Adiabatic Process: Solving a Thermodynamics Problem

[RingNebula57]

Hello everyone!
I've been thinking a lot about this thermodynamics problem , hearing all sorts of opinions but never getting a full rigurous explanation. So we have a cylinder that is placed in vacuum. We insert in the cylinder a monoatomic ideal gas. The gas is trapped inside the cylinder with a piston as shown in the drawing below. The cylinder and the piston are adiabatically isolated.( no heat can escape or enter) Because of the gravitational field and the mass of the piston the gas will remain in the cylinder(in thermodynamic equllibrium). Let's say that the piston has mass m.
Now consider sticking on the piston a ball of mass M>m. Because of the weight of the ball the gas will contract (NOT QUASISTATIC), the piston going down. After this the gas will expand the piston going up, so on and so forth. Will the oscillations of the piston stop? (in other words , will the system ever reach thermodyamic equillibrium)
The interesting thing about this problem is that we can't apply the adiabatic process formula(because the process is not quasistatic) $$P\cdot V^{\gamma}=const$$ . Is sort of like free expansion, where we don't have a defined thermodynamic path in the PV plane.
What do you think?

 

[.Scott]

There are three possible avenues for the oscillations to become dampened: the viscosity of the fluid, heat transfer to or from the gas with the walls of the piston and cylinder, and friction between the piston and the walls of the cylinder.

In your problem statement, you have eliminated the first two of those three. Now that I've mentioned this third possibility (piston friction), I am guessing you'll want to eliminate that one as well.

That would leave with a system that oscillates forever. ... or should I mention possible tidal effects between the moving piston and a potentially inelastic source of the gravity?

 

[Chestermiller]

Damping by gas viscosity is not elominated. In fact, for a frictionless piston, it is the only mechanism.

 

[.Scott]

Damping by gas viscosity is not elominated. In fact, for a frictionless piston, it is the only mechanism.

He specified an ideal gas. So there is no viscosity.

 

[Chestermiller]

He specified an ideal gas. So there is no viscosity.

Says who?

 

[.Scott]

Says who?

The OP: "We insert in the cylinder a monoatomic ideal gas." ... which I suppose is a contradiction in terms. An ideal gas isn't really "atomic".

 

[Chestermiller]

The OP: "We insert in the cylinder a monoatomic ideal gas." ... which I suppose is a contradiction in terms. An ideal gas isn't really "atomic".

Who says an ideal gas has zero viscosity?

 

[.Scott]

Who says an ideal gas has zero viscosity?

It's made up of points that have only elastic collisions. If it's viscous, by what mechanism? I suppose you could have an ideal gas where the points had some sort of attraction to one another - but would that still fit the definition of an ideal gas? I don't think so.

Here's the Google definition: "a hypothetical gas whose molecules occupy negligible space and have no interactions, and that consequently obeys the gas laws exactly"

Here's the Wikipedia definition: "An ideal gas is a theoretical gas composed of many randomly moving point particles whose only interaction is perfectly elastic collision."

 

[Nidum]

http://physics.stackexchange.com/questions/310806/viscosity-of-ideal-gas-from-dimensional-analysis

https://www.slideshare.net/Aapandove/bird-stewart-lightfoot-2002-transport-phenomena-2nd-ed

 

[Chestermiller]

It's made up of points that have only elastic collisions. If it's viscous, by what mechanism? I suppose you could have an ideal gas where the points had some sort of attraction to one another - but would that still fit the definition of an ideal gas? I don't think so.

Here's the Google definition: "a hypothetical gas whose molecules occupy negligible space and have no interactions, and that consequently obeys the gas laws exactly"

Here's the Wikipedia definition: "An ideal gas is a theoretical gas composed of many randomly moving point particles whose only interaction is perfectly elastic collision."

Well, thank you for those various definitions of an ideal gas.

If you want to learn about the mechanisms determining viscosity of gases in the limit of low pressures (i.e., the ideal gas region), see Transport Phenomena by Bird, Stewart, and Lightfoot, Sections 1.3 and 1.4. In section 1.3, a corresponding states plot of reduced viscosity as a function of reduced temperature and reduced pressure is presented. The contour indicated as "low density limit" is a function only of temperature, and represents the behavior in the ideal gas limit. Section 1.4 is entitled Molecular Theory of the Behavior of Gases at Low Density (i.e., the limit of ideal gas behavior). It quantitatively models the viscosity of gases in the ideal gas region using elementary kinetic theory of gases. It then discusses the more rigorous Chapman Enskog theory, based on the Leonard Jones 6-12 potential. It presents tables for applying the results of the Chapman Enskog theory to gases in the ideal gas limit.

 

[.Scott]

Well, thank you for those various definitions of an ideal gas.

If you want to learn about the mechanisms determining viscosity of gases in the limit of low pressures (i.e., the ideal gas region), see Transport Phenomena by Bird, Stewart, and Lightfoot, Sections 1.3 and 1.4. In section 1.3, a corresponding states plot of reduced viscosity as a function of reduced temperature and reduced pressure is presented. The contour indicated as "low density limit" is a function only of temperature, and represents the behavior in the ideal gas limit. Section 1.4 is entitled Molecular Theory of the Behavior of Gases at Low Density (i.e., the limit of ideal gas behavior). It quantitatively models the viscosity of gases in the ideal gas region using elementary kinetic theory of gases. It then discusses the more rigorous Chapman Enskog theory, based on the Leonard Jones 6-12 potential. It presents tables for applying the results of the Chapman Enskog theory to gases in the ideal gas limit.

Does that mean that it is possible for an ideal gas to not be an ideal fluid?

I think part of the problem is that by definition, the ideal gas is made up of point particles that only interact by elastic collision. By that definition, there would never be any interaction among the particles because there would never be any collisions. It would also mean that as the piston dropped, only the particles that actually collided with that moving wall would experience the "heating".

It also means that the compression force would not be communicated to the walls - only to the floor of the cylinder. Hmmm.. we also need to consider a collision between one of these ideal gas particles and the purely adiabatic cylinder wall. Does that collision have to be perfectly elastic - or are we allowed to emit a photon? But if we emit a photon, where will it go. If it interacts with another point, it would violate the ideal gas definition. Sounds like photons are out - or they are in and they are the points in the ideal gas.

Considering only the Y axis, since each particle is communicating the force of the piston to the floor of the cylinder using only elastic interactions, it's hard to imagine the net effect being anything other the elastic.

 

[Chestermiller]

Does that mean that it is possible for an ideal gas to not be an ideal fluid?

I think part of the problem is that by definition, the ideal gas is made up of point particles that only interact by elastic collision. By that definition, there would never be any interaction among the particles because there would never be any collisions. It would also mean that as the piston dropped, only the particles that actually collided with that moving wall would experience the "heating".

It also means that the compression force would not be communicated to the walls - only to the floor of the cylinder. Hmmm.. we also need to consider a collision between one of these ideal gas particles and the purely adiabatic cylinder wall. Does that collision have to be perfectly elastic - or are we allowed to emit a photon? But if we emit a photon, where will it go. If it interacts with another point, it would violate the ideal gas definition. Sounds like photons are out - or they are in and they are the points in the ideal gas.

Considering only the Y axis, since each particle is communicating the force of the piston to the floor of the cylinder using only elastic interactions, it's hard to imagine the net effect being anything other the elastic.

Why don't you read the sections in Bird et al and see if they address your issues satisfactorily? Then get back with us.

 

[.Scott]

Why don't you read the sections in Bird et al and see if they address your issues satisfactorily? Then get back with us.

I did look up "Chapman Enskog theory" and a couple of other terms, but at this point I don't plan on buying any textbooks. Besides, this wasn't "my issue" until you brought them up. I interpreted that as a suggestion that an ideal gas may not be an ideal fluid - but you haven't claimed that. What I did read was about the "ideal gas limit". At this point, I am willing to presume that a gas model was created that in the limit would give you an ideal gas - and that in the limit condition for that model, you may have viscosity. But I would not infer from that that the ideal gas has a viscosity - especially when no mechanism is evident.

That said, you have twice suggested that I read that material. And I take that as a strong indication that I may find value in reading it. If you further assert that this material demonstrates that an ideal gas can have viscosity, that may be enough to put this onto my crowded reading list.

 

[Chestermiller]

I did look up "Chapman Enskog theory" and a couple of other terms, but at this point I don't plan on buying any textbooks. Besides, this wasn't "my issue" until you brought them up. I interpreted that as a suggestion that an ideal gas may not be an ideal fluid - but you haven't claimed that. What I did read was about the "ideal gas limit". At this point, I am willing to presume that a gas model was created that in the limit would give you an ideal gas - and that in the limit condition for that model, you may have viscosity. But I would not infer from that that the ideal gas has a viscosity - especially when no mechanism is evident.

That said, you have twice suggested that I read that material. And I take that as a strong indication that I may find value in reading it. If you further assert that this material demonstrates that an ideal gas can have viscosity, that may be enough to put this onto my crowded reading list.

I think what we are dealing with here is a question of terminology. In engineering, we are taught that, in the limit of low pressures, an ideal gas and a real gas coincide in every aspect of their behavior, including viscosity and temperature-dependence of heat capacity. Otherwise, what is the point of defining and applying the ideal gas model to real physical systems. I am beginning to feel that the entity that is called an ideal gas in Physics is more restrictive, and does not allow for either viscosity or temperature-dependence of heat capacity. So it really doesn't approximate real gas behavior at low pressures in two very important aspects, and would predict, for example, that, in the present problem, the gas deformations and piston oscillations would continue forever (like a spring-mass system in SHM), rather than being damped out by viscosity.

Regarding Bird, Stewart, and Lightfoot, Transport Phenomena, I highly recommend this book. I think you would be very pleased with it. During my engineering career, I referred to this book more than all my other texts combined.

 

[.Scott]

I think what we are dealing with here is a question of terminology. In engineering, we are taught that, in the limit of low pressures, an ideal gas and a real gas coincide in every aspect of their behavior, including viscosity and temperature-dependence of heat capacity. Otherwise, what is the point of defining and applying the ideal gas model to real physical systems. I am beginning to feel that the entity that is called an ideal gas in Physics is more restrictive, and does not allow for either viscosity or temperature-dependence of heat capacity. So it really doesn't approximate real gas behavior at low pressures in two very important aspects, and would predict, for example, that, in the present problem, the gas deformations and piston oscillations would continue forever (like a spring-mass system in SHM), rather than being damped out by viscosity.

I think that is probably the best response to the OP.