How to know when a reversible process between end states exists?

[EE18]

I am continuing to try to understand maximum work reversible processes (and a subset thereof -- Carnot cycles) better. I am here curious about the following system.

(1) Consider one mole of a gas (say, ideal, or van der Waals) which is expanded isothermally, at temperature $T_h$, from an initial volume $V_i$, to a final volume $V_f$. A thermal reservoir at temperature $T_c$, is available.

My question is about how I can know/prove that there exists a way to take the gas (the primary subsystem) reversibly with respect to a composite system consisting of the gas, the reservoir, a reversible work source (RWS), and some set of auxiliary systems (defined as undergoing no net change) from the initial state to the final state of the primary subsystem in this particular way (i.e. along the given isotherm).

I ask because in the context of the following conundrum given to us by Callen in his classic thermodynamics textbook:

(2) A system can be taken from state A to state B (where $S_B= S_A$) either (a) directly along the adiabat $S$ = constant, or (b) along the isochore AC and the isobar CB. The difference in the work done by the system is the area enclosed between the two paths in a P-V diagram. Does this contravene the statement that the work delivered to a reversible work source is the same for every reversible process?

The answer here -- I think -- is that there is no way to reversibly (with respect to the subsystem+surroundings --i.e. subsystem+whatever composite system you consider it as part of) take the primary subsystem to the final state as in b). Perhaps this answer is wrong though, and there is some separate reason that (2) above does not violate the so-called Maximum Work Theorem.

At any rate, the contrast between the two cases discussed above leads to my question: how can I know whether it is possible to take a (sub)system reversibly (again, with respect to some composite system containing the primary subsystem, a reversible heat source (the reservoir in the first example is a particular case), a reversible work source (RWS), and some set of auxiliary systems)? Is it always possible, or are there some changes of state for which this is not possible? If it is always possible, it would seem that my answer to the paradox in (2) is wrong, and I would appreciate any help in resolving it.

 

[Andrew Mason]

The answer here -- I think -- is that there is no way to reversibly (with respect to the subsystem+surroundings --i.e. subsystem+whatever composite system you consider it as part of) take the primary subsystem to the final state as in b). Perhaps this answer is wrong though, and there is some separate reason that (2) above does not violate the so-called Maximum Work Theorem.

There is always a reversible path between two equilibrium states for the system and always a reversible path between two equilibrium states of the surroundings. But they will not be the same paths unless the actual path that resulted in those states was a reversible one.

At any rate, the contrast between the two cases discussed above leads to my question: how can I know whether it is possible to take a (sub)system reversibly (again, with respect to some composite system containing the primary subsystem, a reversible heat source (the reservoir in the first example is a particular case), a reversible work source (RWS), and some set of auxiliary systems)? Is it always possible, or are there some changes of state for which this is not possible? If it is always possible, it would seem that my answer to the paradox in (2) is wrong, and I would appreciate any help in resolving it.

Calculation of entropy change is the integral of heat flow divided by temperature over the reversible path between two equilibrium states. So the total change in entropy is:
$\Delta S = \Delta S_{sys}+\Delta S_{surr}$
where:
$\int_{initial-sys}^{final-sys} \frac{dQ_{rev}}{T}$ over a reversible path for the system and

$\int_{initial-surr}^{final-surr} \frac{dQ_{rev}}{T}$ over a reversible path for the surroundings.

AM

 

[EE18]

There is always a reversible path between two equilibrium states for the system and always a reversible path between two equilibrium states of the surroundings. But they will not be the same paths unless the actual path that resulted in those states was a reversible one.

Calculation of entropy change is the integral of heat flow divided by temperature over the reversible path between two equilibrium states. So the total change in entropy is:
ΔS=ΔSsys+DeltaSsurr
where:
$\Delta S = \Delta S_{sys}+\Delta S_{surr}$
where:
$\int_{initial-sys}^{final-sys} \frac{dQ_{rev}}{T}$ over a reversible path for the system and

$\int_{initial-surr}^{final-surr} \frac{dQ_{rev}}{T}$ over a reversible path for the surroundings.

AM

I am not sure I totally follow your answer. I will rephrase my question: "For a given system, let it be situated as a subsystem (and call it the primary subsystem) in a composite system consisting of primary subsystem, RHS, RWS, and any number of auxiliary subsystems. Does there necessarily exist a reversible process (with respect to the composite system!) which takes the primary subsystem along an arbitrary path in the primary subsystem's thermodynamic configuration space?"

I am not asking whether all paths to the given final state of the primary subsystem (in the composite system's thermodynamic configuration space) need be reversible, but whether one such reversible path (again, in the composite system's configuration space) exists.

 

[Andrew Mason]

I am not sure I totally follow your answer. I will rephrase my question: "For a given system, let it be situated as a subsystem (and call it the primary subsystem) in a composite system consisting of primary subsystem, RHS, RWS, and any number of auxiliary subsystems. Does there necessarily exist a reversible process (with respect to the composite system!) which takes the primary subsystem along an arbitrary path in the primary subsystem's thermodynamic configuration space?"

I am not asking whether all paths to the given final state of the primary subsystem (in the composite system's thermodynamic configuration space) need be reversible, but whether one such reversible path (again, in the composite system's configuration space) exists.

If each component has different initial and final equilibrium states then you have to find a separate reversible path for each component between their respective initial and final states. There is always such a path but the paths will be different.

For example, if you consider an adiabatic isothermal free expansion of a gas into a vaccum, the reversible process for the gas would be an adiabatic quasi-static expansion against an external pressure that is infinitesimally lower than the gas pressure, followed by heat flow into the gas from the surroundings. The reversible process for the surroundings would be the same (negative) heat flow from the surroundings at constant temperature at whatever that temperature is.

AM

 

[EE18]

If each component has different initial and final equilibrium states then you have to find a separate reversible path for each component between their respective initial and final states. There is always such a path but the paths will be different.

For example, if you consider an adiabatic isothermal free expansion of a gas into a vaccum, the reversible process for the gas would be an adiabatic quasi-static expansion against an external pressure that is infinitesimally lower than the gas pressure, followed by heat flow into the gas from the surroundings. The reversible process for the surroundings would be the same (negative) heat flow from the surroundings at constant temperature at whatever that temperature is.

AM

I don't understand the comment about different reversible paths? I am talking about one composite system, and am asking about a reversible path (with respect to that composite system) which takes one particular subsystem (the primary subsystem) from a certain initial state to a certain final state?

 

[Andrew Mason]

I don't understand the comment about different reversible paths? I am talking about one composite system, and am asking about a reversible path (with respect to that composite system) which takes one particular subsystem (the primary subsystem) from a certain initial state to a certain final state?

Can you give us a specific problem?

Thermodynamics deals with systems and their surroundings and processes between equilibrium states. A system is just a particular body or collection of particles of interest having a single thermodynamic equilibrium state. With a composite system (two or more simple systems each of which can have its own thermodynamic equilibrium state) each simple system will have a thermodynamic equilibrium state that can be different than the states of the other simple systems in the composite system. Each simple system has to be analysed separately.

if you are able to determine the initial and final states of each simple system in the composite system you can always find a reversible process between those states for that system.

AM

 

[TSny]

I ask because in the context of the following conundrum given to us by Callen in his classic thermodynamics textbook:

Look at Callen's statement of the maxium work theorem:

The theorem is talking about the net work delivered to the reversible work source (RWS). This work will be the same for all reversible processes that use the same reversible heat source RHS. It is not talking about the work done by the subsystem ("primary system"), which can differ for different reversible processes working with the same RHS.

Consider problem 4.5-1

The answer doesn't depend on which particular reversible process you choose to take the gas from its initial state to its final state. You don't need to pick a specific reversible process to work the problem.

But if you do consider two different reversible processes that take the gas between the initial and final states as described (and that use a thermal reservoir at 300 K), then you generally find that the work done by the gas is different for the two processes (even though the two processes must deliver the same work to the RWS).

For example, if you choose a reversible process such that the state of the gas moves along an isotherm, the work delivered to the RWS will be $300 R \ln 2$. I get that the work done by the gas is $400 R \ln 2$.

If, instead, you choose a reversible process such that the gas first goes through an isochoric process that halves the pressure and then goes through an isobaric process that doubles the volume, the total work delivered to the RWS is still $300 R \ln 2$. But I get that the total work done by the gas is $200 R$.

 

[EE18]

Look at Callen's statement of the maxium work theorem:

View attachment 328321

The theorem is talking about the net work delivered to the reversible work source (RWS). This work will be the same for all reversible processes that use the same reversible heat source RHS. It is not talking about the work done by the subsystem ("primary system"), which can differ for different reversible processes working with the same RHS.

Consider problem 4.5-1

View attachment 328320

The answer doesn't depend on which particular reversible process you choose to take the gas from its initial state to its final state. You don't need to pick a specific reversible process to work the problem.

But if you do consider two different reversible processes that take the gas between the initial and final states as described (and that use a thermal reservoir at 300 K), then you generally find that the work done by the gas is different for the two processes (even though the two processes must deliver the same work to the RWS).

For example, if you choose a reversible process such that the state of the gas moves along an isotherm, the work delivered to the RWS will be $300 R \ln 2$. I get that the work done by the gas is $400 R \ln 2$.

If, instead, you choose a reversible process such that the gas first goes through an isochoric process that halves the pressure and then goes through an isobaric process that doubles the volume, the total work delivered to the RWS is still $300 R \ln 2$. But I get that the total work done by the gas is $200 R$.

I think this gets me very close to understanding. I asked about (2) in my OP on another site weeks ago and, ironically, the answer only came to me yesterday (see my edit here). Does that edit make sense to you? In the specific example you gave, to confirm, is the "delta" of work delivered to the RWS (above or below $W$, the work done directly by the primary subsystem) effected using the heat extracted from the primary subsystem (and "channeled through some auxiliary subsystems")? That is, in the isothermal case for example, some of the work done by the gas needs actually to be redirected as heat to keep the gas isothermal?

But I'm hoping I can bother you for a direct answer to my main question here: "For a given system, let it be situated as a subsystem (and call it the primary subsystem) in a composite system consisting of primary subsystem, RHS, RWS, and any number of auxiliary subsystems. Does there necessarily exist a reversible process (with respect to the composite system!) which takes the primary subsystem along an arbitrary path in the primary subsystem's thermodynamic configuration space?". It seems to me like this is something Callen assumes, but I'm not sure. So, as far as you can tell, is Callen assuming this in his book? Can it be proved?

Thank you again!

 

[TSny]

I think this gets me very close to understanding. I asked about (2) in my OP on another site weeks ago and, ironically, the answer only came to me yesterday (see my edit here). Does that edit make sense to you?

Yes, the edit looks correct to me.

In the specific example you gave, to confirm, is the "delta" of work delivered to the RWS (above or below $W$, the work done directly by the primary subsystem) effected using the heat extracted from the primary subsystem (and "channeled through some auxiliary subsystems")?

Yes, I think you have the right idea here.

That is, in the isothermal case for example, some of the work done by the gas needs actually to be redirected as heat to keep the gas isothermal?

I'm not sure I follow this statement, but you might be thinking correctly. All of the work done by the gas in the reversible isothermal expansion can be considered as contributing to the work delivered to the RWS. However, during the process, the gas must take in an amount of heat equal to the work done by the gas (since there is no change in the internal energy of an ideal gas in an isothermal process). The only heat source in the example of problem 4.5-1 is a thermal reservoir at 300 K. To transfer heat reversibly from the reservoir at 300 K to the gas at 400 K, you can imagine using a Carnot refrigerator as an auxiliary device. The refrigerator takes heat $Q_{RHS}$ from the heat reservoir and uses work $W'$ from the RWS to deliver the necessary heat $Q_{sys}$ to the primary subsystem. The net amount of work delivered to the RWS for the overall process will be the work done by the primary substem minus the work $W'$ that the RWS delivered to the refrigerator.

But I'm hoping I can bother you for a direct answer to my main question here: "For a given system, let it be situated as a subsystem (and call it the primary subsystem) in a composite system consisting of primary subsystem, RHS, RWS, and any number of auxiliary subsystems. Does there necessarily exist a reversible process (with respect to the composite system!) which takes the primary subsystem along an arbitrary path in the primary subsystem's thermodynamic configuration space?". It seems to me like this is something Callen assumes, but I'm not sure. So, as far as you can tell, is Callen assuming this in his book? Can it be proved?

Yes, I think that in principle there will always be a reversible process of the composite system that will take the primary subsystem along the arbitrarily chosen path.

Consider an infinitesimal step along the path of the subsystem. For this step let $dW_{sys}$ be the work done on the subsystem and let $dQ_{sys}$ be the heat added to the subsystem. (Either of these quantities could be negative.) $dW_{sys}$ can be considered as contributing to the net amount of work delivered to the RWS. $dQ_{sys}$ can be arranged to be reversibly delivered to the subsystem by using an auxiliary heat engine or refrigerator that operates between the heat source (RHS) and the subsystem.

I can't see why this wouldn't be possible for any step of the arbitrarily chosen path of the subsystem. But, I could be wrong.

 

[EE18]

Yes, the edit looks correct to me.Yes, I think you have the right idea here.I'm not sure I follow this statement, but you might be thinking correctly. All of the work done by the gas in the reversible isothermal expansion can be considered as contributing to the work delivered to the RWS. However, during the process, the gas must take in an amount of heat equal to the work done by the gas (since there is no change in the internal energy of an ideal gas in an isothermal process). The only heat source in the example of problem 4.5-1 is a thermal reservoir at 300 K. To transfer heat reversibly from the reservoir at 300 K to the gas at 400 K, you can imagine using a Carnot refrigerator as an auxiliary device. The refrigerator takes heat $Q_{RHS}$ from the heat reservoir and uses work $W'$ from the RWS to deliver the necessary heat $Q_{sys}$ to the primary subsystem. The net amount of work delivered to the RWS for the overall process will be the work done by the primary substem minus the work $W'$ that the RWS delivered to the refrigerator.
Yes, I think that in principle there will always be a reversible process of the composite system that will take the primary subsystem along the arbitrarily chosen path.

Consider an infinitesimal step along the path of the subsystem. For this step let $dW_{sys}$ be the work done on the subsystem and let $dQ_{sys}$ be the heat added to the subsystem. (Either of these quantities could be negative.) $dW_{sys}$ can be considered as contributing to the net amount of work delivered to the RWS. $dQ_{sys}$ can be arranged to be reversibly delivered to the subsystem by using an auxiliary heat engine or refrigerator that operates between the heat source (RHS) and the subsystem.

I can't see why this wouldn't be possible for any step of the arbitrarily chosen path of the subsystem. But, I could be wrong.

Beautifully and helpfully answered -- especially the gory details of how you would deal with the required heat delivery reversibly. Thank you so much! The details of Callen Chapter 4 have been escaping me and this has clarified things tremendously.

 

[Andrew Mason]

I think this gets me very close to understanding. I asked about (2) in my OP on another site weeks ago and, ironically, the answer only came to me yesterday (see my edit here). Does that edit make sense to you? In the specific example you gave, to confirm, is the "delta" of work delivered to the RWS (above or below $W$, the work done directly by the primary subsystem) effected using the heat extracted from the primary subsystem (and "channeled through some auxiliary subsystems")? That is, in the isothermal case for example, some of the work done by the gas needs actually to be redirected as heat to keep the gas isothermal?

Not some of the work. All of it. First law: $\Delta U = Q - W$ where W is the work done BY the ideal gas. So if it is isothermal, $\Delta U=0 \text{ and } Q=W$

But I'm hoping I can bother you for a direct answer to my main question here: "For a given system, let it be situated as a subsystem (and call it the primary subsystem) in a composite system consisting of primary subsystem, RHS, RWS, and any number of auxiliary subsystems. Does there necessarily exist a reversible process (with respect to the composite system!) which takes the primary subsystem along an arbitrary path in the primary subsystem's thermodynamic configuration space?". It seems to me like this is something Callen assumes, but I'm not sure. So, as far as you can tell, is Callen assuming this in his book? Can it be proved?

It can be disproved. For simplicity, let's assume the entire composite system is completely isolated from its surroundings. There is always a reversible process with respect to the composite system minus the primary system ("CS-P" ie. the surroundings of the primary system) between any two equilibrium states (ie. of CS-P). Similarly, there is always a reversible path between any two equilibrium states of the primary system.

But if the actual path followed by CS-P was reversible, then the primary system path must also be reversible (ie it cannot be an arbitrary path). That is because in order for the actual path of CS-P to be reversible, all energy transfers must occur in a state of equilibrium with its surroundings ie. the primary system. So both CS-P and P must follow reversible paths. It follows, then, that if the actual path of the primary system, P, is not reversible, the path of CS-P cannot be reversible.

AM

 

[EE18]

Not some of the work. All of it. First law: $\Delta U = Q - W$ where W is the work done BY the ideal gas. So if it is isothermal, $\Delta U=0 \text{ and } Q=W$

It can be disproved. For simplicity, let's assume the entire composite system is completely isolated from its surroundings. There is always a reversible process with respect to the composite system minus the primary system ("CS-P" ie. the surroundings of the primary system) between any two equilibrium states (ie. of CS-P). Similarly, there is always a reversible path between any two equilibrium states of the primary system.

But if the actual path followed by CS-P was reversible, then the primary system path must also be reversible (ie it cannot be an arbitrary path). That is because in order for the actual path of CS-P to be reversible, all energy transfers must occur in a state of equilibrium with its surroundings ie. the primary system. So both CS-P and P must follow reversible paths. It follows, then, that if the actual path of the primary system, P, is not reversible, the path of CS-P cannot be reversible.

AM

I think you and I must somehow be using reversible differently. As far as I know, reversibility is only defined with respect to the entire composite system -- i.e. reversible means isentropic as considered across the entire composite system. Am I misunderstanding you?

 

[Andrew Mason]

Reversible and isentopic are not the same. A reversible adiabatic expansion of an ideal gas is isentropic. But a reversible isothermal expansion of an ideal gas is not isentropic. When applied to an entire composite system, I am not sure the term "isentropic" has any meaning. Different parts of the system experience changes in entropy. If the processes are all reversible, the entropy changes all sum to zero but entropy is not constant in all parts.

AM