The Gibbs Paradox

Last week in this blog, I wrote that the “Gibbs Paradox” deserved an entire entry of its own. Well, here it is. Russ Hobbie and I mention the Gibbs Paradox in a footnote in Section 3.18 (The Chemical Potential of a Solution) in the 4th edition of Intermediate Physics for Medicine and Biology. When calculating the entropy of mixing (where a solute and solvent are intermixed), we derived an expression for the number of ways N particles can be distributed among N sites. If we assume the solute particles are indistinguishable, there is only one way. The footnote then reads

The fact that there is only one mircostate because of the indistinguishability of the particles is called the Gibbs paradox. For an illuminating discussion of the Gibbs paradox, see Casper and Freier (1973).

Fundamentals of Statistical
and Thermal Physics,
by Frederick Reif.

The Gibbs Paradox is examined in more detail by Frederick Reif in his landmark textbook Fundamentals of Statistical and Thermal Physics. (Indeed, Chapter 3 of Intermediate Physics for Medicine and Biology follows a statistical approach similar to Reif’s analysis, and even more similar to the discussion in his introductory textbook—a personal favorite of mine—Statistical Physics, Berkeley Physics Course Volume 5). Reif considers “a gas consisting of N identical monatomic molecules of mass m enclosed in a container of volume V.” When he calculates the entropy, S, of the gas, he obtains

S = N k [ln V + 3/2 ln T + σ]     (7.2.16)

where k is Boltzmann’s constant, T is the absolute temperature, and σ is a constant independent of N, T, and V. He then ends the section with the provocative statement “This expression for the entropy is, however, not correct,” which leads to his discussion (Sec. 7.3) of the Gibbs paradox. Reif continues

The challenging statement at the end of the last section suggests that the expression (7.2.16) for the entropy merits some discussion… [The expression] for S is clearly wrong since it implies that the entropy does not behave properly as an extensive quantity. Quite generally, one must require that all thermodynamic relations remain valid if the size of the whole system under consideration is simply increased by a scale factor α, i.e., if all its extensive parameters are multiplied by the same factor α. In our case, if the independent extensive parameters V and N are multiplied by α, the mean energy… is indeed properly increased by this same factor, but the entropy S in (7.2.16) is not increased by α because of the term N ln V.

Indeed, (7.2.16) asserts that the entropy S of a fixed volume V of gas is simply proportional to the number N of molecules. But this dependence on N is not correct, as can readily be seen in the following way. Imagine that a partition is introduced which divides the container into two parts. This is a reversible process which does not affect the distribution of systems over accessible states. Thus, the total entropy ought to be the same with, or without, the partition in place; i.e.

S = S' + S"     (7.3.1)

where S' and S" are the entropies of the two parts. But the expression (7.2.16) does not yield the simple additivity required by (7.3.1). This is easily verified. Suppose, for example, that the partition divides the gas into two equal parts, each containing N' molecules of gas in a volume V'. Then the entropy of each part is given by (7.2.16) as

S' = S" = N' k [ln V' + 3/2 ln T + σ]

while the entropy of the whole gas without partition is by (7.2.16)

S = 2 N' k [ ln (2 V') + 3/2 ln T + σ] .

Hence

S – 2 S' = 2 N' k ln(2 V') – 2 N' k ln V' = 2 N' k ln2

and is not equal to zero as required by (7.3.1).

This paradox was first discussed by Gibbs and is commonly referred to as the “Gibbs paradox.” Something is obviously wrong in our discussion; the question is what.

Reif then analyzes in more detail the implications of removing the partition between the two sides of the box. He finds that

The act of removing the partition has thus very definite physical consequences. Whereas before removal of the partition a molecule of each subsystem could only be found within a volume V', after the partition is removed it can be located anywhere within the volume V = 2 V'. If the two subsystems consisted of different gasses, the act of removing the partition would lead to diffusion of the molecules throughout the whole volume 2V' and consequent random mixing of the different molecules. This is clearly an irreversible process; simply replacing the partition would not unmix the gases. In this case the increase in entropy in (7.3.2) would make sense as being simply a measure of the irreversible increase of disorder resulting from the mixing of unlike gases [the entropy of mixing that Russ and I calculated].

But if the gases in the subsystems are identical, such an increase of entropy does not make physical sense. The root of the difficulty embodied in the Gibbs paradox is that we treated the gas molecules as individually distinguishable, as though interchanging the positions of two like molecules would lead to a physically distinct state of the gas. This is not so. Indeed, if we treated the gas by quantum mechanics (as we shall do in Chapter 9), the molecules would, as a matter of principle, have to be regarded as completely indistinguishable. A calculation of the partition function would then automatically yield the correct result, and the Gibbs paradox would never arise. Our mistake has been to take the classical point of view too seriously. Even though one may be in a temperature and density range where the motion of molecules can be treated to a very good approximation by classical mechanics, one cannot go so far as to disregard the essential indistinguishability of the molecules.

In a sidenote, Reif adds

Just how different must molecules be before they should be considered distinguishable?… In a classical view of nature two molecules could, or course, differ by infinitesimal amounts… In a quantum description this troublesome question does not arise because of the quantized discreteness of nature… Hence the distinction between identical and nonidentical molecules is completely unambiguous in a quantum-mechanical description. The Gibbs paradox thus foreshadowed already in the last [19th] century conceptual difficulties that were resolved satisfactorily only by the advent of quantum mechanics.

Several good American Journal of Physics articles discuss the Gibbs phenomenon. Pesic examines Gibb’s own writings to trace his thoughts on the issue (“The Principle of Identicality and the Foundations of Quantum Theory: I. The Gibbs Paradox,” American Journal of Physics, Volume 59, Pages 971–974, 1991), and Landsberg and Tranah study in more detail in role of the Gibbs paradox for quantum mechanics (“The Gibbs Paradox and Quantum Gases,” American Journal of Physics, Volume 46, Pages 228–230, 1978). Finally, Casper and Freier (the authors of the paper cited in our footnote) analyze the Gibbs paradox by comparing macroscopic and microscopic points of view (“‘Gibbs Paradox’ Paradox,” American Journal of Physics, Volume 41, Pages 509–511, 1973).

You know, there is a lot of physics in that little footnote on page 68 of Intermediate Physics for Medicine and Biology.