Show that entropy is a state function

[Chestermiller]

I've started thinking about the mechanism of entropy generation in your example and also for free expansion.
In your example the upper plate is moved against the friction due to the viscosity of the liquid. The work that's done is converted to inner energy and increases the temperature of the liquid and generates entropy.

This is correct. There is a mechanistic difference between free expansion of an ideal (compressible) gas and the example I gave of doing work on an incompressible liquid. But viscous dissipation plays a role in both cases.

In free expansion there's no work being done from the outside, no heat is transferred.
The average kinetic energy of these molecules is unchanged since the inner energy is constant (at least for an ideal gas).
When the gas expands it does no work on the surroundings.
The only thing that really happens is that molecules fly around chaotically and move further from each other.
For a real gas there's also viscous friction between the molecules, is that right?

An ideal gas is the limiting case of a real gas at low density, and, even at low density, there are significant numbers of collisions between molecules. (For example, check out the mean free path of air molecules at 1 bar, clearly in the ideal gas region). It is collisions and associated energy transfer between molecules that are responsible for viscous effects. So even an ideal gas exhibits viscous dissipation and is not inviscid. See Bird, Stewart, and Lightfoot for a graph showing the viscosity of gases in the ideal gas limit.

For an ideal gas: There is no potential energy and also no friction between the molecules. The only reason I can see for the entropy increase is the expansion.

As I said, even for an ideal gas there is viscous friction. In the case of free expansion of an ideal gas, even though the gas does no work on its container, the parcels of gas still expand against each other, and this would ordinarily result in expansion cooling. However, because of viscous dissipation in the gas, the expansion cooling is exactly canceled by viscous heating, so there is no net effect on the internal energy of the gas and its temperature. And the viscous heating is ultimately responsible for the entropy increase (since no entropy enters the gas through the insulated boundary of the container).

 

[Chestermiller]

Yes, that would be interesting.

We have the same incompressible viscous fluid as before, being sheared between two infinite parallel plates, but, in this case, rather than both plates being insulated, the lower plate at y = 0 is held at constant temperature To. The system attains a steady state temperature profile and the entropy of the liquid becomes constant. However, entropy is being generated within the fluid both by viscous dissipation and by heat conduction, and this generated entropy is being removed at the lower plate by heat transfer. We are going to solve for the rate of entropy generation and show that it is equal to the rate of entropy removal.

The differential thermal energy balance for this system is described by $$0=k\frac{d^2T}{dy^2}+\mu\left(\frac{V}{h}\right)^2$$To obtain the differential entropy balance equation for the system, we divide this equation by the local absolute temperature T (at arbitrary y), and then do exactly the same mathematical manipulation with the heat conduction term that I did in post #27. What do you obtain when you do this?

 

[Chestermiller]

The only reason I can see for the entropy increase is the expansion.

This is looking at the wrong side of the equation. The volume expansion term $R\ln{(V_2/V_1)}$ is part of the entropy change effect, not the cause of the entropy change. For a closed system, there are only two physical mechanisms that can result in an entropy change during a process: 1. entropy transport across the boundary of the system from heat flow and 2. entropy generation within the confines of the system as a result of viscous dissipation and finite internal heat fluxes. For a reversible process, only the first mechanism is present.

 

[Philip Koeck]

This is looking at the wrong side of the equation. The volume expansion term $R\ln{(V_2/V_1)}$ is part of the entropy change effect, not the cause of the entropy change. For a closed system, there are only two physical mechanisms that can result in an entropy change during a process: 1. entropy transport across the boundary of the system from heat flow and 2. entropy generation within the confines of the system as a result of viscous dissipation and finite internal heat fluxes. For a reversible process, only the first mechanism is present.

I'm not sure what to think about that.
There's the textbook example of calculating entropy change in a free expansion using Boltzmann's formula.
To me that looks like the volume increase, which is proportional to the increase in microstates, is treated as the cause of entropy increase.
Another example that comes to mind is the mixing of two different gases at the same pressure and temperature, which leads to an increase in entropy.
On the other hand, if the gases are the same the entropy doesn't increase although at a molecular level the same thing happens: Molecules fly around collide with each other.
Shouldn't an increase in available space be considered a cause of entropy increase?

 

[Chestermiller]

I'm not sure what to think about that.
There's the textbook example of calculating entropy change in a free expansion using Boltzmann's formula.
To me that looks like the volume increase, which is proportional to the increase in microstates, is treated as the cause of entropy increase.
Another example that comes to mind is the mixing of two different gases at the same pressure and temperature, which leads to an increase in entropy.
On the other hand, if the gases are the same the entropy doesn't increase although at a molecular level the same thing happens: Molecules fly around collide with each other.
Shouldn't an increase in available space be considered a cause of entropy increase?

I'm not familiar with the analysis of free expansion using the Boltzmann formula. But somehow that approach must be consistent with the continuum approach I presented based on viscous dissipation. That probably happens indirectly through the application of gas kinetic theory to derive the gas viscosity.

The mixing of two different gases at the same pressure and temperature involves another entropy generation mechanism that I didn't yet mention (to keep things simple). That mechanism is molecular diffusion, in which the rate of entropy generation is proportional to the square of the concentration gradients of the gases. This is also addressed in Transport Phenomena, but in a later chapter.

As far as an increase in available space being considered a cause of entropy increase, that is not the case if the expansion takes place adiabatically and reversibly (where there is no entropy change). In addition, none of what you are indicating could be considered a cause of entropy change for an irreversible path, since they all exclusively involve looking at alternate reversible paths (for which no entropy generation occurs). What I'm talking about is determining the entropy change directly for an irreversible process without indirectly invoking reversible paths. After all, the original question was related to entropy change as a state property even if the change occurs irreversibly. To understand how that can happen, one first needs to understand how entropy generation plays a role in irreversible processes, and how this entropy generation contributes in such a way that the overall entropy change only depends on the initial and final states. The development in Transport Phenomena clearly lays this out. However, in the present thread, what I've been trying to do is given you a better feel for how entropy generation comes into play.

 

[Philip Koeck]

As far as an increase in available space being considered a cause of entropy increase, that is not the case if the expansion takes place adiabatically and reversibly (where there is no entropy change).

One could argue that in a reversible adiabatic expansion the entropy increase due to volume increase is exactly balanced by the entropy decrease due to cooling.

 

[Philip Koeck]

We have the same incompressible viscous fluid as before, being sheared between two infinite parallel plates, but, in this case, rather than both plates being insulated, the lower plate at y = 0 is held at constant temperature To. The system attains a steady state temperature profile and the entropy of the liquid becomes constant. However, entropy is being generated within the fluid both by viscous dissipation and by heat conduction, and this generated entropy is being removed at the lower plate by heat transfer. We are going to solve for the rate of entropy generation and show that it is equal to the rate of entropy removal.

The differential thermal energy balance for this system is described by $$0=k\frac{d^2T}{dy^2}+\mu\left(\frac{V}{h}\right)^2$$To obtain the differential entropy balance equation for the system, we divide this equation by the local absolute temperature T (at arbitrary y), and then do exactly the same mathematical manipulation with the heat conduction term that I did in post #27. What do you obtain when you do this?

Dividing by the absolute temperature I get:
$$\frac{k}{T}\frac{d^2T}{dy^2}+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=\frac{d}{dy}\left(\frac{k}{T}\frac{dT}{dy}\right)+\frac{k}{T^2}\left(\frac{dT}{dy}\right)^2+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=0$$or, equivalently $$-\frac{d}{dy}\left(\frac{q}{T}\right)+\frac{q^2}{kT^2}+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=0$$

 

[Chestermiller]

One could argue that in a reversible adiabatic expansion the entropy increase due to volume increase is exactly balanced by the entropy decrease due to cooling.

with respect, this makes no sense to me. Bird, et al have identified the only two mechanisms for entropy change in a system as

1. entropy transfer by heat flow across the boundaries of the system

2. entropy generation from irreversible transport processes occurring within the system

In a reversible process, only the first mechanism is significant.

Entropy is a material property of a substance, and is a unique function of its temperature and specific volume, independent of any process path. So, of course, if specific volume changes, it follows that entropy may change. But this has nothing to do with the fundamental mechanism for the change.

Saying that volume change is a cause of entropy change is analogous to saying that change in elevation is a cause of gravitational potential energy change. It is a force acting through a distance that is the fundamental cause (mechanism) of potential energy change.

 

[Chestermiller]

Dividing by the absolute temperature I get:
$$\frac{k}{T}\frac{d^2T}{dy^2}+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=\frac{d}{dy}\left(\frac{k}{T}\frac{dT}{dy}\right)+\frac{k}{T^2}\left(\frac{dT}{dy}\right)^2+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=0$$or, equivalently $$-\frac{d}{dy}\left(\frac{q}{T}\right)+\frac{q^2}{kT^2}+\frac{\mu}{T}\left(\frac{V}{h}\right)^2=0$$

This equation tells us that the net rate of entropy change per unit time (zero, for this steady state situation) is equal to the net local rate of entropy transport (per unit volume) from heat conduction (the first term) plus the net local rate of entropy generation per unit volume (the 2nd and 3rd terms). The latter is equal to the rate of entropy generation from heat conduction (the 2nd term) plus the rate of entropy generation from viscous dissipation.

If we integrate this equation between y = 0 and y = h, we obtain the overall entropy balance per unit area of the plates: $$-\left[\frac{q}{T}\right]_{y=0}=k\int_0^h{\frac{1}{T^2}\left(\frac{dT}{dy}\right)^2dy}-\mu\left(\frac{V}{h}\right)^2\int_0^h{\frac{dy}{T}}$$ The LHS of this equation represents the rate of entropy removal across the lower plate boundary per unit area of the plates, and the right hand side represents the average rate of entropy generation in the fluid per unit area of the plates. Since the net rate of entropy change in this steady state system is zero, all the entropy that is generated must be removed across the lower system boundary. Please see if you can integrate the heat balance equation in post #27, subject to the boundary conditions at the plates, to obtain the temperature profile of the fluid between the plates. Then, please see if you can integrate our overall entropy balance equation to obtain an analytic expression for the RHS of the above equation.

 

[Stephen Tashi]

Entropy is a material property of a substance, and is a unique function of its temperature and specific volume, independent of any process path.

As I understand the main topic of the thread ("Show that entropy is a state function"), the question is whether this can be proven from more fundamental assumptions and definitions - or should it be taken as an assumption (i.e. Assume Entropy is "an extensible property of matter").

The approach implied by "University Physics" is (apparently) to show changes in entropy are the same for any path between two states on a PV diagram and then argue that this somehow implies that a transition that cannot be represented by any path on the diagram produces the same change in entropy.

As I understand the approach of post #27, we assume that any process where the heat $q$ and temperature $T$ along a path can be defined, allows us to define the change in entropy between two situations. In that example, the "path" seems to be a path in space that connects different physical systems (different locations on the rod) rather than a path showing the transition of a single physical system between two states.

For a single physical system, we can consider paths are more general that the set of paths that can be represented on a PV diagram. For example, in what Wikipedia calls a Joule expansion of gas ( https://en.wikipedia.org/wiki/Joule_expansion ) I don't see how to define the volume and pressure for a gas in free expansion, but the gas has a definite total kinetic energy, which allows us to define temperature $T$ as the average kinetic energy where the average is taken over the number of molecules instead of over the (undefined) volume of the gas.

From the viewpoint of teaching thermodynamics, is there a way to prove that entropy change is the same for any transition of a physical system that follows a path on a $(q,T)$ diagram?

 

[Chestermiller]

As I understand the main topic of the thread ("Show that entropy is a state function"), the question is whether this can be proven from more fundamental assumptions and definitions - or should it be taken as an assumption (i.e. Assume Entropy is "an extensible property of matter").

The approach implied by "University Physics" is (apparently) to show changes in entropy are the same for any path between two states on a PV diagram and then argue that this somehow implies that a transition that cannot be represented by any path on the diagram produces the same change in entropy.

As I understand the approach of post #27, we assume that any process where the heat $q$ and temperature $T$ along a path can be defined, allows us to define the change in entropy between two situations. In that example, the "path" seems to be a path in space that connects different physical systems (different locations on the rod) rather than a path showing the transition of a single physical system between two states.

No, this is not what it addresses. (@Philip Koeck, you didn't think really this is what the analysis was addressing, did you?). It addresses the rate of change of entropy for the overall system with respect to time during an irreversible process operating at steady state (which, for this system, is zero). It shows that the net instantaneous rate at which entropy enters and leaves the rod at its two ends is equal to the rate of entropy generation within the rod.

@Stephen Tashi : Please do me a favor. Before commenting further, please see the reference I cited in post # 24: "Chapter 11, Transport Phenomena, Bird, Stewart, and Lightfoot, Problem 11D.1." This shows the details of how to determine the instantaneous rate at which entropy of a system is changing during an irreversible process (not just looking at the initial and final states or even referring to an alternative reversible path between these end states). It provides very powerful insight into the mechanisms responsible for entropy change in irreversible processes. Plus, it also clearly shows why entropy is a state function. Please get back to me with any questions you might have on the BSL development.

 

[Philip Koeck]

As I understand the main topic of the thread ("Show that entropy is a state function"), the question is whether this can be proven from more fundamental assumptions and definitions - or should it be taken as an assumption (i.e. Assume Entropy is "an extensible property of matter").

The approach implied by "University Physics" is (apparently) to show changes in entropy are the same for any path between two states on a PV diagram and then argue that this somehow implies that a transition that cannot be represented by any path on the diagram produces the same change in entropy.
…….…………….

The discussion in "University Physics" shows nicely that all reversible cyclic processes can be approximated by series of Carnot cycles.
They could just as well write that all sufficiently slow (quasistatic) cycles can be approximated by Carnot cycles, just as you say. The only thing that's important for the argument is that the processes can be plotted on a pV diagram.
This obviously means that entropy change must be the same for any quasistatic path between two states, both reversible and irreversible.
The difficult step, as you point out, is to say that this must be due to the fact that this "change of entropy" is actually the change of a state function called entropy. One could still insist that entropy change is something like work or heat, but for some reason it's path-independent as long as the path is quasistatic.

On the other hand, as soon as I know that entropy is a state function I can even calculate its change for fast, irreversible processes like the free expansion, simply by finding a suitable alternative path.

I would say that this interpretation, that entropy is a state function, is very economical and maybe not so far fetched.
I don't know, however, whether it's more like a powerful generalization (or assumption) or whether one is logically or mathematically forced to take this step.
Could one be very annoying and claim that entropy change is the same for all quasistatic paths between two states but different for a fast process between the same states?

 

[Philip Koeck]

with respect, this makes no sense to me. Bird, et al have identified the only two mechanisms for entropy change in a system as

1. entropy transfer by heat flow across the boundaries of the system

2. entropy generation from irreversible transport processes occurring within the system

In a reversible process, only the first mechanism is significant.

Entropy is a material property of a substance, and is a unique function of its temperature and specific volume, independent of any process path. So, of course, if specific volume changes, it follows that entropy may change. But this has nothing to do with the fundamental mechanism for the change.

Saying that volume change is a cause of entropy change is analogous to saying that change in elevation is a cause of gravitational potential energy change. It is a force acting through a distance that is the fundamental cause (mechanism) of potential energy change.

I realize that basic textbooks contain many simplifications and these discussions on PF are really helping me to get past these simplified ideas.
The reason why I thought volume change is a cause of entropy change comes from an example in "University Physics" where they try to show that Carnot's entropy and statistical entropy, given by Boltzmann's formula, are the same thing.
They look at the free expansion and say that if the volume increases by a factor 2 each molecule has twice as many places to be. Then by taking into account the number of molecules you can find the ratio of the number of microstates before and after the expansion and then use Boltzmann's formula to calculate entropy change, which turns out the same as for the non-statistical calculation.

You mention irreversible transport processes. In a free expansion, could one regard the movement of molecules into previously empty space as such a transport process?

 

[Chestermiller]

I realize that basic textbooks contain many simplifications and these discussions on PF are really helping me to get past these simplified ideas.
The reason why I thought volume change is a cause of entropy change comes from an example in "University Physics" where they try to show that Carnot's entropy and statistical entropy, given by Boltzmann's formula, are the same thing.
They look at the free expansion and say that if the volume increases by a factor 2 each molecule has twice as many places to be. Then by taking into account the number of molecules you can find the ratio of the number of microstates before and after the expansion and then use Boltzmann's formula to calculate entropy change, which turns out the same as for the non-statistical calculation.

You mention irreversible transport processes. In a free expansion, could one regard the movement of molecules into previously empty space as such a transport process?

It is more basic than that. The irreversible transport processes occur locally during an irreversible change, and there are only three of them: heat conduction, viscous momentum transport, and molecular diffusion. All of these are related to the movement of molecules.

 

[Chestermiller]

Could one be very annoying and claim that entropy change is the same for all quasistatic paths between two states but different for a fast process between the same states?

If the two end states are thermodynamic equilibrium states, each one must be characterized by a unique distribution of quantum mechanical states (i.e., a unique entropy). So how could the entropy change be different for the fast process between the same two states?

 

[Philip Koeck]

If the two end states are thermodynamic equilibrium states, each one must be characterized by a unique distribution of quantum mechanical states (i.e., a unique entropy). So how could the entropy change be different for the fast process between the same two states?

You almost had me there.
In your argument you assume that entropy is a state function.
If the "entropy change" we calculate, for example using a reversible path between two states, is not the change of a state function, but something more like work or heat, one could get a different value for a path that can't be plotted in a pV-diagram, even though that would be a bit strange, I must admit.

 

[Chestermiller]

You almost had me there.
In your argument you assume that entropy is a state function.
If the "entropy change" we calculate, for example using a reversible path between two states, is not the change of a state function, but something more like work or heat, one could get a different value for a path that can't be plotted in a pV-diagram, even though that would be a bit strange, I must admit.

What is really needed here is the establishment of the equivalence between the quantum mechanical entropy change (clearly a state function) and the thermal entropy change $\int{dq/T}$ for a reversible process.

 

[Philip Koeck]

What is really needed here is the establishment of the equivalence between the quantum mechanical entropy change (clearly a state function) and the thermal entropy change $\int{dq/T}$ for a reversible process.

If I understand correctly you are talking about showing that the entropy given by Boltzmann's formula is the same (apart from a factor and an offset, maybe) as the thermodynamic entropy we're dealing with when we calculate entropy changes.
That would definitely settle my question.
Does anybody know if it's possible to show this equivalence in a general way?

 

[Chestermiller]

If I understand correctly you are talking about showing that the entropy given by Boltzmann's formula is the same (apart from a factor and an offset, maybe) as the thermodynamic entropy we're dealing with when we calculate entropy changes.
That would definitely settle my question.
Does anybody know if it's possible to show this equivalence in a general way?

It is usually done for the specific case of ideal gases in elementary books on Statistical Thermodynamics, but my background in ST is not broad enough to know whether more general representations are around.

 

[Lord Jestocost]

In his lecture "Thermodynamics and Statistical Mechanics", John McGreevy presents a short section "Thermodynamic entropy and statistical entropy are the same":
8.044 Lecture Notes Chapter 4: Statistical Mechanics, via the ...

 

[Stephen Tashi]

If I understand correctly you are talking about showing that the entropy given by Boltzmann's formula is the same (apart from a factor and an offset, maybe) as the thermodynamic entropy we're dealing with when we calculate entropy changes.
That would definitely settle my question.
Does anybody know if it's possible to show this equivalence in a general way?

How many different entropies are there? Their mutual compatibility seems to depend on which entropy is being discussed.

From Boltzmann's Approach to Statistical Mechanics by Sheldon Goldstein

The Second Law is concerned with the thermodynamic entropy, and this is given by Boltzmann's entropy (1), not by the Gibbs entropy(2). In fact, the Gibbs entropy is not even an entity of the right sort: It is a function of a probability distribution, i.e., of an ensemble of systems, and not a function on phase space, a function of the actual state X of an individual system, the behavior of which the Second Law { and macro-physics in general } is supposed to describe.

A stackexchange debate about the topic: https://physics.stackexchange.com/questions/316812/gibbs-vs-boltzmann-entropies

 

[Mirko Vukovic]

Consider the irreversible process of steady state heat conduction occurring along a rod in contact with a reservoir of temperature $T_H$ at x = 0 and a second reservoir of temperature $T_C$ at x = L. The temperature profile along the rod is linear, and given by $$T=T_H-(T_H-T_C)\frac{x}{L}$$and the heat flux q along the rod is constant, and given by: $$q=-k\frac{dT}{dx}=k\frac{(T_H-T_C)}{L}$$ where k is the thermal conductivity of the rod metal.

The temperature profile has been determined by satisfying the differential heat balance equation, given by:
$$k\frac{d^2T}{dx^2}=0$$
The differential entropy balance equation on the rod can be obtained by dividing the differential heat balance equation by the absolute temperature, to yield:
$$\frac{k}{T}\frac{d^2T}{dx^2}=\frac{d}{dx}\left(\frac{k}{T}\frac{dT}{dx}\right)+\frac{k}{T^2}\left(\frac{dT}{dx}\right)^2=0$$or, equivalently $$-\frac{d}{dx}\left(\frac{q}{T}\right)+\frac{q^2}{kT^2}=0$$The second term in this equation represents physically the local rate of entropy generation per unit volume within the rod. If we integrate this equation between x = 0 and x = L, we obtain: $$\frac{q_H}{T_H}-\frac{q_C}{T_C}+\frac{q^2}{kT_HT_C}L=0\tag{1}$$where $q_H=q_C=q$. The first term on the LHS represents the rate of entropy entering the rod per unit area at x = 0, and the second term represents the rate of entropy exiting the rod per unit area at x = L. The coefficient of L in the third term is positive definite, and represents the average rate of entropy generation per unit volume within the rod. Since the rod is operating at steady state, we know that the entropy of the rod is not changing with time. So $L\frac{dS}{dt}=0$, where dS/dt is the average rate of change of entropy per unit volume of the rod. So we can write:
$$0=L\frac{dS}{dt}\gt \frac{q_H}{T_H}-\frac{q_C}{T_C}=-\frac{q^2}{kT_HT_C}L\tag{2}$$Eqn. 2 is equivalent to the Clausius inequality applied to the irreversible heat conduction process in the rod.

Note that, in this very simple case, we have been able to precisely determine the rate of entropy generation within the system. Note also that, Eqn. 1 tells us that the rate of entropy exiting the rod at x = L is just equal to the rate of entropy entering the rod at x = 0, plus the total rate of entropy generation within the rod.

Very nice derivation.

One can rewrite $$-\frac{d}{dx}\left(\frac{q}{T}\right)+\frac{q^2}{kT^2}=0$$ as $$-\frac{d S}{dx}+\frac{S^2}{k}=0$$

But, I am not getting any insights from that equation. Any thoughts or opinions?

Thank you,

 

[Chestermiller]

The parameter q is not the amount of heat (and, even if it were, q/T would not be the entropy). The parameter q is the heat flux (the rate of heat flow per unit cross sectional area).