For the case of all the energy used to do work onto the surrounding atmosphere, coming from the within the expanding system, then dQin= 0 and eqn 4) becomes:

                     dE= -W(lost)  = -(PdV)atm                5)

For the case described by eqn 5) the system's temperature must decline as it does work, in which case one may write:

                         dEsys =(CvdT)sys =- (PdV)atm       6)

The above is enshrined in constructive logic and is certainly simple to follow. Now ask: What happens if we insert eqn 1) into eqn 3)

                  TdS=Td(kIn@) = dE+PdV    7)

 At this point one may ask what is Td(kIn@) in eqn 7)? Seemingly it is now the difference between the system’s energy change and the work done by the system onto its surrounding atmosphere. Perhaps but; what exactly does this have to do with the number of accessible states of either the system, or its surroundings?  It an awkward question that is seemingly avoided in traditional thermodynamics! I personally do not have an answer.

 A plausible answer is that Td(kIn@) represents the energy input as measured by the increase to the system’s number of accessible states corresponds to the system’s energy increase minus the work done. At first one might say that this holds some merit from a logistics perspective, as herein we are saying that the change to accessible states is energy related. But hang on, equation 7) is suppose to concern the expanding system. Hence: Should not the change in the expanding system’s number of accessible states just be a function of that system’s energy change (total or internal energy)? Again I personnally do not understand the tradiational logic but then again I assume that I am not alone.

Of course the lack of clarity is undeniable in eqn 7). If we rewrite eqn 7) then we obtain:

                  (TdS)sys= [Td(kIn@)]sys = dEsys + (PdV)atm    8)

 Now that we have an equation with clarity it now looks like it maybe meaningless

  Reconsider what happens if all the energy required comes from the thermal energy within the expanding system as defined by eqn 7). Based on eqn 7), the expanding system's temperature must decrease, so one might actually think that the number of accessible energy states within the expanding system decreases. It becomes an illogical mess if we try to describe what is happening in terms of entropy. Of course remove entropy we return to eqn 6).  

Yet traditional thermodynamics avoids such constructive logic by also claiming for any expanding system that:

                   TdS = Td(kIn@) = TkIn(V2/V1)    9)

Note: Equation 9) is commonly found in physical chemistry texts. Now eqn 9) claims that isothermal entropy change is a function of volume change rather than energy per say. Herein the traditional argument based upon Boltzmann's entropy being related to randomness. But what does randomness or volume have to do with energy, except for the reality that volume increases of expanding systems signifies work done onto our atmosphere AKAK lost work. 

 The issue becomes this. Eqn 7) & 9) infer different realities than eqn 9). However one rewrites traditional  thermodynamics in terms of a statistical construct must have results that only infer one reality. This is a must if you want it to have any basis on constructive logic. You see traditional thermodynamics is filled with circular logic, just look at the differential shuffle, as inspired by the likes of Gibbs in his famous paper: "Heterogeneity of Homogeneous Substances".

 The big circular issue: There are those who claim that Boltzmann’s brilliant statistical mathematics proves that traditional thermodynamics is correct. However we have just seen one of the many disconnects between real thermodynamics and Boltzmann's brilliance and the very basis of all those statistical arguments. Certainly statistical math was built by equating it to known empirically determined results. In other words it was equated to known results and then claimed to prove those very results.

  The above could become the very definition of circular logic!  I.e. designing a math and then having a constant (k) in eqn 1) such that the math equals empirical data is a fundamental to currently taught statistical thermodynamics. And then to claim that since this math now explains what ewe witness is certainly NOT vested in constructive logic. Those claiming that statistical thermodynamics proves traditional interpretations to be absolutely correct may just be delusional. 

  One may argue that we did not actually deal with statistical thermodynamics in this blog and/or that this author does not fully understand statistical arguments. And to that I agree as it has been decades since I last studied the subject. But that does not diminish any of the simple fundamental arguments presented in this blog. The point remains that your complex statistical consideration must match simpler understandings presented herein, rather than the other way around.   

The above does not alter the reality that Boltzmann's math has helped us understand atomic theory.

   In another blog I discuss that Boltzmann’s constant (k) is not a universal constant as previously presumed, rather it is a constant  which is only valid here on Earth.

  Conclusions:

  We accept that Boltzmann’s mathematical insights were brilliant. However the number of microstates has nothing to do with volume or randomness. It would be simpler if it was a relation to the system’s total energy (AKA: Internal energy). Simply put, increase a systems thermal energy and its number of microstates increases. This will have implications throughout statistical thermodynamics.

  Furthermore we must accept how walls influence all forms of thermodynamics.

  And finally we might need to accept the notion that our world might not be governed by probabilities, at least to the extent that this concept has grown throughout the 20^th and into the 21^st century. This may help the likes of Plank and Mach rest in peace.