For a simple system:

dG = -SdT + VdP

At constant temperature:

dG = VdP

Therefore:

dG/dP = V (no need for partial differentiation), which is a positive quantity.

And:

d2G/dP2 = -BTV, which is non-zero and a negative quantity, because the isothermal compressibility must be positive (required for the stability of the system).

Then, G(P) exhibits positive slope and negative concavity (smaller slope as P increases).

Comparison of New and Old Thermodynamics

1. Logic of the Second Law of Thermodynamics: Subjectivism, Logical Jump, Interdisciplinary Argumentation.

2. New thermodynamics pursues universality, two theoretical cornerstones:

2.1 Boltzmann formula: ro=A*exp(-Mgh/RT) - Isotope centrifugal separation experiments show that it is suitable for gases and liquids.

2.2. Hydrostatic equilibrium: applicable to gases and liquids.

3. The second and third sonic virial coefficients of R143a derived from the new thermodynamics are in agreement with the experimental results.

3.1. The third velocity Virial coefficient derived is in agreement with the experimental data, which shows that the theory is still correct when the critical density is reached.

4. See Appendix Pictures and Documents for details.

Gibbs free energy should be taken as an extensive physical quantity, ie the capacity to do useful work. For a pure compound (no reactions) in a closed system this capacity must come from a difference in T and/or p with the environment. At equilibrium G reaches a minimum, but this not necessarily implies a parabola. Yes, the math has to comply with the physics, not the other way around. Try an ideal gas first ??

Dear Mr. Power-Palm.

In response to your question on Gibbs free energy.

Where from you came to the idea that it is of parabolic shape? (*Are he Gibbs free energy curves of pure substances always parabolic? And if so, can someone point me towards a proof or derivation of this fact*? ).

The format of this RG-page unable to present plot. However, if you appeal to my personal mail, I'll send you plot that shows connection of Gibbs energy and other thermodynamic functions (H -enthalpy, U-free internal energy, etc.). My requisite you can see below.

From this plot, you can clearly see that Gibbs enery directly depends on entropy S -the higher entropy, the lesser Gibbs energy, and visa versa H=G+TS→ G=H-TS.

However, Gibbs energy has another important mathematical feature –on the solution interval (i.e. the enthalpy is given and fixed H=Const), the Gibbs energy is strictly convex function of arbitrary form (particularly, it may be parabolic one). Strict mathematical proof of this statement was first presented by Zeldovich, and you can find this poof in my textbook *A Simple Kinetic* (pages 43-45). By the way, you can see mentioned plot in the same my book – see page 31.

To access this book make the following Cloude procedure:

1. Go to GoogleChrome (or Yandex):

2. Make link https://yadi.sk/d/7w5jSXrB3GBzti and clic.

3. The title *SimpleKinetics.pdf* will appear. Clic and open file. Find pages 43-45 (and 31).

4. Reload this file from Cloud-system to your comp.

Unfortunately, this is Russian version, but the equations are the same at any language, so, it is my hope, that you will understand the proof.

Summary: Gibbs free energy is continuous (i.e. it has first derivatives at any point) convex function of arbitrary form (incl. parabolic shape).

Regards

Vasili DIMITROV, Ph.& Ch.D.

Emeritus Professor of Chemical Physics.

25 Fisherville Rd, #703, North York,

ON, M2R 3B7, Canada.

Tel. home (1)-416-546-8442, cell (1)-647-632-6355.

E-mail: [email protected]

Skype vasilidim

1， The second law of thermodynamics is incorrect. See figure below for details.

2，The system is isothermal and exchanges heat with a large heat source to keep the temperature constant. Only volume changes are discussed.

3，The problem here is that the actual system is balanced and the second law of thermodynamics judges it to be unbalanced.

It can look like a parabola, but in fact the temperature dependence is more complex, both because enthalpic and entropic components. The aggregation state is very important. See theories for ideal gas and perfect crystal.