Application of the first law of thermodynamics

[Squizzie]

Do you think I did?

I'm just asking a simple question about a detail about your example which is quite significant to the understanding of your explanation.

 

[Chestermiller]

I'm just asking a simple question about a detail about your example which is quite significant to the understanding of your explanation.

It is an ideal gas according to the definition of an ideal gas used by chemical engineers.

 

[vanhees71]

Statistical Mechanics?? That applies only to thermodynamic equilibrium conditions, and we have been considering here irreversible processes in which the system passes through non-equilibrium states (not described by statistical mechanics). How does statistical mechanics treat irreversible processes. And when molecular simulation is done, this has to take into consideration molecular interactions. But, as you said (which I disagree with), An ideal gas is defined to be a system of particles that do not interact." So how can statistical mechanics deal with particles that do not interact (and yet describe an irreversible process for such a system)?

I think, Boltzmann would be pretty sad, hearing this. Of course statistical mechanics is by no means restricted to thermostatics.

 

[Chestermiller]

I think, Boltzmann would be pretty sad, hearing this. Of course statistical mechanics is by no means restricted to thermostatics.

So how would Boltzmann have handled a specific problem involving an irreversible process between an initial and final thermodynamic equilibrium state. Please provide an example with details.

 

[vanhees71]

You mean the problem described in #1? I guess, he'd solve the equation of motion for $m$ together with the AC circuit problem. Then the total energy consumed in the resistor is transferred to the water as heat. I don't see, where in this entire problem an ideal gas occurs in the first place.

 

[Chestermiller]

You mean the problem described in #1? I guess, he'd solve the equation of motion for $m$ together with the AC circuit problem. Then the total energy consumed in the resistor is transferred to the water as heat. I don't see, where in this entire problem an ideal gas occurs in the first place.

I think squizzle and I were referring to the irreversible gas expansion problem in posts #16 and forward.

 

[Chestermiller]

@Squizzle You indicate that you are skeptical about the Chemical Engineers' definition of an ideal gas being taken as the limiting behavior of a real gas at low pressures. This includes approaching a non-zero viscosity that is a function only on temperature. I submit that this is a more realistic definition of an ideal gas than the more restrictive definition used by physicists where the ideal gas viscosity is taken as zero. I also submit that the inviscid definition of an ideal gas is not capable of properly analyzing the Joule Thomson effect of an ideal gas flowing through a porous plug with a higher pressure on the upstream side of the plug and an lower pressure on the downstream side of the plug.

Have you ever had a course in fluid mechanics, including viscous Newtonian gases and liquids? I suggest that it is time for you to start learning about this.

 

[Squizzie]

@Squizzle You indicate that you are skeptical about the Chemical Engineers' definition of an ideal gas

My scepticism arises from my inability, despite researching a number of texts_[1][2][3], to identify a Chemical Engineers' definition of an ideal gas that differs from the classical physics definition quoted above.

[1] Smith J. M. (1970) Chemical Engineering Kinetics, Mcgraw Hill
[2] Denn M. M. (2012) Chemical Engineering an Introduction, Cambridge University Press
[3] Backhurst J.R and Harker J. H (2001) Chemical Engineering, Butterworth-Heinemann

 

[Chestermiller]

My scepticism arises from my inability, despite researching a number of texts_[1][2][3], to identify a Chemical Engineers' definition of an ideal gas that differs from the classical physics definition quoted above.

[1] Smith J. M. (1970) Chemical Engineering Kinetics, Mcgraw Hill
[2] Denn M. M. (2012) Chemical Engineering an Introduction, Cambridge University Press
[3] Backhurst J.R and Harker J. H (2001) Chemical Engineering, Butterworth-Heinemann

Transport Phenomena is a book that has stood the test of time, written by the department head and two prominant professors from the chemical engineering department at the university of Wisconsin.

 

[Chestermiller]

Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics: Chapter 2, page 40: "The oscillations of the piston assembly are damped out because the viscous nature of the gas gradually converts gross direct motion of the molecules into chaotic molecular motion. This dissipative process transforms for of the World initially done by the gas in accelerating the piston back into internal energy of the gas. Once the process is initiated, no infinitesimal change in external conditions can reverse its direction; the process is irreversible."

Chapter 2, page 64: "In engineering calculations, gases at pressures up a few bars may often be considered ideal."

 

[Squizzie]

Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics: Chapter 2, page 40: "The oscillations of the piston assembly are damped out because the viscous nature of the gas gradually converts gross direct motion of the molecules into chaotic molecular motion. This dissipative process transforms for of the World initially done by the gas in accelerating the piston back into internal energy of the gas. Once the process is initiated, no infinitesimal change in external conditions can reverse its direction; the process is irreversible."

The example quoted is using "a gas". Not an "ideal gas" as is made abundantly clear in the paragraph following:
"All processes carried out in finite time with real substances are accompanied in some degree by dissipative effects of one kind or another, and all are therefore irreversible." (my emphasis)

 

[Chestermiller]

The example quoted is using "a gas". Not an "ideal gas" as is made abundantly clear in the paragraph following:
"All processes carried out in finite time with real substances are accompanied in some degree by dissipative effects of one kind or another, and all are therefore irreversible." (my emphasis)

Well, what good does it do to use a mathematical model of a substance that does not capture simple the first order picture of how the substance behaves in actual physical situations. Saying that the piston oscillates forever when we know that it would not is just silly, especially when we can easily calculate what the final steady state would be when the oscillation is damped out . Saying that, in the expansion of a gas into half a chamber initially under vacuum, the mechanism for the temperature remaining constant is not related to viscous forces is likewise silly. And in Joule Thomson process, flowing a gas through a porous plug, an inviscid gas model would result in no pressure change while we know that the pressure change in the porous plug is the result of viscous forces; so neglecting viscous stresses would prevent us from modeling the Joule Thomson effect; there would be no Joule Thomson effect without gas viscosity. Saying that the Joule Thomson coefficient for an ideal gas is zero could not be done because the pressure drop being zero would make the Joule Thomson coefficient 0/0.

It all boils down to what you want to use as the definition of an ideal gas. I contend that considering an ideal gas to have viscosity (as real gases have in the limit of low pressures) allows simple calculations and explanations of gas behavior at low pressures which cannot be done with an inviscid ideal gas model.

 

[Squizzie]

Well, what good does it do to use a mathematical model of a substance that does not capture simple the first order picture of how the substance behaves in actual physical situations. Saying that the piston oscillates forever when we know that it would not is just silly, especially when we can easily calculate what the final steady state would be when the oscillation is damped out .

Yes, it would be silly to say that. But that is not what is being said. What I am suggesting is that in the idealised world of frictionless, massless, perfectly insulated cylinders, ideal gas etc., the piston would oscillate forever.
The analysis of such a system can provide an insight into the fundamental properties of pressure, temperature, mass, momentum, energy and, incidentally, the fundamental differences between the various states of matter.
Its practical application has contributed immeasurably to the solution of uncountable practical engineering challenges of the modern industrial and technological world.
I'm sure Boyle, Charles, Gay Lussac, Kelvin, Joule and all the teachers of thermodynamics, and indeed of science generally, would be disappointed to hear you say that this methodology was silly.

Incidentally, you would also have to allow the cylinder's insulation to be less than ideal to allow the "real" viscosity to be able to damp the oscillations.

Thank you for clarifying the real nature of the gas in the experiment.
[EDIT] I know it's a subjective view, but I would suggest the issues of friction, sealing and insulation would rate higher than the minute influence of viscosity as first order omissions from reality.

 

[Chestermiller]

Yes, it would be silly to say that. But that is not what is being said. What I am suggesting is that in the idealised world of frictionless, massless, perfectly insulated cylinders, ideal gas etc., the piston would oscillate forever.
The analysis of such a system can provide an insight into the fundamental properties of pressure, temperature, mass, momentum, energy and, incidentally, the fundamental differences between the various states of matter.
Its practical application has contributed immeasurably to the solution of uncountable practical engineering challenges of the modern industrial and technological world.
I'm sure Boyle, Charles, Gay Lussac, Kelvin, Joule and all the teachers of thermodynamics, and indeed of science generally, would be disappointed to hear you say that this methodology was silly.

Incidentally, you would also have to allow the cylinder's insulation to be less than ideal to allow the "real" viscosity to be able to damp the oscillations.

Thank you for clarifying the real nature of the gas in the experiment.
[EDIT] I know it's a subjective view, but I would suggest the issues of friction, sealing and insulation would rate higher than the minute influence of viscosity as first order omissions from reality.

Would you also suggest that, if indeed your gas is approximated as having zero viscosity, your assumption that the temperature, pressure, and density of the ideal gas in your cylinder are spatially uniform (as the piston oscillates forever) is valid? When you suddenly release your massless frictionless piston from rest, do you think that pressure-, temperature-, and density waves within the gas will develop that propagate in the axial direction along the cylinder?

Let me guess...you're not an engineer, right.

 

[Squizzie]

Let me guess...you're not an engineer, right.

An astute observation. Neither am I a biologist, architect, surgeon, or lawyer. My first love and academic qualification is physics, which is what drew me to this forum.

 

[Frabjous]

My first love and academic qualification is physics, which is what drew me to this forum.

As a physicist, what do you think that you have contributed to the 2 thermodynamic threads that you have been replying to?
Most physicists would be thrilled to learn of ways to generalize the ideal gas law to increase is applicability. You seem more interested in textbook definitions than the phenomena itself.

 

[vanhees71]

I highly recommend Landau+Lifshitz vol. 5 (Stat. Phys. 1) and vol. 6 (hydrodynamics). There you get the physicists' point of view. An ideal gas is, as the name suggests, an idealization. It describes situations of fluid, where the changes are so slow that you can assume that it's always instantaneously in local (!) thermal equilibrium, i.e., there's no heat transfer between fluid cells. This means there's no friction and no dissipation. The result is the perfect-fluid (Euler) equation for hydro, and that's for sure an idealization. In reality you have dissipation. In non-relativistic physics you can describe this in the next step of the approximation by viscous hydrodynamics (Navier-Stokes equation).

This can be studied from the next more fundamental level by employing transport equations like the Boltzmann equation. Hydrodynamics can be derived as an effective description for the gas staying close to local thermal equibrium. At 0th order you get perfect-fluid hydro. There you neglect the collision term completely, because you assume that the fluid cells are instantaneously in thermal equilibrium. It turns out that this neglects all kinds of irreversibility, i.e., you have an adiabatic equation of state. The next step is to take into account deviations from local thermal equilibrium at the linear order. This leads to the occurrence of transport coefficients, i.e., the bulk and shear viscosity and the Navier-Stokes equation.

That's usually sufficient for the non-relativsitic case. In the relativistic case you run into trouble, when doing this naively. For a long time one thus has believed that one needs at least a 2nd-order hydro. Using the relaxation-time approximation for the collision term in the Boltzmann equation you are lead to Israel-Steward 2nd-order hydrodynamics, which is relativistically consistent. In very recent years, it has been figured out, that also a viable first-order Navier-Stokes-like relativistic hydrodynamics is possible by exploiting all kinds of "matching conditions" consistently.

For a detailed introduction to the relativistic case, see

G. S. Denicol and D. H. Rischke, Microscopic Foundations of
Relativistic Fluid Dynamics, Springer, Cham (2021),
https://doi.org/10.1007/978-3-030-82077-0