By Kent W. Mayhew

**Blog Differential Shuffle**

Tradition starts off with the isothermal (*dT*=0), and isobaric (*dP*=0) relation, and equates it to work^{7}:

*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**=dE**system**+(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 19^{th} 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 19^{th} 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.

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