Copied from elsewhere from when I first described the notes:
“These notes are about thermodynamics…
What the top parts of the notes illustrates is called the Gibbs-Duhem equation, and that, through the power of derivatives and Maxwell relations you can use the Gibbs-Duhem equation to prove the equation for the change in entropy, which is usually an equation given to students without context.
The equation is ΔS = n*R*ln(P2/P1) –> change in entropy = moles x gas constant x ln(final pressure/initial pressure) when the temperature is constant, and well, this is the context!
The bottom part of the notes illustrates the relation of F (here, it is Helmholtz free energy, which is usually written as A) to heat, written here as Q. So, it is the relation of F to Q.
Here it is found through an equation of entropy (here, given as S). The crux of the derivation is that E-TS = F, and the kβ term dissolves away, leaving a bare E to join the left-hand side of the equation to create E-TS. What is not mentioned here is that (dlnQ/dβ) = E, or energy.”