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I am quite confused about Entropy.

If we look at the Maxwell-Boltzmann distribution of a gas we see a higher temperature gas is more spread out than a lower temperature gas. The hotter gas covers a much bigger range of velocities.

The hotter gas also has more entropy. Can we say that the hotter gas has more entropy because of this wider range of velocities? I really don't know if they are related or not, but it does seem that the wider range does make it more "disorderly"

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Here, you are relating two concepts: one where the law of thermodynamics are steadfast and logical and one where those laws are meaningless and instead statistics is key.

This is the difference between Thermodynamic Entropy and Informational Entropy.

From the point of view of the Second Law of Thermodynamics, the answer is simple: higher temperature gasses have faster moving particles and thus greater changes in entropy.

From the point of view of Informational Entropy, we care about the probability and positions of particles - at a macroscopic scale. So, if a gas' particles possess larger velocities, they theoretically have a greater number of positions to occupy than a gas with slower particles. This would mean that the higher temperature gas has more entropy than the lower temperature gas.

To answer your question: Yes, you can say the hotter gas has more entropy due to a wider range of velocities, but there are two approaches to concluding so.

I advise that you read up on entropy as it is a misleading topic with the "order" and "disorder" approach versus statistical. "joseph f. johnson" had a spectacular analogy and answer that was understandable and very informational.

Here is the link to his answer: Clear up confusion about the meaning of entropy

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