Gibbs’ free energy is only a rough approximation, which is not the same as the exact value, which is usually computed in such a way that the Gibbs’ equation is a simple sum. The exact values for that exact equation are still unclear, so Gibbs’ free energy is a rough approximation. Gibbs’ free energy is one dimensionless number, and so even a good approximation would not be quite correct. This is the only reason for the discrepancy with Maxwell’s equation.
It’s true that Gibbs’ equation has a few extra terms (there are an extra four energy terms in the calculation, see above). Still, the number of energy terms is so small that they don’t add up to anything noticeable, compared to Maxwell’s equation. For example, the number of energy terms in Gibbs’ equations isn’t a factor of 100. For example, it’s roughly the same as the number of different combinations of energy terms in Maxwell’s equation for example. So the difference between Gibbs’ numbers and Maxwell’s isn’t that large.
In fact, given a small set of energy terms, a few hundred of them do a pretty good job of representing the entire range of possible energy states. It is possible to think of Gibbs’ equation as representing a single potential energy state with lots of particles, while Maxwell’s equation represents a single energy state that has no particles, since it can only have one potential energy state (it’s a potential energy state with no particles).
Now, we see. How does this work for free electron mobility?
If a field is in tension, such as a magnetic field, then a free electron is trapped. However, we do not have free electrons, so there can be no free electron migration. In a tensionless field, there will be free electron mobility, but we have no free electrons (as in the situation when you have no free atoms, or no free atoms that are moving at the same speed, to be able to measure the energy states in the molecules). So, in the situation where there is free electron mobility, there is no free electron movement; therefore, no charge transfer.
Free electron transport is the process by which electrons are emitted from the protons into the other atoms in the molecule. There can be a free electron transport process, if there are some protons/protons in the gas that are moving quickly enough to be free (which is what free transport does in this case, but doesn’t happen in the normal gas and we shouldn’t
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