what is meant by "range of available states" consider a 2 atom gas. If you have 1 Joule of heat in the system then one of the atoms, (let's call him Bob) can have any velocity (vector) up to magnitude $v^2 \le 1J/m$ where #m# is Bob's mass. Whatever energy he doesn't have the other atom (let's call her Alice) must have. If you double the available energy then you double the volume in which Bob's velocity could range. Actually, assuming Alice and Bob have the same mass you'd have a hyper-sphere in 6 dimensional space but you get the point. Increasing energy increases the size of the range of possibilities as to how that energy get's spread out. That must include any system mode that can store energy such as rotational and vibrational degrees of freedom.

The only possible way I can think of to interpret Hewitt's statement is to say it is true for systems of "fundamental particles". For these the intrinsic spin is fixed and there are no effective "internal vibrations" at normal energies. Any motion, including rotating molecules and vibrating crystals, can be resolved as translational motion of the underlying constituent particles.

However even here you can have additional contribution to temperature using the coupling between spin and magnetic fields through the magnetic moments of the particles. There's even a method of refrigeration based on this by way of cycling magnetic fields. https://en.wikipedia.org/wiki/Magnetic_refrigeration

 

[Lujz_br]

Thanks!
About interaction, we read in next paragraph: (Conceptual physics, Twelfth Edition)

"The effect of translational kinetic energy versus rotational and vibrational ki-
netic energy is dramatically demonstrated by a microwave oven. The microwaves
that bombard your food cause certain molecules in the food, mainly water mole-
cules, to flip to and fro and to oscillate with considerable rotational kinetic energy.
But oscillating molecules don’t cook food. What does raise the temperature and
cook the food, and swiftly, is the translational kinetic energy imparted to neigh-
boring molecules that are bounced off the oscillating water molecules. (To picture
this, imagine a bunch of marbles set flying in all directions after encountering
the spinning blades of a fan—also, see page 418.) If neighboring molecules didn’t
interact with the oscillating water molecules, the temperature of the food would be
no different than before the microwave oven was turned on."

In same page:
"Figure 15.3
Particles in matter move in different ways. They move from one place to another, they rotate, and they vibrate to and fro. All these types of motion, plus potential energy, contribute to the overall energy of a substance. Temperature, however, is defined by translational motion."