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Certain unphysical region of the isotherms obtained from of the Van der Waal's equation of state must be excluded because, in this region, volume increases with increasing pressure (i.e. $(\partial P/\partial V)_T>0$) and therefore, violates the thermodynamic stability.

However, in reality, a slightly bigger region is excluded by Maxwell's tie line construction which contains part of isotherms for which $(\partial P/\partial V)_T<0$. But these parts of the isotherms do not immediately seem to be problematic. Is there any physical way of understanding why some thermodynamically stable regions with $(\partial P/\partial V)_T<0$ are also excluded?

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These are metastable regions. They are stable in the sense that $\partial P/\partial V)_T<0$, but when the system is in these regions, it can lower its energy by phase separation.

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  • $\begingroup$ Phase separation, as I understand, happens when the system is abruptly quenched to regions of pressure and temperatures that correspond to the unphysical region negative compressibility, and not those corresponding to the metastable region you mentioned. Isn't it? @fra_pero $\endgroup$
    – SRS
    Commented May 8, 2020 at 15:30
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    $\begingroup$ @SRS, fra_pero is correct. There will be phase separation on the whole interval given by Maxwell's construction. (Actually, it would be more correct to say that Maxwell's construction is valid precisely because phase separation is occurring. That's why it is not valid in mean-field models.) $\endgroup$
    – Yvan Velenik
    Commented May 8, 2020 at 15:44
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    $\begingroup$ In a lab, yes. But it is a metastable state. If you leave the system for long enough, phase separation will occur (when the system reaches its true equilibrium state). You can accelerate this relaxation to its true equilibrium state by, say, shaking the system or perturbing it in other ways. This is why you need, as you say, to increase the pressure very slowly. $\endgroup$
    – Yvan Velenik
    Commented May 8, 2020 at 16:02
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    $\begingroup$ @SRS. I should point out that even the existence of these metastable branches is a nontrivial issue. Namely, one can prove that in lattice gases (with short-range interactions), the pressure has an essential singularity at the transition point. In particular, the pressure cannot be analytically continued beyond this point, which implies that the "metastable branches" are not uniquely defined (there are infinitely many ways to smoothly extend the pressure beyond the transition point (that is, in a $C^\infty$ manner), which are dynamically relevant). $\endgroup$
    – Yvan Velenik
    Commented May 8, 2020 at 16:37
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    $\begingroup$ Thus, the "metastable branches", as such, do not exist in this type of models (and thus, presumably in general fluids with short-range interactions). That they appear in the van der Waals model is because the latter is of mean field type (and thus cannot support phase separation). $\endgroup$
    – Yvan Velenik
    Commented May 8, 2020 at 16:37

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