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For the ideal gas it is easy to show that $1/T$ is an integrating factor, so

$$\delta Q/T = dS$$

is an exact differential.

So far I haven't found a convincing argument that this should apply in general? Can this be proven in general, or is it just an empirical fact that can only be explained satisfactorily from statistical theory?

Qmechanic
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MichaelW
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    links here (possible duplicate) https://physics.stackexchange.com/q/582707/226902 – Quillo Apr 03 '23 at 11:59
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    It is a matter of taste for there are several equivalent formulations/derivations. My preference is the one that follows Gibbs: just postulates it but if you prefer the ones based on Kelvin/Clausius style axioms then probably the clearest derivation is the one using the equivalent Turner-Sears-Kestin axiom; see, (1) Sears: "Modified form of Caratheodory's second axiom", https://doi.org/10.1119/1.1973198, (2) Zemansky: "Kelvin and Caratheodory - A reconciliation", https://doi.org/10.1119/1.1972279, (3) Pau-Chang Lu: "Didactic remarls on Sears-Kestin statement" https://doi.org/10.1119/1.13048 – hyportnex Apr 03 '23 at 13:05
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    The link provided by @Quillo does not really prove anything thermodynamically important because there is always an integrating factor for two variables, that is for any Pfaffian $\tilde p = f(x,y)dx+g(x,y)dy$ with smooth functions $f,g$ there is always a $\lambda (x,y)$ such that $\lambda \tilde p= dq$. This not true in general for more than three variables, but it is true for the functions in thermostatics. – hyportnex Apr 03 '23 at 13:12
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    In https://physics.stackexchange.com/questions/62177/integrating-factor-1-t-in-2nd-law-of-thermodynamics, @JanLalinsky summarizes the original proof of Clausius and you can find the same in more detail in the books by Fermi or Pippard. – hyportnex Apr 03 '23 at 13:20
  • @hyportnex There is mathematical theorem that there is always integrating factor in 2D space, but meaning of this factor is not clear. The thermodynamic considerations allow us to interpret this factor as 1 over gas/absolute temperature. – Ján Lalinský Apr 04 '23 at 00:42
  • @JanLalinsky I completely agree with you and this is what was missing in the referenced post https://physics.stackexchange.com/questions/582707/proof-that-frac1t-is-the-integrating-factor-for-dq, for it only showed that for two variables you can have an integrating factor. This is unlike Zemansky who clearly proves the universality of such integrating factor in thermodynamics. – hyportnex Apr 04 '23 at 01:07
  • In "Kelvin and Caratheodory - A reconciliation" Zemansky comments that "The fact that a system of two independent variables has a reversible dQ that always admits an integrating factor regardless of the second law is of course interesting, but its importance in physics is not established until it is shown that the integrating factor is a function of temperature only, and that it is the same function for all systems." – hyportnex Apr 04 '23 at 01:08
  • @hyportnex Oh that's interesting. Thank you. – Ján Lalinský Apr 04 '23 at 11:26
  • @JanLalinsky I have always believed that the transitivity axiom of thermal equilibrium (0th Law) combined with the general existence of integrating factor for 2-variable function should be enough to prove the universal existence of isentropic (reversible adiabatic) surfaces. This by breaking up all interactions ${T,x_1,x_2,...}$ of a "system" into individual single interactions $[T,x_1,x_2,...] = [T, x_1]\bigcup [T, x_2]\bigcup ..= [y_1, x_1]\bigcup [y_2, x_2]\bigcup ..$, where $y_k$ is the characteristic intensive of the $k^{th}$ interaction, each possessing an integrating factor. – hyportnex Apr 04 '23 at 14:32
  • For each homogeneous piece of the body this "decomposition" is to be understood both by location and interaction. That such decomposition is possible ("$-2^{nd}$ Law"?) is a separate (most) primitive postulate but I think this assumption always underlies any macroscopic argument. – hyportnex Apr 04 '23 at 15:06
  • @hyportnex do you have a more detailed written memoir on this idea? I'm interested in derivations of existence of entropy, and its history. To me, it's not entirely clear that any change of state where multiple variables change simultaneously can be equivalently replaced by a set of steps where only two variables change at a time. – Ján Lalinský Apr 04 '23 at 15:21
  • Let us continue this discussion in chat. – hyportnex Apr 04 '23 at 15:40
  • @JanLalinsky Now I remember. Actually, I got the idea originally reading Planck: Introduction to theoretical physics - Theory of heat pp56-65, para:43-47, see https://archive.org/details/theoryofheatbein007775mbp/page/n69/mode/2up. There Planck proves the existence of the entropy function for a collection of simple bodies characterized by $\theta_k, V_k$ with $\theta$ and $V$ are the empirical temperature and volume, resp. I am only adding to this that I think the same argument also holds for any more complicated bodies as long we can separately control their individual extensives. – hyportnex Apr 05 '23 at 16:51

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