Temperature and degrees of freedom (split from Hypothesis about temperature)

[KJW]

On 3/18/2024 at 10:35 AM, sethoflagos said:

The 'independence' you claim relies on the fact that statistically the rotational and vibrational transfers average out to zero over a sufficiently large number of collisions.

Yes, pressure and temperature are statistical quantities. I never said that non-translational motion had no effect on each individual collision. But the universality of the ideal gas law is a testament to the statistical independence of pressure and temperature on non-translational motion.

 

 

On 3/16/2024 at 9:23 AM, sethoflagos said:

Quote

One other thing: In the ideal gas law, pressure, which is proportional to temperature

Only in one very special case: that of a theoretical constant volume thermodynamic process.

Why are you saying that constant volume is a theoretical very special case?

 

 

On 3/16/2024 at 9:23 AM, sethoflagos said:

More general cases show P, T dependence involving the ratio of specific heats, (~7/5 for diatomic gases) which per force requires consideration of all available thermal degrees of freedom, not just the three translational ones.

Can you elaborate on this?

 

 

Edited March 19 by KJW

[sethoflagos]

1 hour ago, KJW said:

Yes, pressure and temperature are statistical quantities. I never said that non-translational motion had no effect on each individual collision. But the universality of the ideal gas law is a testament to the statistical independence of pressure and temperature on non-translational motion.

This simply doesn't follow. The IDE is not universal, particularly for higher pressures, and says nothing of the nature of particla collisions. A bit of algebraic rearrangement gives:

C_VT = C_PT - PV which is equivalent to U = Q + W.

This demonstrates that it's simply a convenient reformulation of the First Law. Obviously, C_V and C_P include all available thermal degrees of freedom on an equivalent footing.

1 hour ago, KJW said:

Why are you saying that constant volume is a theoretical very special case?

All practical thermal processes involve bulk expansion or compression. A truly constant volume operation would be rather difficult to achieve in practice.

1 hour ago, KJW said:

Can you elaborate on this?

Gas processes operate within a spectrum that ranges between two idealised endpoints: the isothermal (PV = constant) and the isentropic (PV^k = constant) where k is the ratio of specific heats C_P/C_V. As a general rule, the faster a process occurs, the more nearly it approaches the isentropic endpoint. This is a strong indicator that all thermal degrees of freedom are immediately available to momentum exchange.

Everybody needs some kind of mental picture to help get their heads around such physical processes, and as you've pointed out, equipartition allows estimation of pressure and temperature from consideration of linear momentum alone. But to extrapolate from this an independence of rotational and vibrational modes is to confuse correlation with causation I think. At least it rattles the mental pictures others have which may be no bad thing, but you're going to get some kickback for that.   

[KJW]

On 3/20/2024 at 12:12 AM, sethoflagos said:

On 3/19/2024 at 9:35 PM, KJW said:

Yes, pressure and temperature are statistical quantities. I never said that non-translational motion had no effect on each individual collision. But the universality of the ideal gas law is a testament to the statistical independence of pressure and temperature on non-translational motion.

This simply doesn't follow. The IDE is not universal, particularly for higher pressures

By "universality", I meant universal with respect to different gases, including gases that differ in their heat capacity. I had already said that I view the ideal gas law as a low-pressure limit, so "higher pressures" don't apply. And even if one does consider the van der Waals equation, the difference compared to the ideal gas law are non-zero volume of the molecules and intermolecular forces. Neither of these relate to the rotations of molecules or vibrations within molecules.

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

says nothing of the nature of particle collisions.

Why are you fixated on particle collisions? Particle collisions are irrelevant to the point I was making.

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

 

A bit of algebraic rearrangement gives:

C_VT = C_PT - PV which is equivalent to U = Q + W.

A bit of algebraic rearrangement from what?

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

A truly constant volume operation would be rather difficult to achieve in practice.

We are talking about gases... simply place it in a sealed borosilicate glass flask.

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

All practical thermal processes involve...

...

Gas processes operate...

Why are you mentioning "processes"? Processes, are also not relevant to the point I was making.

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

This is a strong indicator that all thermal degrees of freedom are immediately available to momentum exchange.

This seems to be where the error in your interpretation of what I said occurs. I never said that energy doesn't transfer between all the degrees of freedom. In particular, I never said that energy doesn't transfer between translational and non-translational degrees of freedom. I also never said that if the temperature is changed, that this only affects the translational degrees of freedom. What I did say is this:

On 3/16/2024 at 12:23 AM, KJW said:

One thing should be mentioned: Only the translational modes of molecular motion contribute to the temperature. Different substances have different heat capacities because they absorb energy in all their modes, but only the translational modes increase the temperature, thus the more modes that are available to the molecule, the more energy that is absorbed for a given increase in temperature.

That the same ideal gas law applies to argon, nitrogen, carbon dioxide, water, and ethane proves the point I was making. That all these gases have different heat capacities, particularly at higher temperatures, also proves the point I was making.

 

 

On 3/20/2024 at 12:12 AM, sethoflagos said:

But to extrapolate from this an independence of rotational and vibrational modes is to confuse correlation with causation I think.

I never mentioned "causation". The word I used was "contribute". Ultimately, what I said rests on a definition of "temperature". That's why I mentioned a gas thermometer earlier. Thermometers measure temperature and so must have a definition of temperature in their design. And gas thermometers are based on the ideal gas law, thus defining temperature in terms of the translational motion only.

 

 

[sethoflagos]

On 3/22/2024 at 11:44 AM, KJW said:

 

On 3/19/2024 at 3:12 PM, sethoflagos said:

 

A bit of algebraic rearrangement gives:

C_VT = C_PT - PV which is equivalent to U = Q - W.

A bit of algebraic rearrangement from what?

Case under consideration (for illustrative purposes), constant pressure heating/cooling of a substance for which internal energy is a function of temperature only.

PV = RT                                     (per kilomole basis)

By Mayer's relation R = C_P - C_V

PV = (C_P - C_V)T

Take partial derivatives and substitute appropriate values for constant pressure process,

VdP + PdV = C_PdT_P - C_VdT_V 

 0     +  dW  =   dQ   -   dU

Hence:

dU = dQ - dW

Obviously, for a constant pressure process, P is anything but proportional to T.

On 3/22/2024 at 11:44 AM, KJW said:

What I did say is this:

On 3/15/2024 at 3:23 PM, KJW said:

One thing should be mentioned: Only the translational modes of molecular motion contribute to the temperature. Different substances have different heat capacities because they absorb energy in all their modes, but only the translational modes increase the temperature, thus the more modes that are available to the molecule, the more energy that is absorbed for a given increase in temperature.

That the same ideal gas law applies to argon, nitrogen, carbon dioxide, water, and ethane proves the point I was making. That all these gases have different heat capacities, particularly at higher temperatures, also proves the point I was making.

Per the above, your 'proof' seems merely a tedious repetition of the patently false.

 

On 3/22/2024 at 11:44 AM, KJW said:

 

On 3/19/2024 at 3:12 PM, sethoflagos said:

A truly constant volume operation would be rather difficult to achieve in practice.

We are talking about gases... simply place it in a sealed borosilicate glass flask.

Last time I checked, even borosilicate glass had a finite Young's modulus and non-zero thermal expansion coefficient.

 

On 3/22/2024 at 11:44 AM, KJW said:

 

On 3/19/2024 at 3:12 PM, sethoflagos said:

says nothing of the nature of particle collisions.

Why are you fixated on particle collisions? Particle collisions are irrelevant to the point I was making.

What do temperature and pressure even mean when there are no particle inteeractions to communicate them?

On 3/22/2024 at 11:44 AM, KJW said:

Why are you mentioning "processes"? Processes, are also not relevant to the point I was making.

In context, it's any defined path between different thermodynamic states. Such as changes in temperature and pressure. Or is your conjecture also confined to conditions of thermodynamic equilibrium only?

 

Edited March 25 by sethoflagos
grammar

[KJW]

15 hours ago, sethoflagos said:

your 'proof' seems merely a tedious repetition of the patently false.

Perhaps you can state precisely where I have made my error. I believe that what I have said has been misinterpreted, and I seem to be unable to get my interpretation across. I would find it instructive to know precisely what you think I am saying.

 

 

15 hours ago, sethoflagos said:

Or is your conjecture also confined to conditions of thermodynamic equilibrium only?

I don't regard my statement as a "conjecture", and I am quite surprised by the controversy it has generated. But yes, what I said assumes thermodynamic equilibrium because it is only under conditions of thermodynamic equilibrium that a proper definition of temperature is assured. That's not to say that what I said can't be applied to broader conditions, but that was never my intention.

 

Edited March 26 by KJW

[sethoflagos]

2 hours ago, KJW said:

Perhaps you can state precisely where I have made my error. I believe that what I have said has been misinterpreted, and I seem to be unable to get my interpretation across. I would find it instructive to know precisely what you think I am saying.

Rather than state them, I'll quote them:

On 3/15/2024 at 3:23 PM, KJW said:

One thing should be mentioned: Only the translational modes of molecular motion contribute to the temperature. Different substances have different heat capacities because they absorb energy in all their modes, but only the translational modes increase the temperature, thus the more modes that are available to the molecule, the more energy that is absorbed for a given increase in temperature.

If you'd stated that this observation was limited to the monatomica gases and low temperature hydrogen, then no one would have batted an eyelid. However, as presented it appeared that you intended this as definitive and universal. 

When this idea was immediately blown out of the water by @exchemist's query about the temperature of solids, you might have considered retracting. But instead you shifted your ground to the following argument:   

On 3/15/2024 at 11:42 PM, KJW said:

One other thing: In the ideal gas law, pressure, which is proportional to temperature, depends only on the translational motion of the gas molecules. Neither rotational motion, nor vibrational motion affects the pressure and therefore the temperature of an ideal gas.

With the sole exception of the theoretical ideal constant volume process, this isn't even true for the monatomic gases. So far from strengthening your case, it actually weakens it.

3 hours ago, KJW said:

I don't regard my statement as a "conjecture", and I am quite surprised by the controversy it has generated.

It certainly comes across as an article of faith. Hence the degree of kickback perhaps. 

The only instance I can think of where thermodynamics treats translational and internal degrees of freedom differently is in the theoretical analysis of diffusivity coefficients (both thermal and mass). Which sort of makes sense since diffusion is hardly likely to happen without some translational motion. Not particularly relevant to my own areas of study, but if you're interested they get a mention in:

https://en.wikipedia.org/wiki/Thermal_conductivity_and_resistivity

https://en.wikipedia.org/wiki/Mass_diffusivity