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The Entropy Mystery of Ideal Gases: What surprising changes occur when gases are mixed?
In the world of thermodynamics, entropy is considered an important measure of the degree of chaos or disorder in a system. However, when we mix ideal gases together, entropy behaves in unexpected and bizarre ways, leading to discussions of the "Gibbs Paradox." This paradox explores the non-additivity of entropy caused by gases that do not consider the indistinguishability of particles, making entropy appear anomalous in some cases and even seem to violate the second law of thermodynamics.
Gibbs' paradox makes us rethink, can entropy really increase when two gases are mixed?
Background of Gibbs Paradox
Between 1874 and 1875, physicist Josiah Willard Gibbs investigated a very simple but thought-provoking experiment. He imagined two identical containers of an ideal gas. When their partitions were removed, the gases encountered mixing, which resulted in unexpected changes in entropy. Although the state does not change in nature, the entropy of the mixture is predicted to be more than twice the entropy of the individual gases according to the non-additive entropy formula, forming the core of the paradox.
Through these thoughts, we seem to have reached the boundary of entropy and mixing, but what is the physical meaning behind this?
Calculation of entropy: state of ideal gas
In classical mechanics, for an ideal gas, we define its state as a specific energy U, volume V, and N particles. Each particle has a certain momentum and position, and this setup transforms the state description of the gas into points in a six-dimensional phase space. However, according to quantum mechanics, many states of phase space are actually limited. Especially when the energy is precisely fixed, we may even only obtain a certain state, and then reduce the entropy to zero.
Solving Gibbs' Paradox: The Importance of Indistinguishability
To resolve this paradox, we need to reconsider the indistinguishability of particles. Gibbs' method at that time ignored the mutual identification of particles. Even at the microscopic level, the states of each particle randomly arranged should actually be counted as the same. All this led to the important discovery of correct calculation of entropy, ensuring that entropy can be additious in the case of large samples.
If particle indistinguishability is properly defined, how will our framework be reshaped as a result?
Revelation from the Mixed Paradox
Another important issue is closely related to the hybrid paradox. The core of this paradox lies in how we deal with the change in entropy when gas A and gas B are mixed. If the two gases are slightly different, the entropy of the mixed system will increase. And when they are the same, the change in entropy disappears. This brings us to the definition of entropy itself: it depends to some extent on our ability to observe and discern. Under the premise that the difference between gases cannot be detected, the calculation method of entropy becomes relatively flexible.
Conclusion
Overall, Gibbs's Paradox and the Mixing Paradox remind us of some profound truths about entropy. The nature of entropy is somewhat subjective, and understanding a particular system depends on whether the particles are identifiable. As we delve deeper into these paradoxes, we may discover, is the definition of entropy accurate enough to truly describe our physical world?
