# On the So-Called Gibbs Paradox, and on the Real Paradox

## Abstract

**:**

_{Class}is not an extensive function of the variables E, V, N [3].

_{I}= N ln 2

_{II}= 0

_{I}, when the two components A and B are distinguishable, to ∆S

_{II}when they become ID. The fact is that one never observes any intermediary value between ∆S

_{I}and ∆S

_{II}. The fact that ∆S changes discontinuously as one changes the ID continuously, is viewed as a paradox. However, there is no paradox here, and there was no allusion to any paradox in Gibbs writings. There are many examples that a discontinuous transition follows a continuous change of a parameter. For instance the fact that the density of water changes discontinuously when the temperature changes continuously, say between 90C to 110C is not viewed as a paradox. Furthermore, the presumed continuous change in the extent of ID of the particles is now recognized as, in principle, invalid. Particles are either distinguishable or ID-there are no intermediate values of indistinguishability [6].

**Figure 2.**Continuous change of colors, labels or shapes; The objects change from different to identical. However they remain distinguishable.

But if such considerations explain why the mixture of gas-masses of the same kind stands on different footing from mixtures of gas-masses of different kinds, the fact is not less significant that the increase of entropy due to mixture of gases of different kinds in such a case as we have supposed, is independent of the nature of the gases.

If we should bring into contact two masses of the same kind of gas, they would also mix but there would be no increase in entropy.When we say that when two different gases mix by diffusion, and the entropy receives a certain increase, we mean that the gases could be separated and brought to the same volumeby means of certain changes in external bodies, for example, by the passage of a certain amount of heat from a warmer to a colder body. But when we say that when two gas masses of the same kind are mixed under similar circumstances, there is no change of energy or entropy, we do not mean that the gases which have been mixed can be separated without change to external bodies. On the contrary, the separation of the gases is entirely impossible.

- Mixing two different gases causes increase in disorder.
- Increase in disorder is conceived as an increase in entropy.Therefore from 1 and 2 it follows that:
- Mixing causes an increase in entropy.

## References and Notes

- Gibbs, J.W. Scientific papers; Vol. I Thermodynamics, Longmans Green: New York, 1906; p. 166. [Google Scholar]
- Ben-Naim, A. Statistical thermodynamics based on Information. A farewell to Entropy.; Scientific- World: Singapore, 2007. [Google Scholar]
- Hill, T.L. Introduction to Statistical Thermodynamics; Addison Wesley: Reading, Mass, 1960. [Google Scholar]
- Ben-Naim, A. Molecular Theory of solutions; Scientific-World: Singapore, 2006. [Google Scholar]
- Note that if one uses the purely classical partition function for ideal gases, one does not get the results (1) and (2) which are consistent with experiments. We assume here that (1) and (2) were derived from the classical limit of the quantum mechanical partition function, i.e., after introducing the correction due to the ID of the particles.
- See for example reference 2.
- Gibbs (1906) page 167
- Ben-Naim, A. Is mixing a thermodynamic process? Am. J of Phys.
**1987**, 55, 725–733. [Google Scholar] [CrossRef] - Ben-Naim, A. Entropy Demystified. The second law of thermodynamics reduced to plain common sense.; World Scientific: Singapore, 2007. [Google Scholar]

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**MDPI and ACS Style**

Ben-Naim, A.
On the So-Called Gibbs Paradox, and on the Real Paradox. *Entropy* **2007**, *9*, 132-136.
https://doi.org/10.3390/e9030133

**AMA Style**

Ben-Naim A.
On the So-Called Gibbs Paradox, and on the Real Paradox. *Entropy*. 2007; 9(3):132-136.
https://doi.org/10.3390/e9030133

**Chicago/Turabian Style**

Ben-Naim, Arieh.
2007. "On the So-Called Gibbs Paradox, and on the Real Paradox" *Entropy* 9, no. 3: 132-136.
https://doi.org/10.3390/e9030133