Molecular Vibration and Gas Temperature

[Wilko]

I posed this thought experiment elsewhere, so the tone may seem a bit odd. It's based on what I've been reading on physical chemistry, atmospheric radiation and thermal physics. Just wondering if my interpretation is on the money or if I've made any incorrect assumptions.

"Time for another random thought experiment. Imagine a “balloon” filled with air, it's not essential but we can also say its a closed system in a vacuum to make it simpler. The balloon is a spherical membrane completely transparent to infrared radiation at 4.3 microns (one of the absorption peaks for CO2, Sodium Chloride windows are transparent to IR as it turns out.) Then you illuminate the “balloon” with infrared radiation at 4.3 microns.

What happens then? Glad you asked….A CO2 molecule inside the balloon will absorb an IR photon, become vibrationally excited, undergo molecular collisions, gain a small amount energy in collisions, lose a small amount of energy in collisions, and then emit an IR photon almost identical to the one it absorbed, it’s a very small perturbation based on how much vibrational energy was gained from, or lost as translational energy during the collisions for this particular CO2 molecule we have been so intently following. I'm ignoring Doppler broadening and the like for simplicity's sake and because I'm mainly interested in the interaction between translational and vibrational energy.

As it turns out, the vibrational energy is much larger than the energy involved in collisions, so it’s quite hard for the collisions to affect vibration even if they could interact directly.

On average though, over many molecules, many absorption/emission cycles and many collisions the amount of translational kinetic energy stays the same, and the amount of energy present as radiation stays the same. The radiation does get a little ‘messier’ (spectral broadening) but it never undergoes wholesale conversion into translational kinetic energy.

PHEW! So the average translational energy of the molecules in the gas stays the same, even when we illuminate it with infrared radiation. That’s important because average translational kinetic energy is what we’re defining when we talk about the temperature of a gas, particularly in the real world. Vibrational energy is a temporary form of internal energy that isn’t really relevant when we talk about the macroscopic properties of a gas like Temperature, Pressure and Volume; those qualities are very specifically derived from average translational kinetic energy of the molecules in the gas.

SO, if we illuminate our “balloon” with infrared radiation at 4.3 microns, we would expect any increase in the temperature of the air inside to manifest itself as an increase in pressure, which we could measure as a change in the volume of the balloon (offsetting the pressure increase somewhat, I do realize) So, can the balloon expand by illuminating it with infrared at 4.3 microns? Based on what I’ve read and understand, I can’t see how."

Fun stuff, anyone got some extra insight to offer?

 

[alxm]

As it turns out, the vibrational energy is much larger than the energy involved in collisions, so it’s quite hard for the collisions to affect vibration even if they could interact directly.

You contradict yourself here since you just asserted that collisions change the energy of the emitted photon.

Anyway the fact is they do interact. A molecule that absorbs a photon and goes to a higher vibrational state can return to a lower vibrational state 'non-radiatively', in other words by dissipating the energy as translational/rotational energy. They're not decoupled.

PHEW! So the average translational energy of the molecules in the gas stays the same, even when we illuminate it with infrared radiation.

So it follows this is false.

That’s important because average translational kinetic energy is what we’re defining when we talk about the temperature of a gas, particularly in the real world. Vibrational energy is a temporary form of internal energy that isn’t really relevant when we talk about the macroscopic properties of a gas like Temperature, Pressure and Volume; those qualities are very specifically derived from average translational kinetic energy of the molecules in the gas.

This is not true at all. These properties are derived from the statistical-mechanical description of the system, the partition function. And the vibrational energy is most certainly part of that.

Fun stuff, anyone got some extra insight to offer?

"Molecular thermodynamics" by McQuarrie is a good book.

 

[peteratcam]

That’s important because average translational kinetic energy is what we’re defining when we talk about the temperature of a gas, particularly in the real world. Vibrational energy is a temporary form of internal energy that isn’t really relevant when we talk about the macroscopic properties of a gas like Temperature, Pressure and Volume; those qualities are very specifically derived from average translational kinetic energy of the molecules in the gas.

To reinforce what alxm says. The heat capacity of a substance is most certainly a macroscopic property which certainly depends on the vibrational and rotational degrees of freedom.

 

[Wilko]

Cheers guys, I am familiar (not intimately) with the concept of specific heat capacity, and that to accurately describe the total internal energy of a system, in this case, a volume of gas, you have to account for all the degrees of freedom, translational, vibrational and rotational. But "heat" is quite different to "Temperature", and I was quite specifically referring to Temperature. So the question still stands, what's the relevance of molecular vibration when it comes to calculating or measuring temperature in a gas?

Alxm, why would the average translational energy change for the whole of the gas, when illuminated with infrared radiation? Is there an asymmetry that says the vibrationally excited molecule is more likely to lose energy rather than gain it during molecular collisions? There could well be, but I haven't read about it.

The texts I have read stated that molecular collisions affect rotational levels strongly, vibrational levels weakly, and electron levels not at all. I am not denying there is an exchange of energy, just that it ought to be equally likely to occur in both directions, in which case the average amount of energy existing as translational, vibrational and radiation forms shouldn't change. If there is an asymmetry, so be it, I don't mind!

Alxm, I may not have read widely enough, but all the explanations I have read describe spectral broadening occurring due to molecular collision, I haven't seen too much on total non-radiative dissipation of the absorbed energy...just to complicate things further, what about conservation of the absorbed photon's original momentum?

It [total non-radiative dissipation of the absorbed energy] could occur, after all spontaneous emission can occur as the result of molecular collisions, but all the explanations of the process I have read so far describe absorption, excitation, small exchanges of energy in collisions, and emission with slight spectral broadening. Kind of photon in, photon out.

In any case, based on what you guys understand, do you think the "balloon" will expand?

 

[Wilko]

Alxm, gotcha, partition function, that's going to be fun. Yikes.

I already understand there's a Maxwellian distribution of particle velocities...

Whats your take on it, what "deep physical quality" are we measuring when we measure temperature (of a gas)?

Molecular Thermodynamics, will do.

 

[peteratcam]

Cheers guys, I am familiar (not intimately) with the concept of specific heat capacity, and that to accurately describe the total internal energy of a system, in this case, a volume of gas, you have to account for all the degrees of freedom, translational, vibrational and rotational. But "heat" is quite different to "Temperature", and I was quite specifically referring to Temperature. So the question still stands, what's the relevance of molecular vibration when it comes to calculating or measuring temperature in a gas?

The concept of temperature would only be useful if it is a reproducible thing. Whatever my definition of temperature, it has to be that if I measure the temperature of an isolated system now, and come back a day later and measure again, they should be the same. For this to hold, the system must have come to equilibrium. So if I zap my gas with an IR laser, then quickly stick a thermometer in, you will get some reading. After a bit, I stick the thermometer in again, and I get a different reading. I do this again a bit later, and the reading is different again, but not very much. Eventually, the reading on the thermometer stays the same.

With hindsight, my first reading wasn't really a reading of the temperature, even though it came from sticking a thermometer in and reading the scale. Only the temperature readings after a long time should be taken as 'temperature'. The other readings are non-reproducible. An operational definition of temperature to do with sticking a thermometer in and reading it, is a bad definition of temperature. The idea that translational energy is special is wrong and follows from a bad definition or understanding of temperature.

I am not denying there is an exchange of energy, just that it ought to be equally likely to occur in both directions, in which case the average amount of energy existing as translational, vibrational and radiation forms shouldn't change. If there is an asymmetry, so be it, I don't mind!

Play this games for me:
make two columns, Vib and Trans.
Write the number 100 in the column vib, and 0 in trans, like this:

Vib | Trans
100 | 0

Now toss a coin - if it is heads, subtract one from vib, and add one to trans.
If it is tails, subtract one from trans and add it to vib, and write the next row.
ONE CAVEAT: you can't let any column get lower than zero - if that happens, toss the coin again.
eg:
Vib | Trans
100 | 0
99 | 1
98 | 2
99 | 1
98 | 2
97 | 3
...
Do this for ages.

Question: Is it equally likely to shift 1 unit of energy in either direction?
Question: Is the average of the first 10 rows different from the average of the last 10?

 

[conway]

Wilko's questions are better than he is being given credit for. It is not easy to describe a mechanism whereby energy gets converted from light to heat. At least two recent threads have petered out without shedding any real light on this topic (albeit related to solids) including one last week titled "Transparency??". (In that thread, the question is turned on its head...if we normally expect light to be converted to heat when it strikes a solid, how do we explain the cases where it passes right through?)

Saying that the modes are coupled doesn't explain how they are coupled.

 

[Wilko]

Pete (hope you don't mind me taking that liberty) the coin flip idea is an interesting way to model the exchange of energy between vibrational and translational states, but I think the design could be improved a bit, forgive me if this sounds condescending. For one thing, what you've described seems to be a model of the same two molecules interacting over and over again, whereas in the gas we can expect the excited CO2 molecule to collide with many different molecules, the tallying up on the translational side seems to imply that the outcome of future unrelated collisions are affected by previous collisions, let me think it through some more. In any case, the starting translational energy, while smaller than the vibrational energy involved is still non-zero.

This "survey" is conducted from the point of view of the excited CO2 molecule.

Vib | Trans
Collision 1 (starts with the excited CO2 molecule and some starting amount of translational energy shared between the two interacting molecules)
100 | 10
99 | 11

Collision 2 (same CO2 molecule, different colliding molecule, same amount of translational energy in the interaction as last time)
99 | 10
100 | 9

Collision 3 (same excited CO2 molecule slightly less translational energy shared between the two colliding molecules, because not all molecules in the gas have the same translational energy, although it would still make sense even if we just used some averaged quantity)
100 | 8
101 | 7

And so on and so on, until the excited CO2 molecules emits a photon. I've tried to find info on the lifetime of the absorption/emission cycle and the number of collision events but haven't found much.

Then you'd do this for a large number of CO2 molecules.

More to follow on spectral broadening.

 

[Wilko]

Spectral broadening gives us some clues as to how energy is conserved and exchanged between the vibrational and translational states. I understand that in the broadest sense there's Natural Broadening, Doppler Broadening and Pressure Broadening. I'm just going to consider one aspect of Pressure Broadening.

I am grossly oversimplifying things, but in a nutshell, an excited molecule will not necessarily emit a photon of the same energy it absorbed because it can gain or lose (transform) small amounts of vibrational energy during collisions with other molecules, that's essentially the origin of Pressure Broadening. The most likely frequency/energy of the emitted photon is still that of the absorption peak. However there is a Lorentzian distribution of possible values which is symmetric around the absorption peak, this could only be possible if the excited molecules were equally able to gain or lose vibrational energy during collisions with other molecules.

 

[Wilko]

Anyway, since we're gambling with coins, whose money is on the balloon expanding? Any thoughts on the outcome of the thought experiment?

 

[Wilko]

Let me clarify something else, I'm not suggesting you can't get more "heat" moving through the gas when you illuminate it with IR; the thought experiment is very carefully designed to see if it will increase Volume or Pressure. I understand quite specifically that the definition of Temperature in the Ideal Gas Laws is different to that described by the Kinetic Theory of Gasses.