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I am working through some notes of Gould and Tabochnik here and I am confused by their argument showing that entropy does not change in a quasi-static adiabatic process. First, entropy is defined to be an additive function of the state which is not allowed to decrease in an isolated system. Then a bit later there is the following paragraph

For pure heating or cooling the increase in the entropy is given by $$dS = \left(\frac{\partial S}{\partial E}\right) dE.\qquad (2.68)$$ In this case $dE = dQ$ because no work is done. If we express the partial derivative in (1.68) in terms of $T$, we can rewrite (1.68) as $$dS = dQ/T \qquad \text{(pure heating)}. (2.69)$$ We emphasize that the relation (1.69) holds only for quasistatic changes. Note that (1.69) implies that the entropy does not change in a quasistatic, adiabatic process.

But (2.69) (mis-labelled as 1.69) only applies when no work is done. But certainly work might be done in a quasi-static adiabatic process, so how can (2.69) possibly apply?

Instead, is the following argument valid? A quasi-static process is one which passes through a sequence of equilibrium states. In each equilibrium state, the entropy has to be at its maximum value. Thus, the entropy cannot increase in passing from one equilibrium state to another. So overall the entropy can't change in a quasi-static process either.

But I don't think this argument can be right either, because then the same argument would apply to any quasi-static process, so why do we need "adiabatic" as well?

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so how can (2.69) possibly apply?

In fact the relation $$ dS = \frac{dQ}{T} $$ is valid for any quasi-static change, not only for adiabatic change. This is because the relation is actually basis of definition of thermodynamic entropy.

Instead, is the following argument valid? A quasi-static process is one which passes through a sequence of equilibrium states. In each equilibrium state, the entropy has to be at its maximum value. Thus, the entropy cannot increase in passing from one equilibrium state to another. So overall the entropy can't change in a quasi-static process either.

No, it is not valid. When you read "entropy attains its maximum value" the latter is meant as maximum $for~given~internal~energy~and~volume$. When the system passes through different equilibrium states, its energy and volume change and the corresponding maximum entropy changes too.

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