Second Law of Thermodynamics and Entropy

       One concept that is often talked about  and rarely understood well is the notion of entropy and the second law of thermodynamics. It is common place to say that the second law states that entropy always increasing or remains constant paired with the statement that entropy is a measure of disorder. This is in fact correct but the problem is that it is stated and talked about carelessly before students have a firm grounding in Statistical Mechanics let alone Quantum Statistical Mechanics.  This is a problem not because anything wrong is being taught but because (as is often the case) very subtle and hard concepts are introduced before one has the right intellectual machinery to fully wrap one's mind about the concept; as a result teachers and professors often resort to very vague statements. Confusion is compounded when students are then given concrete problems from introductory textbooks where  these highfalutin statements can't be readily gleaned and in fact never seem to apply.
     The sad truth is that if we were to properly learn these concepts it is better for the concept of entropy to remain mysterious and simply be talked about as merely a state function. The interpretation of entropy as a measure of disorder requires a proper discussion of statistical mechanics precisely the thing that is not done. In fact, the fact that Entropy always increases can be derived by thinking carefully about Carnot cycles and engines. Recall that Clausius coined the term "entropy" from apparently a greek word, " entropia" which according to wiktionary means " a turning towards".
     The reader should be confused at this point if my point has been made. Why? Well, if entropy is a measure of disorder why did Clausius choose this word when he defined entropy. It seems as though it has nothing to do with disorder. The answer is this; our formulation of the second law only makes sense after accepting Boltzmann's work which came after Clausius' work. This implies there is a formulation of the second law that does not require the definition or formulation of entropy. It is this understanding that takes the back sit in these vague discussions and plays the important role when a lot of introductory thermal physics concepts are problems are being taught.
    It must be stressed that I am not trying to imply that something wrong is being taught, rather I am making a pedagogical point how we should be learning the material. It will be the goal of the next few posts to introduce another way of talking about the Second law of thermal dynamics one that makes no specific reference to entropy as a measure of disorder.