SELF

26

S.B. Karavashkin and O.N. Karavashkina

While the conventional derivation of statistic regularity for entropy does not account this basic aspect that reflects Yavorsky’s refinement [10] (we gave it in the beginning of item 5) that the terms ‘work’ and ‘heat’ have a dual sense: on one hand, it separates these concepts, but at the same time they have much in common, as in both cases this corresponds to a definite quantity of transferred energy. Consequently, in both cases, if the system’s temperature changed, the distribution function and the internal energy of the system change, too. This is natural, although the statistical physicists approach this issue otherwise.

“Now consider the change of subsystem’s energy in a more general case, when it interacts with surrounding bodies (medium) and exchanges with them the energy in the immediate contact… As during the whole time of process the system is in equilibrium, the probability distribution is determined by the Gibbs equilibrium distribution (this is why they needed the process to be quasi-static – Authors).

For the full change of the average energy, we can write

where w_i is the Gibbs distribution with the temperature equal to that of thermostat, and _i is the subsystem’s energy in the ith state. This last (the temperature – Authors), however, has not to be constant during this process” [11, p. 390].

This last stipulation says, though such separation in variation is formally admissible, the temperature of system is not time-constant, irrespectively of quasi-static pattern of process. While the author interprets the summands in the right part of (45) so: “The first summand in (45) still expresses the work done on the system. The second summand represents the part of energy change of the system interacting with the medium that is not relevant to the change of exterior parameters. In other words, the second summand in (45) is the change of average energy of the system that arises due to immediate energy transfer from the particles of medium to the particles of system, without change of exterior fields either bodies’ relative location. This part of energy change we will call the quantity of heat supplied to the system, denote it dQ. Then

The formula (46) is the energy conservation law for thermal processes” [11, p. 390].

At the same time, according to the Gibbs distribution law, w_i is the function of _i . And if in the first summand of (45) the energy _i changes, w_i will change inevitably. This points us, it is incorrect to mere formally vary the product w_ii while the multipliers are interrelated. The same in the second summand of (45). If w_i = f(_i), if this regularity was analytical, the function _i = f  ^-1(w_i) will exist, too, and this also corroborates the conclusion that such varying is illegal. To vary the full energy of the system, we have to do it so:

and in this equality we should not separate the summands into the work of external forces and heat supply, as both summands of the right part of (47) will change in case of some or other way to supply the energy to the system, though in each case in its own way, and we have to determine it in the specific conditions of modelling.

Otherwise it appears that for some small quantity of heat Q , on the basis of (45) and (46) we have to write the following chain of inequalities [11, p. 391]:

First of all we see that in (48) the work of external forces does not turn into zero because of the second term in the right part. Furthermore, in the second summand of right part of (48), though we vary in _i , in the exponential term this variable also appears due to the substitution of (40) into (48), i.e., by-passing the variation. And as the author points, when supplying the heat energy to the system, the static temperature of the system has to grow, but the parameter appears in the second term of the right part, by-passing the variation. And in this case the condition of quasi-static process is unable to retain the temperature of the system as whole. When lessen the introduced energy, the temperature growth will fall in proportion. If we neglect the temperature growth, we will have to automatically neglect the introduced heat, and vice versa. But (48) does not reflect this basic aspect and in this equation the introduced heat does not cause the temperature change of the whole system. At the same time, despite some crucially important parameters are present in the derivation, avoiding the known rules of modelling, the author in his further transformation of (48) thinks natural the following identity:

in which he varies already in the temperature but does not vary in (_i) .

Nothing of surprise that in the further transformation the author comes to the following expression [11, p. 391]:

and furthermore, “The formula (50) shows that in a quasi-static process the quantity of heat received either lost by the system can be represented as

where is the variation of some function

It is obvious that   is the full differential of the expression in brackets,

where const means an arbitrary constant. The function got the name of system’s entropy” [11, p. 394].

Basing on (53), we can easily show the interior discrepancies that followed from this chain of incorrect operations. Noting that is a full differential, determine the variation of . Yield

Substituting now into (54) the value of energy E and  E   from (46), yield

With it, even if we take the work of external forces and the value Q zero, the expression will take the following form:

and we already may not equalise to zero the second summand in (56), as [11, p. 370]

and we summate in all states of the system. This means, if at least for one subsystem the probability of existence with higher energy changes in retaining the probabilities for other subsystems, the value Z will inevitably change.

Comparing now (57) with (51), we see that even in a particular case, when the work done on the system was absent, these expressions are different. This means, the substitution that the author made in (53) was illegal, nothing to say of his generalisation of the problem, when the difference in definitions increases even more.

Thus, we see that the statistic description of entropy, even for ideal thermodynamic systems, is based on the illegal processing the regularities that determine the physical model. Together with the above conclusion of incorrect idea of entropy in nonlinear physical models and models having the internal sources, we can unambiguously conclude: entropy is a fully incorrect concept in thermodynamics, and its generalisation onto macro-scale systems is even more incorrect, as in such systems there are revealed the features ignored in the lab models.