Multiple definitions for equilibrium

[burakumin]

During my studies I failed to understand thermonynamics and compared the whole subject to black magic. This frustrated me a lot. Years later I tried to restudy it by myself reading sources with less conventional approaches. I had finally come to believe I could make sense of it. But re-reading certain articles (for example on wikipedia) leave me confused even about basic concepts like thermodynamical equilibrium. I feel bit discouraged.

I can find at least two distinct definitions for thermodynamical equilibrium depending on the sources. Sometime these definitions are even introduced in the same article as implicitly equivalent. A system is in equilibrium iff:
1) it does not exchange energy or matter with its surrounding and its macroscopic properties (state functions) does not change over time;
2) it is as a uniform temperature (a situation called thermal equilibrium) a uniform pressure (mechanical equilibrium) and constant concentrations (chemical equilibrium)

Let's consider a simple system that consists of two gases in two distinct containers that have adabiatic, impermeable and unmovable boundaries between each other and with their surroundings. Let's suppose those gases are chemically different, have different temperatures and different pressures.

Is such a system in equilibrium or not ? To me it perfectly fits definition 1 (which I had always thought as the valid definition of equilibrium) but clearly fails to fullfil conditions of definition 2. So I cannot see how someone could say these definitions are equivalent

 

[PaulDirac]

It is not a simple system as you said. It consists of two distinctive systems that have nothing to do with each other. If they brought into thermal contact through a permeable wall then they'd change energy and/or matter. After a second, they would come into equilibrium and thereafter both definitions you remarked would be satisfied though. But what you just mentioned is not a system as a whole, so none of above conditions makes sense here.

 

[burakumin]

Sorry I used "simple" with a non technical meaning. I just meant "trivial", "not complicated to understand"

I may be completely out of it but I've never ever seen any connexity condition on the definition of "system", neither in thermodynamics nor in other fields of physics. To me a system was just the result of an abstract and arbitrary division of the world into two regions, the main region of interest being the system, the other the surrounding.

Now if the connexity condition is really necessary, I think I can describe other systems that I would call connex and that still match definition 1 but not definition 2 (but in a subsequent post because I think I need to make a diagram and to understand how to attach it :-) ).

To admin: I've made a mistake, my intend was to post my message in "classical physics". Would it be possible for someone to move the thread ? Thank you

 

[WannabeNewton]

You are referring to two different notions of thermodynamic equilibrium so I suspect that is your confusion. They are not equivalent. 1 is saying if system A is in equilibrium and system B is in equilibrium and the two systems interact then they will be in equilibrium with one another when their generalized forces are equal. 2 is saying a system is in equilibrium in the first place if its macrostates are uniform. As such, the part of 1 that talks about the time independence of the system's macrostates should really belong in 2 as phase space averages and time averages coincide.

It is worth noting that the statement in 1 about no exchange of particles in equilibrium is certainly not true in general systems. You can have fluctuations in particle number in a system in equilibrium with a heat bath through the grand canonical ensemble. These particle fluctuations die out only in the so called thermodynamic limit. An example where it doesn't is a gas of photons.

 

[burakumin]

Hi NewtonWannabe

I'm not sure I understand you. Are you referring to the fact that "thermodynamic equilibrium" can be used to refer to either a specific situation of a single system or an equivalence relation between systems ("to in thermodynamic equilibrium with") ?

For now I'm focusing on first case (what it means for one system to be in thermodynamic equilibrium). So according to you do I have to consider that for the system I described in my first post, only one of the definition makes sense ? Do you consider like PaulDirac that it cannot be called a (single) system ?

 

[burakumin]

To PaulDirac

As promised here is a new experiment: View attachment th2.pdf

Let's suppose I have an impermeable and unmoveable boundary box (black in the diagram). Let's suppose it is also adiabatic except on two opposite sides (dashed line) where it is in thermal contact with thermal reservoirs of respective temperatures T1 and T2. The box contains two regions separated by an adiabatic, impermeable but moveable wall (grey) and each region is filled with a gas.

The moveable wall is going to stabilize after some time, right ? In this experiment no part is isolated but still at the end the content of the box won't have a uniform temperature but will contain two subsystems with their own temperature. Am I wrong to call the content of the box a system ?

 

[WannabeNewton]

So according to you do I have to consider that for the system I described in my first post, only one of the definition makes sense ? Do you consider like PaulDirac that it cannot be called a (single) system ?

Yes.

 

[burakumin]

Sorry but I'm even more confused. In that case I don't even understand how to recognise a "system".

What is the number of "systems" in my second example (with the diagram) ?

Other situation: if i have two gases in separate boxes and i open a hole so that they mix. Do I have to consider that before the opening of the hole, I must speak of "2 systems" and after I have one system ?

 

[256bits]

hello burakumin,
A system can be anything you want it to be -your house, the human body, a rrom in a house, a part of your body such as the heart, a car, the engine in the car, one cylinder in the engine, the earth, the Earth and the moon, the atmosphere of the earth. There can be an open system, also called a control volume, a closed system, and an isolated system. Depending upon which one chosen, mass and/or energy and anything else can cross the system boundary, or not.

The system boundary, and type of system, is chosen so that the system can be analyzed, otherwise what's the point. An open system is expected to have mass and enegry transfer across the boundary, and thus equilibrium conditions are not being analyzed. A closed system does not have mass transfer across the boundary, so one can analyze, if so desired, for an equilibrium condition such as the closed system being in thermal and mechanical equilibrium with its surroundings. And internal chemical if need be.
An isolated system does not have any mass or energy transfer with its surroundings, so here one might want to analyze to a state of mechanical, thermal and chemical equilibrium, in other words a homogenous state, but not necessarily so.

So, the first system you have chosen compising the 'boxes' and the surroundings outside the sytem, seems to be in mechanical, chemical and thermal equilibrium and satisfies definition 1 and 2. Remember you did say it had adabiatic, impermeable and unmovable boundaries.The catch-22 that you face is that you are switching systems on the fly, to then asking about an analysis of the fluids within the boxes being in equilibrium, where a different answer can come about.

Do not change systems on the fly, but stick to one and only one from beginning to end.
Now, hopefully, this will help you to make sense of the two previous answers given to you.

 

[burakumin]

A system can be anything you want it to be

That is exactly what I've always thought but that is not what PaulDirac says. So either you or he is wrong.

The system boundary, and type of system, is chosen so that the system can be analyzed, otherwise what's the point

From a pragmatical point of you, I agree that we generally prefer "systems that can be analysed". However in my (maybe naive opinion) a law of physics does not care if the situations it describes can or cannot be analysed by humans. If its applicability has restrictions, the law must explicitly include them. So if the notion of thermodynamics equilibrium only apply to "connex objects" (whether they can be called systems or not), it must be stated explicitely or the statement is (once again to my very humble opinion) wrong.

So, the first system you have chosen compising the 'boxes' and the surroundings outside the sytem, seems to be in mechanical, chemical and thermal equilibrium and satisfies definition 1 and 2

I'm certainly stupid but how could it satisfy definition 2 while its temperature for example is not uniform.

The catch-22 that you face is that you are switching systems on the fly, to then asking about an analysis of the fluids within the boxes being in equilibrium, where a different answer can come about.

I'm not switching the systems. I'm considering the union of the boxes as a unique single system, which is considered incorrect by PaulDirac, but apparently not by you if "A system can be anything I want it to be". Am I missing something ?

But what you just mentioned is not a system as a whole, so none of above conditions makes sense here.

As I said I read some unconventional sources. "www.ams.org/notices/199805/lieb.pdf" is one of them. I quote:

The notion of Cartesian product, Γ1×Γ2, corresponds simply to the two (or more) systems being side by side on the laboratory table; mathematically it is just another system (called a compound system), and we regard the state-space Γ1×Γ2 as being the same as Γ2×Γ1. Points in Γ1×Γ2 are denoted by pairs (X,Y), as usual. The subsystems comprising a compound system are physically independent systems, but they are allowed to interact with each other for a period of time and thereby to alter each other’s state.

So do you consider Lieb's article is using improper terminology ?

 

[DrClaude]

I might have a slightly different perspective here, but I never like the expression "in equilibrium" by itself. I think it more proper to qualify a system as "in equilibrium with," followed by, e.g., "another system" or "the environment." You could also say that it is "in equilibrium with itself," or more properly that it is in "internal equilibrium" (which corresponds to the 2nd definition in the OP).

In the OP, I would agree that you can consider the two gases as a single system, ans there is definitely no internal equilibrium. If the partition was removed, internal equilibrium would be achieved, but would it make sense to say that the system is "in equilibirum", since it would certainly not be in equilibrium with the environment since there is no way for the system to exchange even heat with the environment.

 

[256bits]

This is what you asked in your original post. highlighting in red

Is such a system in equilibrium or not ? To me it perfectly fits definition 1 (which I had always thought as the valid definition of equilibrium) but clearly fails to fullfil conditions of definition 2. So I cannot see how someone could say these definitions are equivalent

You analyze using one, and only one, of the following system boundaries at a time.
Never switch between a chosen boundary to another in mid-course.
1. Equilibrium with the surroundings: System surrounds the outside surface of the boxes
The walls are adiabatic, meaning that there is no heat flow in out out. For this to happen the outside wall of the box(s) has to be at the temperature of the surroundings. Ergo the boxes are in thermal equilibrium with the surroundings.
Also, there are no unbalanced forces on the box(s), and there is no chemical reaction of the boxes with the environment, and no mass flowing in or out of the box(s), hence mechanical and chemical equilibrium between the box(s) and the surroundings.

In other words, one does not care what is happening within the boxes, since it is impossible to tell with the given conditions that the walls are impermeable , adiabatic, unmovable.

Definition 1 and 2 apply

2. The gases within the boxes: system surrounds the gas on the inside surface of the container
For either gas 1 or gas 2,
Again the walls are adiabatic, so the inner temperature of the wall is at the gas temperature.
Same with mechanical and chemical.
Each gas is in thermal, mechanical and chemical equilibrium with the inside of the container.

Definition 1 and 2 apply

3. Both gases: System boundary emcompasses both gases and the wall between.
( Put both boxes side by side with a common wall for explanation )
The wall is impermeable, adiabatic, and immovable, as per your conditions.
Impermeable - no mass flow
Adiabatic - no heat flow, ergo wall on gas 1 side is at temperature of gas 1
. - same for gas 2
Immovable - the wall takes up the pressure exerted by gas1 or gas 2

Definition 1 and 2 apply.

You might say wait a minute, there is a pressure differential, a temperature differential, and chemical differential between the boxes. Those differentials are taken up by the wall through the given conditions, putting everything in "equilibium".


See if you can do your 2nd example.

 

[burakumin]

I might have a slightly different perspective here, but I never like the expression "in equilibrium" by itself. I think it more proper to qualify a system as "in equilibrium with," followed by, e.g., "another system" or "the environment." You could also say that it is "in equilibrium with itself," or more properly that it is in "internal equilibrium" (which corresponds to the 2nd definition in the OP).

The distinction you are introducing is, from what I understand, the one that was previously discussed: "to be in internal equilibrium" versus "to be in equilibrium with". But I'm not sure this distinction is that important as it seems one can use one of the concept to define the other:
- A is in equilibrium with B if the system A + B is in internal equilibrium
- X is in internal equilibrium if any pair of subsystems are in equilibrium with each other

You might say wait a minute, there is a pressure differential, a temperature differential, and chemical differential between the boxes. Those differentials are taken up by the wall through the given conditions, putting everything in "equilibium".

I agree with you on the situation and on your analysis (which is reassuring me as I considered my two example to be actual cases of equilibrium). My problem is more about the phrasing of equilibrium and I cannot help but thinking that definition 2 is utterly disatisfying and has lead me to much confusion in the past. If we consider that for equilibrium uniformity of temperature (for example) only apply to what could be called "a thermally connex component of the system", a precise and careful definition should be provided before defining the equilibrium itself (for example a subpart surrounded by thermal barrier but with no internal thermal barrier). Maybe such definitions do exist but I've never found them.

See if you can do your 2nd example.

I consider that the two gases are individually in equilibrium and their union is also in equilibrium as a whole. No exchange of matter. Each of the two non-adiabatic walls has a unique temperature on both its sides (respectively T1 and T2) and the only moveable wall has stabilized so that pressures in both gases equates. Do we agree?

Anyway, thank you for help 256bits. It has been valuable. However if I could make a last global remark, the diversity of diverging opinion in this thread (in particular concerning the applicability of the concept of "system") is a bit disturbing.

 

[vortextor]

Maybe here is a hidden problem with the definition of the "system"
Instead of:
"system is a thing (or collection of)"
we are using (tacitly) a more specific one:
"system is a thing etc. for which the definitions 1 and 2 are valid"
So, something broadly circular.

 

[burakumin]

Maybe here is a hidden problem with the definition of the "system"
Instead of:
"system is a thing (or collection of)"
we are using (tacitly) a more specific one:
"system is a thing etc. for which the definitions 1 and 2 are valid"
So, something broadly circular.

I really hope you are wrong. Because I would consider this as a terrible idea.