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Why when we have free expansion of a gas in a chamber, there is no energy increase? I understand why in phenomenological way but when I try to use thermodynamic equations I cannot understand the reason why energy doesn't change when entropy changes.

Now when a gas (who was in one half of the chamber) expands in the whole chamber, that means that we have an irreversible process, which is characterised by an entropy increase. Or we can say that now that the dimensions of the space in which the non-interacting gas particles can be found changed, then (since we know that the energy of a particle in a box is a function of quantum number's $n_x,n_y, n_z$ and the dimensions of the box) the energy of an individual particle will change (probably is reduced but I am not too sure) then the same happens for all the particles and subsequently for the whole gas. Because the particles have more "options" for the position in which they can be found, then doesn't that mean that we will have more micro states? And in equilibrium, the system goes the the macrostate with the higher multiplicity, which means with the higher amount of microstates, which have the same energy as the macrostate in equilibrium, and for the entropy that means :

$S=kln\Omega$ that it's value will increase. But if we move from one macrostate (when the particles are in one half of the box) to another one (where they are in the whole chamber and particles change energy but also entropy changes because we have a change in the multiplicity, otherwise entropy wouldn't change), isn't that the same as saying that the system goes from one state with an energy A to another one with energy B, how is it then that the energy for a free expansion is constant ?

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The internal energy doesn't change because we assume we are able to maintain adiabaticity. It is a condition your force on your system. It is very simple to think of expansion processes where the energy changes: for example, if you heat up a balloon, it will eventually pop because you have supplied the air molecules inside with energy.

The total number of microstates corresponding to the macrostate with the fixed energy $E$ does increase, which is why the entropy also does increase. The number of microstates has nothing to do with the energy -- only the entropy.

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  • $\begingroup$ how does the entropy increase when the nr. of microstates doesnt? $\endgroup$
    – imbAF
    Oct 23, 2021 at 16:53
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    $\begingroup$ The number of microstates does increase. In particular, if I think of my gas as a classical ideal gas (so that the phase space is the momentum-position space), then my energy is just p^2/2m, independent of position --- so the increase in the possible positions of the gas molecules increases the number of accessible microstates directly. $\endgroup$
    – r_phys
    Oct 23, 2021 at 16:57

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