Gibbs Paradox

4 Key excerpts on "Gibbs Paradox"

• 

  eBook - ePub

  Gibbs' Entropic Paradox and Problems of Separation Processes

  • Eugene Barsky(Author)
  • 2017(Publication Date)
  • Elsevier

    (Publisher)

  Chapter 3

  The Gibbs Paradox and Attempts of Its Solution

  Abstract

  The notion of thermodynamic probability of a state ensures the number of microstates of a system, which provides the understanding of statistical entropy and ensemble of the system.

  Gibbs defines microstates for different and identical particles in a different way. On this basis, the entropy of a mixture of gases of different particles becomes greater than that of the sum of components before their mixing, and this increase is stepwise. It is considered that entropy is a peculiar energy, and therefore the existence of this discontinuity generates a paradox connected with the violation of the conservation law. Persistent attempts of solving this paradox during one and a half centuries have not given an unambiguous result. The solution of the Gibbs Paradox has been sought for experimentally. This search was based on the determination of the value of gas mixing work. Numerous unsuccessful attempts of such kind led to the acceptance of the fact that it is impossible to substantiate entropy jump within the framework of thermodynamics. Therefore, scientists try (sometimes) to come to an agreement that the entropy jump is absent while mixing different gases, but this is not proved in any way. The attempts of solving this paradox within the bounds of quantum physics statistics also have not given any results. Operational and informational approaches to the paradox based on entropy presentation as energy also have not given any results. The kinetic theory of gases looks especially problematic at an industrial separation of a mixture of gases, where the efficiency, from the standpoint of existing notions, is from 10

  − 9

  to 10−

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Handbook of Engineering and Non-Equilibrium Thermodynamics

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  Sevick et al. using optical tweezers apparatus. Gibbs Paradox In statistical mechanics, a simple derivation of the entropy of an ideal gas based on the Boltzmann distribution yields an expression for the entropy which is not extensive (is not proportional to the amount of gas in question). This leads to an apparent paradox known ________________________ WORLD TECHNOLOGIES ________________________ as the Gibbs Paradox, allowing, for instance, the entropy of closed systems to decrease, violating the second law of thermodynamics. The paradox is averted by recognizing that the identity of the particles does not influence the entropy. In the conventional explanation, this is associated with an indistin -guishability of the particles associated with quantum mechanics. However, a growing number of papers now take the perspective that it is merely the definition of entropy that is changed to ignore particle permutation (and thereby avert the paradox). The resulting equation for the entropy (of a classical ideal gas) is extensive, and is known as the Sackur-Tetrode equation. Poincaré recurrence theorem The Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The Recurrence theorem apparently contradicts the Second law of thermodynamics, which says that large dynamical systems evolve irreversibly towards the state with higher entropy, so that if one starts with a low-entropy state, the system will n ever return to it. There are many possible ways to resolve this paradox, but none of them is universally accepted. The most typical argument is that for thermodynamical systems like an ideal gas in a box, recurrence time is so large that for all practical purposes it is infinite.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Problems in the Philosophy of Science

  • Lev D. Beklemishev(Author)
  • 2000(Publication Date)
  • Elsevier Science

    (Publisher)

  It is always dangerous to extrapolate from a state of partial knowledge to conceptual regions too far remote from physical data! On the other hand, I admit that there is a host of reasons to which one can appeal if one wishes to adopt a critical attitude towards the H-theorem ; moreover, this theorem has never been rigorously derived and the range of its validity is therefore far from clear. 10. We next turn to the often-cited Gibbs entropy paradox or discontinuity paradox of Gibbs, which also arises in the domain of thermophysics and is ‘parasitic’ on Boltzmann’s entropy equation. Envisage two separated vessels containing different gas molecules, that is, of the species A and the species B. For the sake of con- venience we assume the two volumes to be equal, to have the same pressure, and hence to contain equal numbers of molecules. The entropy of gas A is SA and that of gas B is SB. From some established equations we can then derive the entropy of the mixture, if gases A and B are assumed to be ideal : SAB =SA +SB + 2Nk log 2, where N is the number of molecules. Verbally expressed: the entropy of the mixture of the two ideal gases exceeds total entropy of the separated gases by a factor 2Nk log 2. Workers in this field, re- flecting adequate acquaintance with the topic, would express the main argument more lucidly by using the physicist’s vernacular in the following way, In phenomenological thermodynamics the final entropy of a A BUDGET OF PARADOXES IN PHYSICS 189 mixture of two different yet chemically noninteracting gases XAB must, in concordance with the second law, be greater than the initially given entropy SA +SB of the unmixed gases. For the entropy of mixing we get AS=SAB - (SA +SU). Now, we posit that the two gases A and B are different but have the same number of molecules N . Therefore: AX=2Nk log 2, for A # B. And the increment AX is positive, as was to be expected.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Entropic Invariants of Two-Phase Flows

  • Eugene Barsky(Author)
  • 2014(Publication Date)
  • Elsevier Science

    (Publisher)

  P is the system probability, which is always equal or less than unity.

  Here, we outline some problems connected with entropy to be solved for generating a similar notion for two-phase flows.

  3. Problematic aspects of entropy

  There remains one unclear question connected to entropy dimension. It is well grounded that while forming the macroscopic entropy, Clausius ascribed to it a dimension equal to energy divided by temperature according to Eq. (1.1) , because he could do nothing else. However, many people perceive entropy as energy, as mentioned above. Recall once more that entropy according to Boltzmann is:

  H = k log φ and k =

  R

  N A

  where R is the absolute gas constant R   =  8.31  J/mol, and N A is the Avogadro number

  N A

  = 6.023 ·

  10 23

  1 mol

  .

  Hence, the Boltzmann's constant dimension gives, in essence, the dimension of the absolute gas constant. This reduces Boltzmann's entropy to Clausius' entropy, and, from the standpoint of physics, these parameters are identical. However, there is a latent drawback in this dimension, which often manifests itself in a paradoxical way.

  We can illustrate this by two examples connected with mixing and, especially, separation of gases. We conceive two isolated identical volumes with different numbers of gas particles (N 1 , N 2 ) that are different in these two volumes, but identical within each volume (Figure 1.1 ). Following Lorentz, we can assume that in each volume the entropy equals zero, because the probability of the presence of particles in each volume P   =  1 and log 1  = 

  Sign up to read

  Learn more about book