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The Clausius statement of the second law is:

"Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time."

This is typically illustrated in a manner similar to the diagram labelled "(a)" below: Text

Now note that in the quoted Clausius’ statement above, he only states that "some other change" must occur at the same time as the heat flow from cold to hold. He does not say that the change necessarily has to be a positive net work on the device (i.e $W_{net,in}>0$). So in theory, if I am to take the above statement as truth, then surely it should be possible to construct a device (a refrigerator) as shown in the diagram above labelled "(b)"?

That is, a device which transfers heat from a colder body to a hotter body and does a positive net work on the surroundings during the cycle. This clearly seems wrong but it appears to me that it doesn't violate the second law (as it is stated above) and it doesn't violate the first law either since we still have that $\Delta U = 0$ for the cycle. Is this all correct? Is it possible to create a device operating on a cycle which transfers heat from a cold source to a hot source and at the same time does positive net work on its surroundings (i.e. $W_{net,out}>0$)? It seems somewhat intuitive to me that this shouldn't be possible. If it isn't possible, then shouldn't the Clausius statement be updated to say that "Heat can never pass from a colder to a warmer body without a positive net work input into the device causing the heat flow"?

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That is, a device which transfers heat from a colder body to a hotter body and does a positive net work on the surroundings during the cycle.

Although this statement of the second law is rather imprecise, the device you describe (which I will call a refrigeration engine) is indeed forbidden by it.

We already know of functioning devices which can take work and convert it to thermal energy, which we can generically call heaters. So, with your refrigeration engine, you could couple it with a heater in the hot reservoir to further increase the heat in the hot reservoir. Then you would wind up with a combined system that violates Clausius’ statement.

Now, since we know that heaters exist, and by Clausius’ statement the combined refrigeration engine/heater cannot exist, we can conclude that the refrigeration engine cannot exist.

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    $\begingroup$ Thanks. This is a really great answer. As is the one above. If i could accept both I would but yours is more or less exactly what I was after. $\endgroup$ Aug 8 '21 at 19:53
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If you assume Clausius' statement to be simply what you quoted you are in principle right: it does not explicitly forbid positive work in a refrigerator (which is forbidden by thermodynamics as we now know it). But that version of the statement is, let us say, unprecise or incomplete, and apparently Clausius himself was dissatisfied with it. The problem being that it lacks mathematical description: what is meant by "some other change"?

The point is, that it is not "any other change" that has to happen (which would include positive work, as you say). it is rather meant to be "it does not occurr spontaneously: something else must happen for the process to take place".

Unfortunately, I could not find Clausius' original version of the principle. German wikipedia states it in a slightly less misleading way, as "there is no change of state whose only result is the flow of heat from a cold to a warm source". In this version, it is clear that the point is not that "something else also has to happen for the process to take place" - which seems to indicate that as long as "anything else" happens, heat can flow in that direction - but rather that "you need to do something in order for the process to take place". That "something" that you need to do is work - as clearly stated by the successive formulations of the 2nd principle - for which Clausius himself played a fundamental role! And it is then found out that that something has to be work on your side, not any kind of work.

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  • $\begingroup$ Thanks for the great response! Okay I see what you mean. Doesn't this imply that the clausius statement (in the form I used in my post) contains less useful information than the statement "Heat can never pass from a colder to a warmer body without a positive net work input into the device occurring at the same time" because this latter statement specifies the exact kind of "other change" that has to occur. Surely the later statement is a more complete form of the 2nd law given this extra info it carries? Also which successive formulations are you referring to? The KelvinP statement? $\endgroup$ Aug 8 '21 at 13:23
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    $\begingroup$ Yes the latter is a more complete statement - it does not mean Clausius' is wrong, but you must be careful in understanding what it really means. And yes, I was referring to the Kelvin statement and several others you can find here: en.wikipedia.org/wiki/Second_law_of_thermodynamics $\endgroup$
    – JalfredP
    Aug 8 '21 at 13:36
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    $\begingroup$ This is a nide answer (+1). I've made a couple of minor edits. $\endgroup$
    – Gert
    Aug 8 '21 at 15:17
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    $\begingroup$ @SalahTheGoat That's what I've been taught, haha! A refrigerator, aka a heat pump requires positive INPUT of mechanical energy to pump heat from the cold sink (e.g. fridge inside) to the hot sink (back of the sink). $\endgroup$
    – Gert
    Aug 8 '21 at 16:23
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    $\begingroup$ And it makes sense: heat machines (like steam or internal combustion) draw heat from hot to cold and then produce mech. work. That's the exact opposite of a heat pump! $\endgroup$
    – Gert
    Aug 8 '21 at 16:52

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