The Differential Shuffle
Tradition starts off with the isothermal (dT=0), and isobaric (dP=0) relation, and equates it to work:
W=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
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
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 said, that for some processes this may not be the case.
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)
Traditional thermodynamics rewrites eqn 6), as the “enthalpy relation”:
dH=TdS+VdP eqn 8)
Applying similar logic, traditional rewrites: TdS in the following manner:
TdS=d(TS)-SdT eqn 9)
Therefore, eqn 8) becomes:
dH=d(TS)-SdT+VdP eqn 10)
Which can be rewritten as:
d(E=TS)=-SdT+VdP eqn 11)
Define “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).
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)
Define Gibbs free energy as:
G=E-TS+PV eqn 16)
By inserting eqn 16) into eqn 15), traditional thermodynamics 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)
Traditional thermodynamics is 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 beheld with such relevance is that it was equated to the lost work as deduced by 19th century heat engines, e.g. Carnot cycle. The equating of: W=TdS, 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 Lord Kelvin’s Second Law of Thermodynamics. 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.
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.
Understandably, processes whereupon pressure and volume must be treated equally become easier to comprehend by considering the whole, i.e. we treat pressure and volume equally.
Our new way of thinking is not that different after all. We start with the ability of a system to do work:
TS=E+PV eqn 18)
For any process, change to the ability to do work is defined by the a new general law:
d(TS) = dE+d(PV) eqn 19)
For infinitesimal changes we can rewrite eqn 19) as:
TdS+SdT=dE+PdV+VdP eqn 20)
Or if you prefer:
W=dE+PdV+VdP eqn 21)
Deriving all relations based upon the general law (eqn 20) ultimately results in the same equations, when the same conditions are applied. I.e. holding two parameters constant in eqn 20), results in the same series of Maxwell equations.