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I'm wondering about entropy, free energy and disorder. I see entropy defined along the lines of "the amount of energy within a system that is unavailable to be used for useful work". So, the energy that is available to do work, ie. the Free energy, is given by the internal energy minus the entropy contribution (using the Hemholtz free energy rather than Gibbs just for simplicity).

Also, I am aware that the entropy is associated with the level of disorder of a system. So, say we had a mixture of two gases confined in a box. Now say that gas A was all initially up in one corner, and gas B filled the rest of the box - but to make things equal these gases are similar in that particles of A don't repel each other any greater than they repel gas B, or any greater than B repels B. Also, we could say that every particle is initially positioned on some imaginary cubic lattice, so that it might look something like

[tex] \left(\begin{array}{cccc}A&A&B&B\\A&A&B&B\\B&B&B&B\\B&B&B&B\end{array}\right) [/tex]

and then some time passes until they have randomly ended up all positioned at one of these imaginary points again, and it has evolved to something like

[tex] \left(\begin{array}{cccc}A&B&B&B\\B&B&B&A\\B&B&A&B\\A&B&B&B\end{array}\right) [/tex]

Clearly the disorder has increased, in other words the entropy has increased. However, if all the particles repel each other equally and are at the same density throughout the system, just like how it started, then there would be no difference in pressure on any of the walls at either instant, right?

Well, where has the energy available to do work gone due to the disorder?

I'm imagining "useful work" as the ability to move a wall as if it was a piston, thus increasing the volume and lowering the pressure. Surely the gas particles could push the wall no differently when they are all mixed up compared to when they were separate, assuming there has been no temperature change just because the particles have rearranged themselves?

What am I missing here?

Also, going back to the Helmholtz free energy, it's written as F = U - TS, where U is the internal energy (ie. the potential energy due to the Coulomb interactions of the particles, right) and TS is the entropic term. I always wondered about the TS term, why does it have that form? I mean, S is measured in Joules per Kelvin, so multiplying it by the temperature makes the dimensions correct to be used in that equation, but why was it decided that TS is the amount of entropic energy present in the system?

I'm aware that increases in entropy go hand in hand with increases of temperature, but... still I'm not sure... looking at the units you could call the change in entropy "the change in energy per unit temperature", so multiplying it by the temperature gives you a change in energy, but what energy is this?

Also, I am aware that the entropy is associated with the level of disorder of a system. So, say we had a mixture of two gases confined in a box. Now say that gas A was all initially up in one corner, and gas B filled the rest of the box - but to make things equal these gases are similar in that particles of A don't repel each other any greater than they repel gas B, or any greater than B repels B. Also, we could say that every particle is initially positioned on some imaginary cubic lattice, so that it might look something like

[tex] \left(\begin{array}{cccc}A&A&B&B\\A&A&B&B\\B&B&B&B\\B&B&B&B\end{array}\right) [/tex]

and then some time passes until they have randomly ended up all positioned at one of these imaginary points again, and it has evolved to something like

[tex] \left(\begin{array}{cccc}A&B&B&B\\B&B&B&A\\B&B&A&B\\A&B&B&B\end{array}\right) [/tex]

Clearly the disorder has increased, in other words the entropy has increased. However, if all the particles repel each other equally and are at the same density throughout the system, just like how it started, then there would be no difference in pressure on any of the walls at either instant, right?

Well, where has the energy available to do work gone due to the disorder?

I'm imagining "useful work" as the ability to move a wall as if it was a piston, thus increasing the volume and lowering the pressure. Surely the gas particles could push the wall no differently when they are all mixed up compared to when they were separate, assuming there has been no temperature change just because the particles have rearranged themselves?

What am I missing here?

Also, going back to the Helmholtz free energy, it's written as F = U - TS, where U is the internal energy (ie. the potential energy due to the Coulomb interactions of the particles, right) and TS is the entropic term. I always wondered about the TS term, why does it have that form? I mean, S is measured in Joules per Kelvin, so multiplying it by the temperature makes the dimensions correct to be used in that equation, but why was it decided that TS is the amount of entropic energy present in the system?

I'm aware that increases in entropy go hand in hand with increases of temperature, but... still I'm not sure... looking at the units you could call the change in entropy "the change in energy per unit temperature", so multiplying it by the temperature gives you a change in energy, but what energy is this?

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