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One has taken dW = -PdV which means that the
system is IN EQUILIBRIUM all along. But the process is not necessarily
reversible - the system is allowed to absorb, slowly though, heat from
surroundings at a higher temperature.
By the way, there are easy ways of checking whether the
entropy is a state function for more complex systems - it isn't of course.
Let us test eq. /2/ by considering an analogous situation. We
have a large
group of people each of which possesses a certain amount of money. Let
M = SUM n_i m_i /6/
be the total amount of money in the group and n_i is the number of
people whose
money is m_i. Through exchanging money with the outside world M slightly
changes and the change can be expressed in two ways:
dM = SUM m_i dn_i + SUM n_i dm_i /7/
dM = SUM m_i dn_i /8/
In my view /8/ is correct whereas /7/ is meaningless.
If so and if the analogy
is correct, statistical thermodynamics rests on a wrong foundation. The reason
is that heat is regarded as essentially different from work.