Maybe you and I have different classical definitions of entropy. I quote: “∫ dQ/T = 0”. dQ would be the change in transfer of heat with external reservoirs, ant T would be temperature. Heat moving around within a system does not count towards dQ, therefore entropy doesn’t change no matter how the particles are configured.
Clausius went on and said there were such things as irreversible processes, which is in my opinion wrong, and refined the formula to an effective ∫ dQ/T > 0. Even so, both formulas are true.
Clausius said entropy of a closed system can only stay the same or increase. I say it never changes at all. You seem to be saying that it can increase or decrease. Somebody has to be wrong about something.
I’ve reproduced the relevant section of his paper in the spoiler below[1].
[SPOILER]According to this, the second fundamental theorem in the mechanical theory of heat, which in this form might appropriately be called the theorem of the equivalence of transformations, may be thus enunciated:
If two transformations which, without necessitating any other permanent change, can mutually replace one another, be called equivalent, then the generation of the quantity of heat Q of the temperature t from work, has the equivalence-value
Q
─
T
and the passage of the quantity of heat Q from the temperature t[SUB]1[/SUB] to the temperature t[SUB]2[/SUB], has the value
Q(1/T[SUB]2[/SUB] - 1/T[SUB]1[/SUB]),
wherein T is a function of the temperature, independent of the nature of the process by which the transformation is effected.
If, to the last expression, we give the form
Q/T[SUB]2[/SUB] - Q/T[SUB]1[/SUB],
it is evident that the passage of the quantity of heat Q, from the temperature t[SUB]1[/SUB] to the temperature t[SUB]2[/SUB], has the same equivalence-value as a double transformation of the first kind, that is to say, the transformation of the quantity Q from heat, at the temperature t[SUB]1[/SUB] into work, and from work into heat at the temperature t[SUB]2[/SUB]. A discussion of the question how far this external agreement is based upon the nature of the process itself would be out of place here; but at all events, in the mathematical determination of the equivalence-value, every transmission of heat, no matter how effected, can be considered as such a combination of two opposite transformations of the first kind.
By means of this rule, it will be easy to find a mathematical expression for the total value of all the transformations of both kinds, which are included in any circular process, however complicated. For instead of examining what part of a given quantity of heat received by a reservoir of heat, during the circular process, has arisen from work, and whence the other part has come, every such quantity received may be brought into calculation as if it had been generated by work, and every quantity lost by a reservoir of heat, as if it had been converted into work. Let us assume that the several bodies K[SUB]1[/SUB], K[SUB]2[/SUB], K[SUB]3[/SUB], &c., serving as reservoirs of heat at the temperatures t[SUB]1[/SUB], t[SUB]2[/SUB], t[SUB]3[/SUB], &c., have received during the process the quantities of heat Q[SUB]1[/SUB], Q[SUB]2[/SUB], Q[SUB]3[/SUB], &c., whereby the loss of a quantity of heat will be counted as the gain of a negative quantity of heat; then the total value N of all the transformations will be
N=Q[SUB]1[/SUB]/T[SUB]1[/SUB] + Q[SUB]2[/SUB]/T[SUB]2[/SUB] + Q[SUB]3[/SUB]/T[SUB]3[/SUB] + &c. . = Σ Q/T
It is here assumed that the temperatures of the bodies K[SUB]1[/SUB], K[SUB]2[/SUB], K[SUB]3[/SUB], &c. are constant, or at least so nearly constant, that their variations may be neglected. When one of the bodies, however, either by the reception of the quantity of heat Q itself, or through some other cause, changes its temperature during the process so considerably, that the variation demands consideration, then for each element of heat dQ we must employ that temperature which the body possessed at the time it received it, whereby an integration will be necessary. For the sake of generality, let us assume that this is the case with all the bodies; then the forgoing equation will assume the form
N = ∫ dQ/T,
wherein the integral extends over all the quantities of heat received by the several bodies.
If the process is reversible, then, however complicated it may be, we can prove, as in the simple process before considered, that the transformations which occur must exactly cancel each other, so that their algebraical sum is zero.
For were this not the case, then we might conceive all the transformations divided into two parts, of which the first gives the algebraical sum zero, and the second consists entirely of transformations having the same sign. By means of a finite or infinite number of simple circular processes, the transformations of the first part must admit of being made in an opposite manner, so that the transformations of the second part would alone remain without any other change. Were these transformations negative, i.e. from heat into work, and the transmission of heat from a lower to a higher temperature, then of the two the first could be replaced by transformations of the latter kind, and ultimately transmissions of heat froma lower to a higher temperature would alone remain, which would be compensated by nothing, and therefore contrary to the above principle. Further, were those transformations positive, it would only be necessary to execute the operations in an inverse manner to render them negative, and thus obtain the foregoing impossible case again. Hence we conclude that the second part of the transformations can have no existence.
Consequently the equation
∫ dQ/T = 0
is the analytical expression of the second fundamental theorem in the mechanical theory of heat.[/SPOILER]
~Max
[1] Clausius, M. R. (1856, August). On a modified Form of the second Fundamental Theorem in the Mechanical Theory of Heat. London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, Series 4, 12(77), 92-98. Retrieved from London, Edinburgh and Dublin Philosophical Magazine and Journal of Science : Free Download, Borrow, and Streaming : Internet Archive