What are entropy and reversible processes really? I'm confused about the concepts of entropy and reversible processes.
Before explaining the definition of entropy, they used the term "reversible process" without defining it formally. Then, when entropy was defined, I was told the definition of a reversible process is $\Delta S=0$.
However, in the book Thermodynamics, Kinetic Theory and Statistical Thermodynamics, by F.W. Sears and G.L. Salinger, they define the variation of entropy of a reversible cycle as $\Delta S \equiv \frac{dQ}{T}$. Using this, they can evaluate the entropy variation of reversible processes.
They give the example of an adiabatic process which has zero change in entropy. Does this mean all adiabatic processes are reversible?  
They also calculate the non-zero entropy variations in other reversible processes.
They calculate the variation of entropy in irreversible processes by constructing a reversible process, (i.e. since entropy is a state function and only depends on the state and not on the path).
My questions are:


*

*What is the formal definition of reversible processes, without needing to define the entropy and then proving the equivalence between the two definitions?

*Is the variation of entropy of reversible processes equal to zero? If so, then why do they (Sears and Salinger) calculate it? 

*Assuming the fact that $\Delta S=0$ in reversible processes, why do they construct a reversible process in order to evaluate the variation of entropy on non-reversible processes?


Thanks in advance. (I know there are other questions about this topic, but I don't think they clarify the distinction between entropy as it relates to reversible processes.)
 A: *

*The practical definition of a reversible process is one for which the system (no mass entering or leaving) passes through a continuous sequence of thermodynamic equilibrium states.  

*No, the entropy change of a system does not have to be zero for a reversible process.  But, the entropy change for the combination of a system plus its surroundings does have to be zero for a reversible process.  So, it depends on whether we are talking about the system or about the combination of the system and its surroundings.

*Since entropy is a function of state, all we need to know is the conditions at the two thermodynamic equilibrium end states to determine the entropy change between these states.  The only way we have of determining the entropy change is to conceive of (i.e., dream up) a reversible path between the same two end states, and calculate the integral of dq/T for the reversible path.  Any reversible path will do, since all reversible paths will give the exact same value for the integral.  However, if we have an irreversible process, we will not get the correct answer for the entropy change if we integrate over the irreversible path, since a reversible path is required to determine the entropy change.
