when we mix 2 components of ideal gas. for example we mix the flow from pure A and the flow from pure B(A and B is ideal gas). if both temperature and pressure are constant, then entropy change will be expressed as;
Sfinal-Sinitial=-Rc[yA *ln(yA)+yB*ln(yB)] (S:total entropy per mole Rc:gas constant, yi: mole fraction of component i(i=A or B)
I can't understand why this expression doesn't apply to the situation that we mix 2 equal flows. such like mixing A and A, or mixing B and B.
For example, Let's suppose that we mix the pure A flow(let it called flow 1) and the pure A flow(let it called flow 2). Although 2 flows are constituted of equal material. it seems to me that the entropy change will be expressed such as:
Sfinal-Sinitial=-Rc[y1 *ln(y1)+y2*ln(y2)],(In this condition yi is defined as mole fraction of A which came from flow i(i=1 or 2))
But my textbook illustrate entropy change equals 0, naturally
Two A molecules can not be distinguished. Their exchange does not produce a new microstate.
Every real material absorbs and emits radiation. This applies, in particular, to gases. The classical theory of ideal gases ignores the interaction of the gas particles with thermal radiation. Thus, whenever this interaction is an essential part of the phenomenon, the classical theory is bound to lead to unrealistic conclusions.
If interaction with thermal radiation is taken into account, an ideal gas performs work even when it expands in a vacuum, provided that the vacuum is filled by thermal radiation (which is always the case when the vacuum is bounded by material at a temperature different than absolute zero). This produces temperature changes both in the free expansion of the ideal gas and in its adiabatic mixing with another ideal gas [A. Paglietti, Continuum Mech. Therm. 24, 201-210 (2012)].
It is true that mixing of two identical gases which are in the same initial state before mixing does not produce any entropy change. This is simply because the process does not produce any change in the state of two gases before and after they are joined together –whether the gases are ideal or not. But then the question arises that even the smallest difference beween the two gases will produce, upon mixing, the finite change of entropy that you refer to in your formulae.
This inconsistency of the classical theory is sometimes referred to as Gibbs’ paradox. The paradox can be resolved if the gas interaction with thermal radiation is taken into account. In this case a small difference in the property of the two ideal gases produces a proportional small change (as opposed to a given finite change) in their entropy after mixing.
This is known as the Gibbs paradox (which he formulated around 1875), and has a long and interesting history. You can read https://en.wikipedia.org/wiki/Gibbs_paradox to get an impression. There are numerous papers and textbooks claiming the paradox is resolved by quantum mechanics. Here are a few quotes:
"The resolution of this paradox is, in the last instance, another example of the successes of quantum mechanics in getting rid of some unsolved problems of classical mechanics, even though in this case there were strong arguments in favor of the generic densities." (D. ter Haar, Elements of Statistical Mechanics, p. 140.)
"It is not possible to understand classically why we must divide Σ(E)
by N! to obtain the correct counting of states. The reason is inherently quantum mechanical. Quantum mechanically atoms are inherently indistinguishable." (K. Huang, Statistical Mechanics, p. 153.)
"The resolution of Gibbs paradox lies in quantum mechancs [...] Identical particles must be counted in a different way from distinguishable particles (they have different ”statistics”). This difference between identical and distinguishable particles persists even in the classical limit and leads to a resolution of the Gibbs paradox." (From L.E. Reichl, A Modern Course in Statistical Physics, p. 68)
But E.T Jaynes (The Gibbs Paradox, p 17) makes a more interesting remark, in which he does not at all agree with the previous authors (after showing that the classical approach allows for an additional function dependent on N only in the entropy):
"Therefore it might appear that the shortcoming of classical statistical mechanics was not any failure to yield an extensive entropy function, but only its failure to determine the numerical value of the chemical constant. But even this criticism is not justified at present; the mere fact that it is believed to involve Planck’s constant is not conclusive, since e and c are classical quantities and Planck’s constant is only a numerical multiple of e2/c. We see no reason why the particular number 137.036 should be forbidden to appear in a classical calculation. Since the problem has not been looked at in this way before, and nobody has tried to determine that constant from classical physics, we see no ground for confident claims that it can or cannot be done." As far as I know, no one took up this challenge.
A relatively recent paper is: H. Peters, Demonstration and resolution of the Gibbs paradox of the first kind, Eur. J. Phys., 35, (2014), 015023.
Many times it was claimed that the paradox was resolved (or that it is not a paradox at all), but papers keep appearing about it. So, have fun exploring the topic.
Here is a list of 60 papers on the subject: http://gvanderzwan.nl/sites/lectures/GibbsParadox/node2.html
Entropy: A concept that is not a physical quantity
https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity
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