Ok, I feel it's time to (sort of) wrap up this thread with a discussion of Clausius 4th, 5th, and 6th papers (
http://www.humanthermodynamics.com/Clausius.html#anchor_116), in which he obtains the analytical expression for entropy we are all exposed to in introductory classes. This will conclude my analysis of the historical literature; we have covered quite a bit of material already!
First I'd like to thank everyone who participated in making this thread a useful one- I am taking a lot of the material here and turning it into some decent lecture notes.
Ok- in 1854 Clausius opens his fourth paper with
"In my memoir “On the Moving Force of Heat, &c.”, I have shown that the theorem of the equivalence of heat and work, and Carnot’s theorem, are not mutually exclusive, by that, by a small modification of the latter, which does not affect its principle, they can be brought into accordance. With the exception of this indispensable change, I allowed the theorem of Carnot to retain its original form, my chief objection then being, by the application to the two theorems to special cases, to arrive at conclusions which, according as they involved known or unknown properties of bodies, might suitably serve as proofs of the truth of the theorems, or as examples of their fecundity.
Clausius doesn't cite work by anyone else (here or anywhere). He first states his 'first theorem':
"Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the one being always proportional to that of the other."
But we know that this is incomplete: the amount of work that heat may produce in a process is also proportional to the temperature; in a cycle, the temperature *difference* between hot and cold. Clausius continues:
"The forces which here enter into consideration may be divided into two classes: those which the atoms of a body exert upon each other, and which depend, of course, upon the nature of the body, and those which arise from the foreign influences to which the body may be exposed. According to these two classes of forces which have to be overcome, of which the latter are subject to essentially different laws, I have divided the work done by heat into interior and exterior work.
He writes this as (in modern notation)
\dot{E} = JQ - p\dot{V}
Clausius then considers P = P(V, T) and E = E(V, T), and obtains
J \Lambda - p = \frac{\partial E}{\partial V}
JK = \frac{\partial E}{\partial T}
And so his 'first theorem' is what we have seen several times already:
\frac{\partial p}{\partial T} = J(\frac{\partial \Lambda}{\partial T}-\frac{\partial K}{\partial V})
Next, he states a 'second theorem':
"The theorem, as hitherto used, may be enunciated in some such manner as the following:
In all cases where a quantity of heat is converted into work, and where the body effecting this transformation ultimately returns to its original condition, another quantity of heat must necessarily be transferred from a warmer to a colder body; and the magnitude of the last quantity of heat, in relation to the first, depends only upon the temperature of the bodies between which heat passes, and not upon the nature of the body effecting the transformation.
"In deducing this theorem, however, a process is contemplated which is too simple a character; for only two bodies losing or receiving heat are employed, and it is tacitly assumed that one of the two bodies between which the transformation of heat takes place is the source of the heat which is converted into work. Now by previously assuming, in this manner, a particular temperature of the heat converted into work, the influence which a change of this temperature has upon the relation between the two quantities of heat remains concealed, and therefore the theorem in the above form is incomplete.
[...]
"Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.
"Everything we know concerning the interchange of heat between two bodies of different temperature confirms this; for heat everywhere manifests a tendency to equalize differences of temperature, and therefore to pass in contrary direction, i.e. from a warmer to colder bodies. Without further explanation, therefore, the truth of this principle will be granted."
Consider what Clausius just asserted: a certain process is forbidden to occur under any circumstance, and he does not offer any quantitative proof.
Clausius continues:
"On considering the results of such processes more closely, we find that in one and the same process heat may be carried from a colder to warmer body and another quantity of heat transferred from a warmer to a colder body without any other permanent change occurring. In this case we have not a simple transmission of heat from a colder to a warmer body, or an ascending transmission of heat, as it may be called, but two connected transmission of opposite characters, one ascending and the other descending, which
compensate each other. It may, moreover, happen that instead of a descending transmission of heat accompanying, in the one and the same process, the ascending transmission, another permanent change may occur which has the peculiarity of not being reversible without either becoming replaced by a new permanent change of a similar kind, or producing a descending transmission of heat. In this case the ascending transmission of heat may be said to be accompanied, not immediately, but immediately, by a descending one, and the permanent change which replaces the latter may be regarded as a compensation for the ascending transmission.
"Now it is to these compensations that our principle refers; and with the aid of this conception the principle may be also expressed thus: an
uncompensated transmission of heat from a colder to a warmer body can never occur. "
(emphasis mine)
Again, Clausius make a definitive claim that a certian process can *never* occur, but does not provide any justification. However, he does go on to calculate something useful:
"If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generations of the quantity of heat C of the temperature t from work, has the equivalence-value:
C/T
"and the passage of the quantity of heat Q from the temperature t1 to the temperature t2, has the equivalence-value:
C(1/T2-1/T1)
"wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.
That's confusing! He said T is a *function* of the temperature, not "the" temperature. As it happens (luckily for Clausius), his function coincides with Kelvin's definition of the absolute temperature T =J/\mu. But for now, 'T' is not 'temperature'.
Clausius then analyzes a series of thermal reservoirs and the flow of heat from the first to the final. This is simply \sum \frac{C}{T}. Passing to the continuum limit, and considering a cyclic process, Clausis obtains:
"The equation:
N = \int \frac{dC}{T} = 0
is the analytical expression, for all reversible cyclical processes, of the second fundamental theorem in the mechanical theory of heat."
Clausius then considers irreversible processes:
"If we represent the transformations which occur in a cyclical process by these expressions, the relation existing between them can be stated in a simple and definite manner. If the cyclical process is reversible, the transformations which occur therein must be partly positive and partly negative, and the equivalence-values of the positive transformations must be together equal to those of the negative transformations, so that the algebraic sum of all the equivalence-values become = 0. If the cyclical process is not reversible, the equivalence values of the positive and negative transformations are not necessarily equal, but they can only differ in such a way that the positive transformations predominate. The theorem respecting the equivalence-values of the transformations may accordingly be stated thus: The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing."
Clausius then simply writes, for "every cyclical process which is in any way possible." (not just reversible):
\int\frac{dC}{T} \geq 0.
Hopefully, you can see why there has been so much confusion about what the entropy 'really is'. Specifically, Clausius has made a series of vague statements in order to write a formula, presented without proof or derivation. There really is no logical foundation to Clausius' work.
Skipping ahead to Clausius' 9th paper, he picks up where he left off:
"The other magnitude to be here noticed is connected with the second fundamental theorem, and is contained in equation (IIa). In fact if, as equation (IIa) asserts, the integral:
\int\frac{dC}{T}.
"Vanishes whenever the body, starting from any initial condition, returns thereto after its passage through any other conditions, then the expression dC/T under the sign integration must be the complete differential of a magnitude which depends only on the present existing condition of the body, and not upon the way by which t reached the latter. Denoting his magnitude by S, we can write
dS = dC/T
"or, if we conceive this equation to be integrated for any reversible process whereby this body can pass from the selected initial condition to its present one, and denote at the same time by So the value which the magnitude S has in that initial condition,
S = S_{0} + \int\frac{dC}{T}
"This equation is to be used in the same way for determining S as equation (58) was for defining U. The physical meaning of S has already been discussed in the Sixth Memoir.
"we obtain the equation:
S - S_{0} = \int\frac{dC}{T}
"We might call S the
transformation content of the body, just as we termed the magnitude U its thermal and ergonal content. But as I hold it to be better terms for important magnitudes from the ancient languages, so that they may be adopted unchanged in all modern languages, I propose to call the magnitude S the
entropy of the body, from the Greek word τροπη, transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy; for the two magnitudes to be denoted by these words are so nearly allied their physical meanings, that a certain similarity in designation appears to be desirable.
[...]
"For the present I will confine myself to the statement of one result. If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat:
1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum."
Here (finally) are the laws of thermodynamics written down in a form that most of us have seen. I also want to note that I have read the 6th paper as closely as I could, I cannot guess what Clausius meant by "The physical meaning of S has already been discussed in the Sixth Memoir." I believe he meant "uncompensated" processes, but I wouldn't exactly call that a 'physical meaning'.
So, we have covered the development of Thermodynamics from 1822 (Carnot's initial report) through 1865 (Clausius's naming of 'entropy' and two laws of thermodynamics). Sadly, the field did not progress much over the next 100 years; as a result, many textbooks (most of which were first written in the 1950s and 1960s) relay the subject as it was written 100 years previously; that is without any coherent logic and mathematical structure. Fortunately (for us) during the past 20 years or so, the foundational elements of thermodynamics (heat, temperature, entropy) are being re-examined and refined, and there have been several excellent reviews posted on this thread.