I This gas work violates 2nd law of thermodynamics?

[Standard]

Let's assume that TA=TB
Piston A will be pushed back into the vacuum.

Let's assume that the second piston (gas B) is held to prevent it from doing the same thing as piston A.

Now if piston A moves forward.
When piston A reaches piston B, the temperature of A has changed, so TA<TB, because gas A has expanded (it now occupies a larger volume).

So the pressure of piston B will prevent piston A from advancing and so piston B will not be pushed back, but on the contrary piston B will push piston A back on contact when piston B is released.

 

[Chestermiller]

What is the exact word-for-word statement of this problem?

 

[aliinuur]

What is the exact word-for-word statement of this problem?

the statement of the problem: Heat transfer on its own, without external work, what is impossible by the 2nd law.

 

[Chestermiller]

the statement of the problem: Heat transfer on its own, without external work, what is impossible by the 2nd law.

That is specified by the Clausius inequality. The entropy generated by a process in a closed system is positive or zero.

 

[aliinuur]

That is specified by the Clausius inequality. The entropy generated by a process in a closed system is positive or zero

does that mean definiton with 2 bodies should be abandoned in favor of entropy definiton?

 

[berkeman]

does that mean definiton with 2 bodies should be abandoned in favor of entropy definiton?

What in the world are you saying / asking?

 

[Demystifier]

@aliinuur A statement of the 2nd law relevant in your case is this: https://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Clausius_statement
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

Here the crucial part is "without some other change ...". In your example there is also some other change involved, so your example does not violate the 2nd law as stated above.

 

[Philip Koeck]

@aliinuur A statement of the 2nd law relevant in your case is this: https://en.wikipedia.org/wiki/Second_law_of_thermodynamics#Clausius_statement
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.

Here the crucial part is "without some other change ...". In your example there is also some other change involved, so your example does not violate the 2nd law as stated above.

Just to expand a bit: The other thing that changes is that the potential energy that's stored in the device due to the pressure difference between gas A and vacuum is used up when gas A is allowed to expand.

Actually it's not even clear that any heat is transferred between the gases in the process. The expansion of A and the compression of B could both be adiabatic.

 

[Chestermiller]

does that mean definiton with 2 bodies should be abandoned in favor of entropy definiton?

I don't understand what you are asking.

 

[Demystifier]

does that mean definiton with 2 bodies should be abandoned in favor of entropy definiton?

Unlike others, I understood that you are talking about different formulations of the 2nd law. Some formulations don't explicitly talk about entropy, but about heat, which are equivalent because $dQ=TdS$. But AFAIK no formulation talks about two bodies.

 

[Chestermiller]

Unlike others, I understood that you are talking about different formulations of the 2nd law. Some formulations don't explicitly talk about entropy, but about heat, which are equivalent because $dQ=TdS$.

dQ=TdS only for a reversible path.

 

[Demystifier]

dQ=TdS only for a reversible path.

It doesn't make sense. For a reversible path $dS=0$ (if by $S$ one means entropy of the whole system), hence $dQ=0$, hence the content of $dQ=TdS$ is trivial for a reversible path. For $dQ=TdS$ to be non-trivial, it has to be applicable for non-reversible processes as well.

 

[Chestermiller]

It doesn't make sense. For a reversible path $dS=0$ (if by $S$ one means entropy of the whole system), hence $dQ=0$, hence the content of $dQ=TdS$ is trivial for a reversible path. For $dQ=TdS$ to be non-trivial, it has to be applicable for non-reversible processes as well.

dS is not generally zero for a reversible path. Consider reversible isothermal expansion.

 

[Chestermiller]

There have been many statements of the Second Law over the years, couched in a complicated language and multi-word sentences, typically involving heat reservoirs, Carnot engines, and the like. These statements have been a source of unending confusion for students of thermodynamics for over a hundred years. What has been sorely needed is a precise mathematical definition of the Second Law that avoids all the complicated rhetoric. The sad part about all this is that such a precise definition has existed all along. The definition was formulated by Clausius back in the 1800s. (The following is a somewhat fictionalized account, designed to minimize the historical discussion, and focus more intently on the scientific findings.) Clausius wondered what would happen if he evaluated the following integral over each of the possible process paths between the initial and final equilibrium states of a closed system:
$$I=\int_{T_i}^{T_f}{\frac{dQ}{T_{int}}}$$where $T_{int}$ is the temperature at the interface with the surroundings. He carried out extensive calculations on many systems undergoing a variety of both reversible and irreversible paths and discovered something astonishing: For any closed system, the values calculated for the integral over all the possible reversible and irreversible paths (between the initial and final equilibrium states) is not arbitrary; instead, there is a unique upper bound to the value of the integral. Clausius also found that this observation is consistent with all the “word definitions” of the Second Law.

Reference: https://www.physicsforums.com/insights/understanding-entropy-2nd-law-thermodynamics/

 

[Philip Koeck]

TL;DR Summary: Gas by doing negative work(losing heat), say accelerating a piston violates 2nd law of thermo, because kinetic energy of piston can be used to make other gas hotter, thus in summary one gas gets colder, other one gets hotter and this by simple piston!!!!

View attachment 328132

I'd really like to see some more details from the OP.
For example, why is gas A loosing heat and where is there a transport of heat in this process?

 

[Demystifier]

dS is not generally zero for a reversible path. Consider reversible isothermal expansion.

I don't get it. Reversible means the final state is identical to the initial one, right? The identical states have the same $S$, hence the overall change of entropy during the whole path is $\Delta S=0$. Hence either (i) $dS=0$ at all parts of the path, or (ii) $dS>0$ at some parts of the path and $dS<0$ at other parts. But $dS<0$ contradicts the 2nd law, which rules out (ii). What remains is (i) $dS=0$ during the whole path. Do I miss something?

 

[aliinuur]

I'd really like to see some more details from the OP.
For example, why is gas A loosing heat and where is there a transport of heat in this process?

heat of gas A goes to kinetic energy of the piston A, heat transfered via pistons, but there is no external work or something like that

 

[Philip Koeck]

heat of gas A goes to kinetic energy of the piston A, heat transfered via pistons, but there is no external work or something like that

Gas A is not necessarily giving off any heat. If the expansion is adiabatic then Q = 0, for example. However the temperature decreases!

If the expansion is somewhere between adiabatic and isothermal then heat actually flows into gas A. Even so gas A would get colder.

If the expansion is isothermal then heat flows into gas A to balance the work done during expansion and the temperature is constant.

You could find an expansion where heat leaves the gas, but that would have to be steeper than an adiabatic curve in a PV-diagram.

Maybe this helps: For an ideal gas inner energy is proportional to temperature. So if the temperature of gas A drops this just means some of its inner energy has been used to do expansion work. It doesn't necessarily mean there was a flow of heat.

Concerning work that's put in: Some work was obviously needed to get the device into the state shown on the left.

 

[Chestermiller]

I don't get it. Reversible means the final state is identical to the initial one, right? The identical states have the same $S$, hence the overall change of entropy during the whole path is $\Delta S=0$. Hence either (i) $dS=0$ at all parts of the path, or (ii) $dS>0$ at some parts of the path and $dS<0$ at other parts. But $dS<0$ contradicts the 2nd law, which rules out (ii). What remains is (i) $dS=0$ during the whole path. Do I miss something?

Reversible CYCLE means that the final state of the system is identical to the initial one. Reversible path between state A and state B of the system does not require that states A and B of the system are the same.

 

[Demystifier]

Reversible CYCLE means that the final state of the system is identical to the initial one. Reversible path between state A and state B of the system does not require that states A and B of the system are the same.

Fine, the states $A$ and $B$ are not the same. But do they have the same entropy, namely, is $S(B)=S(A)$? You seem to be saying that one entropy can be larger, e.g. $S(B)>S(A)$. But then in the path $B\to A$ we have $dS<0$, which contradicts the 2nd law. So how can such a path exist?

 

[Chestermiller]

Fine, the states $A$ and $B$ are not the same. But do they have the same entropy, namely, is $S(B)=S(A)$? You seem to be saying that one entropy can be larger, e.g. $S(B)>S(A)$. But then in the path $B\to A$ we have $dS<0$, which contradicts the 2nd law. So how can such a path exist?

What process are you considering here?

 

[Demystifier]

What process are you considering here?

Any process with an alleged reversible cycle you referred to.

 

[Chestermiller]

Let’s focus on a reversible isothermal expansion or compression of an ideal gas, ok @Demystifier?

 

[Demystifier]

Let’s focus on a reversible isothermal expansion or compression of an ideal gas, ok @Demystifier?

OK. So can you answer my question in #50 now?

 

[Chestermiller]

OK. So can you answer my question in #50 now?

$$S_B-S_A=R\ln(V_B/V_A)$$$$S_A-S_B=R\ln(V_A/V_B)$$$$\Delta S_{cycle}=0$$So what's the problem?

 

[Demystifier]

$$S_B-S_A=R\ln(V_B/V_A)$$$$S_A-S_B=R\ln(V_A/V_B)$$$$\Delta S_{cycle}=0$$So what's the problem?

During the contraction (i.e. when the volume decreases), the entropy decreases. This contradicts the 2nd law during the contraction. Of course, it doesn't mean that the gas cannot be contracted. But it means that there is some additional change of entropy (in the environment) not captured by the equations above, so that $\Delta S_{\rm total}> 0$ even during the contraction.

You are really saying that entropy of a subsystem (the gas) can decrease. I agree, but I point out that the entropy of the full system (gas + environment) cannot decrease.

What really is important is the following. Suppose that the full system consists of several subsystems, each with its own temperature $T_i$. Then for each subsystem we have $dQ_i=T_idS_i$. The second law is
$$\sum_i dS_i\geq 0$$
or in terms of heats
$$\sum_i \frac{dQ_i}{T_i} \geq 0.$$
In particular, if all subsystems have the same temperature $T_1=T_2=\cdots=T$, then the last inequality simplifies to
$$\sum_i dQ_i\geq 0.$$
In this case we also have
$$\sum_i dQ_i=T\sum_i dS_i$$
or shortly
$$dQ=TdS$$
where $dQ=\sum_i dQ_i$ and $dS=\sum_i dS_i$.

 

[Chestermiller]

During the contraction (i.e. when the volume decreases), the entropy decreases. This contradicts the 2nd law during the contraction.

This is not a contradiction of the 2nd law. It satisfies the Clausius inequality for a reversible process applied to the system.

Of course, it doesn't mean that the gas cannot be contracted. But it means that there is some additional change of entropy (in the environment) not captured by the equations above, so that $\Delta S_{\rm total}> 0$ even during the contraction.

The > sign applies to an irreversible process path. For a reversible process path, the entropy change of system plus surroundings is zero.

You are really saying that entropy of a subsystem (the gas) can decrease. I agree, but I point out that the entropy of the full system (gas + environment) cannot decrease.

Correct. So what.

What really is important is the following. Suppose that the full system consists of several subsystems, each with its own temperature $T_i$. Then for each subsystem we have $dQ_i=T_idS_i$. The second law is
$$\sum_i dS_i\geq 0$$
or in terms of heats
$$\sum_i \frac{dQ_i}{T_i} \geq 0.$$
In particular, if all subsystems have the same temperature $T_1=T_2=\cdots=T$, then the last inequality simplifies to
$$\sum_i dQ_i\geq 0.$$

The = sign applies only to a reversible change in all the subsystems. If the path is irreversible, then the temperature in one or more of the subsystems must be non-uniform and the T in dQ/T for that subsystem must be taken as the temperature at the interface between the subsystem and its neighbors through which the heat dQ flows (see Fermi, Thermodynamics and Moran et al, Fundamentals of Engineering Thermodynamics).

 

[Demystifier]

@Chestermiller I still have a feeling that I'm missing something. Let me try this way. You said that the formula $dQ=TdS$ is valid only for reversible processes. If so, is there a generalization of this formula valid for all processes?

 

[Chestermiller]

@Chestermiller I still have a feeling that I'm missing something. Let me try this way. You said that the formula $dQ=TdS$ is valid only for reversible processes. If so, is there a generalization of this formula valid for all processes?

For a closed system, the entropy change of the system for an irreversible or a reversible process path is $$\Delta S=\int{\frac{dQ}{T_B}}+\sigma$$where $T_B$ is the temperature at the interface between the system and surroundings and $\sigma$ is the entropy generated within the system during the process due to irreversibility; for an irreversible path $\sigma>0$ and, for a reversible path, $\sigma=0$. Also, for a reversible path, the temperature within the system is uniform at all points along the path and equal to the interface temperature at all points along the reversible path. Furthermore, for a reversible path, the heat variations dQ along the path do not have to match those along irreversible path, nor do the boundary temperature variations along the path.

Here is how to find the entropy change for an irreversible process path: https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/

 

[Philip Koeck]

TL;DR Summary: Gas by doing negative work(losing heat), say accelerating a piston violates 2nd law of thermo, because kinetic energy of piston can be used to make other gas hotter, thus in summary one gas gets colder, other one gets hotter and this by simple piston!!!!

View attachment 328132

I've tried to simplify the problem, hopefully retaining its essence, and do an analysis, but I ran into some difficulties. Maybe someone can tell me how to proceed.

If I have a closed cylinder as in the sketch but only 1 piston between gas A and gas B, so no vacuum, then I don't need to think about collisions between pistons.
Initially I have the following situation:
Gas A occupies 1/3 of the cylinder on the left, gas B occupies 2/3 on the right and they are separated by a piston. The number of moles (n) is the same for both and remains constant.

For the starting values of volume, pressure and temperature we have:
V_B = 2 V_A
P_A = 2 P_B
T_A = T_B = T

There is no heat transfer between a gas and anything else (piston, cylinder or the other gas).
The piston is extremely heavy so that expansion and compression will be quasi-static.

If I release the piston gas A will expand very slowly and gas B will be compressed. The sum of volumes is constant.

This process should stop when the pressures of gas A and B are the same.
Gas A will have cooled off and gas B will have heated up.

Here's my problem:
If I treat the expansion of A and the compression of B as reversible adiabatic, then the whole process will be reversible since nothing happens in the surroundings.
(I checked that analytically, just to be sure, and if I use expressions for reversible adiabatic processes ( P V^γ = constant, for example), I get ΔS = 0 for both gases (with some rounding errors).

However, by just looking at it, I would swear the process must be irreversible.

How would you calculate the final temperatures and volumes of the gases and the entropy-changes?