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Ideal Gases and Solutions
Published in Jeffrey Olafsen, Sturge’s Statistical and Thermal Physics, 2019
While the pressure of each gas (its partial pressure) has halved in this mixing process, the total pressure, which is the sum of the partial pressures, is unchanged. What if both gases are the same? Then nothing is changed by the removal of the partition; the entropy before and after its removal must be the same. In classical theory this result is a paradox, since one could imagine the two gases being made more and more alike, until it became impossible to distinguish them by their properties; yet the increase in entropy due to mixing is independent of how different the two gases are. Quantum theory, which draws a clear distinction between distinguishable and indistinguishable particles, resolves the paradox. If the two gases are different, a molecule of one is distinguishable from a molecule of the other, and entropy is increased by mixing. If the gases are truly identical, the molecules are indistinguishable, and there is no entropy increase. For example, suppose that each gas is isotopically pure, but the two gases consist of different isotopes of the same element. In this case the molecules of one gas are distinguishable from those of the other (though not by chemical means), and there is an increase in entropy on mixing. However, if we start with the same isotopic mixture on either side of the partition, the entropy of mixing is already present before the partition is removed and does not increase further. Since the entropy of mixing exists even in pure thermodynamics, Gibbs’ paradox illustrates the fact that even this classical theory requires the concept of indistinguishability, which derives from quantum theory, to make it self-consistent.4
Gibbs paradox of some thermodynamic properties in one-dimensional Gross-Pitaevskii equation
Published in Yuli Rahmawati, Peter Charles Taylor, Empowering Science and Mathematics for Global Competitiveness, 2019
We have presented several quantities experiencing the Gibbs paradox and solved the problem by treating a set of condensates as a set of indistinguishable oscillators. To reach our goal, we initially make the condensate partition function to be extensive by defining the harmonic volume associated with the inverse-cube of the average geometric trapping frequency. In this case, we perform the scaling transformation to the partition function with the dimensionless parameter, in which the inverse cube of average geometric frequency is included.
Gibbs’ paradox according to Gibbs and slightly beyond
Published in Molecular Physics, 2018
Indeed, as illustrated in Figure 1, two cases can a priori be distinguished: either the gases to be mixed are different (case A in Figure 1) or they are identical (case B in Figure 1). In either case, one finds that, according to Equation (3), the entropy of mixing reads where the final volume Vfin of the box for each gas is double that of the initial volume Vin they were in Vfin = 2Vin. In case A, the result of Equation (5) is consistent with the standard thermodynamics result one can obtain by considering the mixing process as a free expansion of either gas into a volume of double its initial size [19]. In case B, however, we see from Figure 1(B) that there is no change in the thermodynamic state of the system upon mixing and, therefore, the corresponding entropy change should be zero. As it stands, Equation (3) thus leads to some predictions at odds with the theory of thermodynamics. This particular inconsistency is often referred to as Gibbs’ paradox of mixing. More generally, the Gibbs’ paradox can also refer to the technical root cause of Gibbs’ paradox of mixing; namely the fact that the free energy (and entropy) described by Equation (3) are not extensive quantities contrary to their counterparts in thermodynamics. In fact, the failure of case B is easily seen to be attributable to the fact that, according to Equation (3), SIG(N, V, β) ≠ 2SIG(N/2, V/2, β). The technical resolution of this problem is surprisingly very simple: one introduces a slightly different partition function that reads and defines equivalently a free energy and entropy, which for an ideal gas give (upon using the Stirling approximation) and, The mixing entropy between two gases as in Figure 1 becomes then where ρin and ρfin are, respectively the initial and final number densities of the gases undergoing the mixing process. In case A, ρin = 2ρfin and Equation (8) gives back an entropy of mixing of kBln 2 per particle. In case B, the number density is unchanged between the final and initial states, i.e. ρin = ρfin and therefore Equation (8) gives zero mixing entropy. Equivalently, one finds that , illustrating that the Gibbs’ paradox has been dealt with.