Blog Differential Shuffle
Tradition starts off with the isothermal (dT=0), and isobaric (dP=0) relation, and equates it to work7:
TdS=dE+PdV eqn 1)
Where W is work done, dE is internal energy change, P is isobaric pressure, dV is volume change. T is isothermal temperature and dS is entropy change
Commentary: The above all seems fine from a mathematical perspective but what does it actually mean because in eqn 1) both the isothermal entropy change (TdS) and internal energy change (dE) concern an isothermally expanding system, while the irreversible work (PdV) is done onto the surrounding atmosphere i.e. Lost Work.
Certainly clarity would be provided by rewriting eqn (1) as
(TdS)system=dEsystem+(PdV)atmosphere eqn 1)
Continuing with traditional logic (?): Based upon eqn 1), in terms of internal energy change (dE), we obtain:
dE=TdS-PdV eqn 2)
When transforming either eqn 1) or eqn 2), most texts will use the following relation:
PdV = d(PV)-VdP eqn 3)
A more precise analysis would write:
d(PV)=PdV+VdP+dPdV eqn 4)
For infinitesimal changes: dPdV<<<PdV and/or VdP, then changes as described in eqn 3) approximates changes as described eqn 4). It must be stated, that this may not be the case for all processes.
Continuing with the traditional: Combining eqn 2) with eqn 3), gives:
dE=TdS-d(PV)+VdP eqn 5)
Collecting the terms, then eqn 5) can be rewritten:
d(E+PV)=TdS+VdP eqn 6)
Traditional thermodynamics defines “enthalpy” as:
H=E+PV eqn 7)
Commentary: At this point one should scratch their head because if PdV in eqn 1) is the work done to the surrounding atmosphere then PV in eqn 7) must also have to do with the surrounding atmosphere yet the term enthalpy traditionally concerns the system in question and not its surroundings. In other words enthalpy as defined by eqn 7) verges on being meaningless!
Back to traditional thermodynamics which then often rewrites eqn 6), and calls it the “enthalpy relation”:
dH=TdS+VdP eqn 8)
Traditional thermodynamics rewrites: TdS in the following manner:
TdS=d(TS)-SdT eqn 9)
Commentary: There is nothing wrong with going from eqn 8 to 9) except for the minor fact that no one knows what entropy (S) really means!
Therefore, eqn 8) becomes:
dH=d(TS)-SdT+VdP eqn 10)
Which can be rewritten as:
d(E=TS)=-SdT+VdP eqn 11)
“Helmholtz free energy” as:
F=E-TS eqn 12)
Consequentially, eqn 11) can be rewritten as:
dF=-SdT-PdV eqn 13)
Helmholtz free energy change is for changes in temperature (T) and volume (V). However it derivation is fraught with illogical consequences. Okay let logic be dammed and let us continue.
Again, traditional starts off with eqn 1):TdS=dE+PdV. Applying the transformations for d(TS) and d(PV), as given by eqn 3) and eqn 9) respectively, gives:
dE=d(TS)-SdT-d(PV)+VdP eqn 14)
Eqn 14) can be rewritten as:
d(E-TS+PV)=-SdT+VdP eqn 15)
Commentary: Again one vested in logic may query: What does d(E-TS+PV) really mean if E, and TS are the system, while PV is its surroundings. Oh ya I forgot let are playing logic be dammed. Anyhow this differential shuffle is such eloquent math who cares about its logic.
Tradition next defines Gibbs free energy as:
G=E-TS+PV eqn 16)
Commentary: I wonder what E-TS+PV really means, guess it doesn’t matter because entropy has no meaning either
Tradition next inserts eqn 16) into eqn 15), and obtains:
dG=-SdT+VdP eqn 17)
Changes to Gibbs free energy [eqn 17)] applies to processes that are both isometric (dV=0) and isentropic (dS=0?): Isentropic really depends upon your interpretation of S (See my entropy blog)
It is interesting that whether one considers the above to be blundering eloquent abstract math, or something else, the truth is the traditional approach may have gotten Gibbs free energy sort of right but for completely the wrong reasons see my blog on Physical Chemistry.
Okay forgetting the above discussed minor voids in logic, let us look at a broader picture. Thermodynamics may be unique in its use of differentials! It starts with a part: PdV, from which the whole: d(PV) is then subtracted, obtaining the other parts: VdP. Certainly logical dictates that if one started off with the whole: d(PV), one could then deduce the parts: PdV & VdP!!!
The reason that eqn 1) is traditionally beheld with such relevance is that it was equated to , thus used to explain the lost work as deduced by 19th century heat engines, e.g. Carnot cycle. This, was a mental progression to Clausius’s understanding that ST gives energy under the constraint of lost work. Of course lost work meant that the Carnot engine could not return to its original state without an influx of energy, leading to the second law of thermodynamics whether you claim it is Lord Kelvin’s or Clausius’s deduction.
It all would be so humorous if it were not for the fact that the second law and entropy, both took on a demigod status, and the 150 yrs of indoctrination that has followed. Arguably the roots of the second law is one of perpetual motion does not occur in nature. We can now argue that perpetual motion does not exist here on Earth because all systems must interact with our atmosphere, but this is no longer a universally applicable proposition.
Due to the elevated status of entropy in terms of isobaric isothermal work eqn 1) became the first equation in thermodynamics. Although
lacking clarity entropy (S) was construed so that its relation to both volume (V) and internal energy (
The net result being the indoctrination of the cumbersome array of differential equations 1) through 17), all embedded with circular logic. The fact that statistical mechanics is accepted as the inarguable proof behind traditional thermodynamics, speaks more of the power of statistics, then the science’s logic. And of course the equating of Boltzmann’s constant (k) so that it explains empirical data here on Earth just reinforces what is said.
The simplest explanation for this is our new perspective that lost work: W=PdV, signifies the ideal work required to displace the Earth’s atmosphere against gravity. If only our 19th century scientists had realized how useful expanding systems tend to displace our atmosphere, then who knows. Well we certainly know that the science would be simpler, as it would have been based upon constructive logic rather than some dance of partial derivatives.