Implications to Statistical Thermodynamics
By Kent W Mayhew
Certainly when properly used statistical thermodynamics should provide insights to various phenomena. We all can agree that statistical ensembles can be used to describe the actions of vast numbers of interacting particles i.e. atoms, molecules. And we have the likes of Boltzmann to thank for this. Even so such ensembles were based upon elastic collisions, and with our improved understanding that intermolecular collisions are inelastic, means that at most levels how we write the statistical sciences, is in need of a rethink. Which will be left to others who possess the skills to do so.
A result of believing that irreversibility can only be explained using statistical arguments i.e. entropy and the second law, has led to claim that the probability construct is the only way to properly comprehend our universe. This author believes otherwise namely that probabilities give results rather than not reasons, hence our universe can be better understood using logical constructs. Moreover, if you believe that your language (no matter what language you speak) is the only language that explains what you witness then you may be in dire need of a reality check. Remember statistical mathematics is nothing more than a language, albeit a complex eloquent one.
However, issues with statistical thermodynamics go beyond irreversibility and inelastic collisions. And this is one of logic. When constructive logic is used to formulate a theory then you will be able to move in any direction fom any point in its structured analysis and at all times arrive at constructive logical results. This is not the case for our traditional understanding of thermodynamics along with statistical thermodynamics, as we are about to demonstrate!
Implications of our New Understanding to Statistical Thermodynamics
A fundamental equation to statistical thermodynamics is entropy (S) defined in terms of the number of microstates (@) i.e.
Note: There are other plausible explanations as to why logarithmic functions belong in thermodynamics other than the accepted statistical reasoning.
Arguably eqn 1) is a valid equation because Boltzmannís constant (k) was designed (or if you prefer equated) so that the ideal gas law is also valid. That being:
When contemplating an expanding system, we are often taught the following isobaric isothermal relation:
Throughout this website I have discussed that the only logical approach for dealing with eqn 3) is to consider that dE = system energy change (AKAK Internal Energy), while PdV = Lost work into our atmosphere. Even so the isothermal entropy (TdS) remains troublesome. Instead compare eqn 3) to what was discussed concerning the first law, i.e.:
dQin = dEsys + (PdV)atm = dEsys + PatmdVsys 4)
Certainly the RHS of the above two equations are the same hence eqn 3) is basically a version of the first law whereby one considers that TdS equates to the energy input dQin.
If you do not understand why eqn 3) lacks clarity while eqn 4) has clarity then see parameters
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 systemís 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 systemís number of accessible states corresponds to the systemís 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 systemís number of accessible states just be a function of that systemís 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 Boltzmannís 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 Boltzmannís constant (k) is not a universal constant as previously presumed, rather it is a constant which is only valid here on Earth.
We accept that Boltzmannís 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 systemís 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 20th and into the 21st century. This may help the likes of Plank and Mach rest in peace.
Copyright Kent W. Mayhew