For magnetic systems, Rushbrooke inequality is a direct consequence of the thermodynamic relation between CH, CV and isothermal susceptibility, their positivity, and the definition of the critical exponent alpha as [controlling the behavior of CH as function of the reduced distance from the critical temperature..
In the case of fluid system, the usual definition of alpha refers to the constant volume specific heat (CV).
However, the role played by CV in the thermodynamic relation between CP, CV and isothermal compressibility is not the same as CH. Some additional hypothesis has to be made in order to derive the R. inequality for fluid systems or am I missing something trivial ?
I do not think it is as simple as that. Just following the same derivation one could conclude that the critical exponent of CPshould appear in a Rushbrooke-like inequality. The point I make is that the usual definition of alpha is through the critical behavior of CV, not CP. The two specific heats have no obvious necessity to have the same exponent.
From the second law, entropy is concave in the arguments U, V, N. Transform this according to Legendre wrt U and V. You get the Gibbs potential G(T, p, N). This is convex (or possibly concave, depending on choice of normalisation. In any case, it has a definite convexity). This means that the matrix of second derivatives of G wrt T and p, is always positive definite. This implies trivially that the diagonal elements are always positive, and less trivially so, that the determinant is always positive. This latter claim is the Rushbrooke inequality.
The same reasoning goes through with other variables: start from the extensive ones and entropy as your the thermodynamic potential, Legendre transform everything but one, and you get a convex thermodynamic potential leading to a Rushbrooke inequality,
@F.Leyvraz: I think that this suggestion too is at least not complete. From the condition on the hessian of G(p,v) (v being V/N), one can easily derive, once again the condition
CP >= T a2/(KTv) where a is the thermal expansion coefficient and KT the isothermal compressibility. I.e. the inequality that can be easily obtained from teh esecond equation writen by Gert van der Zwan.
But from this inequality one can derive immediately only a condition involving the critical exponent of CP. How to get alpha ?
From your suggestion it would be natural to look at the equivalent of the condition on the hessian for a suitable Legendre transform. The Helmholts free energy would look a natural candidate. Unfortunately it is neither convex nor concave, as function of the two variables T and v. It is concave function of T and convex function of v. So, the hessian is neither positive not negative definite.
I repeat, maybe I am overlooking something simple. But the problem is not the derivation of the inequality for CP.
From the second relation written by Gert Van der Zwan, knowing that Cp >=0, then you can derive that Cv>= -T/V (d V/dT)^2 / K_T, hence -alpha'
I might be wrong, but it seems to me that the proper analogue of CH for fluids is neither Cv nor Cp, but Cmu; with mu being the chemical potential,
and Cmu being the derivative of the grand canonical energy:
UGC (V,T,mu) = U-mu N; with respect to the temperature at constant volume and constant chemical potential: U = internal energy, and N=number of particles (or moles)
Cmu = (d UGC / d T)mu,V
This is, at least ( If I am not wrong), what one finds when translating the Ising models into gas lattice representations. Of course, I am aware that Cmu is something that probably cannot be measured directly in the lab.
@Noé G. Almarza : The point is that the exponent alpha is by definition the exponent associated to Cv in a fluid.
@Franzese: Cv>= -T/V (d V/dT)^2 / K_T cannot be used to derive the inequality.
On the one hand, every positive quantity is >= -T/V (d V/dT)^2 / K_T, so in the same way, just writing K_T >= -T/V (d V/dT)^2 / K_T one could prove that
gamma'+beta>=1, which is definitely false for exact 2D Ising model.
On the other hand, it is mathematically impossible to obtain the inequality for the critical exponents from a thermodynamical inequality where a positive quantity is >= a negative one, since one has to pass through the log of those quantities in order to prove the small lemma required (see Stanley's book).
For the moment, I am looking at the experimental literature to better understand the situation. If I'll find a reasonable explaination, I'll write it here.
Sorry for my late reply (Reserachgate did not alerted me about your replies). Giorgio is correct about the fact that my derivation involving a negative quantity is useless (because we need positive arguments for log functions). Sorry about that. However, Gert's last observation gives one possible answer. But there is also another (see below).
1. Gert is correct about the fact that the second relation he wrote implies that C_p cannot diverge slower than C_V near T_c, hence you can demonstrate the inequality for the critical exponent of C_p (in this case there are no negative quantities involved) and then use Gert's observation to reach the final relation with alpha.
2. You can derive the exact Rushbrooke "equality" using the static scaling hypothesis, stating that the Gibbs free energy is a generalized homogeneous function, applied to a fluid system (following paragraphs 11.2 and 11.3 in Stanley's book) with which it is possible to show that, since there are only two relevant field, then there are only two independent critical exponent, from which the Rushbrooke relation and all the other scaling laws (Widom, Griffiths, ...) can be derived.
I hope you can use these considerations to answer your doubt.
A short update about this discussion. We can relax the condition about Cp and Cv. To have the relation satisfied for alpha' (exponent of Cv) is not necessary that Cp>=Cv. It is sufficient that the relation hold for exponent of Cp (let's call it alpha''), i.e. alpha''+2 beta gamma' >=2 implies alpha'+2 beta gamma' >=2 (regardless if alpha''
Wasn't this issue addressed for PVT systems in an open access article published by Elsner (2014) in Engineering 6, 789-826?
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