Confusion about the entropy of mixing

In summary: If you have a basic understanding of quantum mechanics, you can go deeper into equilibrium statistical mechanics, but it's not required.In summary, Gibb's paradox in statistical mechanics is a seeming discrepancy between thermodynamics and the calculation of entropy when two identical boxes are combined. While thermodynamics suggests the combined entropy remains the same, a statistical mechanics viewpoint indicates a significant increase in entropy due to the increase in potential microstates. Can anyone help resolve this discrepancy?
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sha1000
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TL;DR Summary
I'm seeking clarity on a seeming discrepancy between thermodynamics and statistical mechanics concerning the calculation of entropy when two identical boxes are combined. While thermodynamics suggests the combined entropy remains the same, a statistical mechanics viewpoint indicates a significant increase in entropy due to the increase in potential microstates. Can anyone help resolve this?
Hello everyone,

I am seeking some clarification regarding a question related to thermodynamics and statistical mechanics. My understanding is that when we combine two identical boxes with the same ideal gas by removing the wall between them, the resulting system's entropy stays the same. Essentially, the total entropy of the new system is the summation of the entropies of the original two boxes (i.e., Stot = S1 + S2 = 2S1 or 2S2).

However, from the standpoint of statistical mechanics, it appears that this entropy increase might not be as straightforward. Let's consider that we have N1 particles in a volume V1, which results in an entropy of S1. If we duplicate this system with a partition in place, we can simply double the entropy. But, if we remove the partition, we're left with 2N1 particles in a volume of 2V1. My confusion arises from the fact that when calculating the number of microstates in this new system, the entropy seems to increase significantly due to the doubled number of particles in the doubled volume.

Could anyone shed some light on this apparent discrepancy between these two views?

Thank you in advance for your help!
 
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sha1000 said:
TL;DR Summary: I'm seeking clarity on a seeming discrepancy between thermodynamics and statistical mechanics concerning the calculation of entropy when two identical boxes are combined. While thermodynamics suggests the combined entropy remains the same, a statistical mechanics viewpoint indicates a significant increase in entropy due to the increase in potential microstates. Can anyone help resolve this?

Hello everyone,

I am seeking some clarification regarding a question related to thermodynamics and statistical mechanics. My understanding is that when we combine two identical boxes with the same ideal gas by removing the wall between them, the resulting system's entropy stays the same. Essentially, the total entropy of the new system is the summation of the entropies of the original two boxes (i.e., Stot = S1 + S2 = 2S1 or 2S2).

However, from the standpoint of statistical mechanics, it appears that this entropy increase might not be as straightforward. Let's consider that we have N1 particles in a volume V1, which results in an entropy of S1. If we duplicate this system with a partition in place, we can simply double the entropy. But, if we remove the partition, we're left with 2N1 particles in a volume of 2V1. My confusion arises from the fact that when calculating the number of microstates in this new system, the entropy seems to increase significantly due to the doubled number of particles in the doubled volume.

Could anyone shed some light on this apparent discrepancy between these two views?

Thank you in advance for your help!
If the gases in the two parts of the box are of identical particles (atoms/molecules), then the entropy doesn't change. If they are not identical there's mixing entropy. You get this right within statistical mechanics, using quantum theory. Anyway, quantum statistical mechanics is simpler than classical. If you know enough quantum theory, it's thus easier to learn statistical physics starting from quantum many-body theory and, for equilibrium, the maximum-entropy principle and take the classical limit to get the results of classical statistics. The same also holds for off-equilibrium statistical mechanics, where you can derive the Boltzmann(-Uehling-Uhlenbeck) equation via the Kadanoff-Baym equations of quantum many-body theory.
 
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Related to Confusion about the entropy of mixing

What is entropy of mixing?

Entropy of mixing is a measure of the increase in entropy when two or more different substances are combined. It quantifies the degree of disorder or randomness that results from mixing different particles or molecules.

Why is there confusion about the entropy of mixing?

The confusion often arises because entropy of mixing can be counterintuitive. People struggle to understand why mixing two substances, even without any energy exchange, results in an increase in entropy. Additionally, the statistical mechanics underlying the calculations can be complex and non-intuitive.

How is the entropy of mixing calculated?

The entropy of mixing for ideal gases or ideal solutions can be calculated using the formula: ΔS_mix = -nR(Σ x_i ln(x_i)), where n is the total number of moles, R is the gas constant, and x_i is the mole fraction of each component. This formula is derived from the principles of statistical mechanics.

Does the entropy of mixing always increase?

Yes, for ideal solutions and gases, the entropy of mixing always increases. This is because mixing results in a greater number of possible microstates for the system, leading to higher entropy. However, in non-ideal cases, interactions between particles can complicate this picture, but generally, entropy still tends to increase.

What are some common misconceptions about the entropy of mixing?

Common misconceptions include the belief that entropy of mixing is always zero for identical substances or that it is only relevant for gases. In reality, entropy of mixing is zero for identical substances, but it is also relevant for liquids and solids, provided they can mix at a molecular level. Another misconception is that entropy of mixing is always a simple additive property, ignoring the complexities introduced by non-ideal behavior.

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