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There are 11 gaseous elements and two liquid elements at standard temperature and pressure. The rest are solid. Can phase be predicted from quantum mechanical principles?

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There are laws that were derived by Maxwell, which allow us to make phase diagrams (P-Chem), however these are not QM. –  JoeHobbit Jun 26 '11 at 22:53
    
@Joe: Can the critical temperatures for the CPTs not be calculated quantum mechanically? –  Will Vousden Jun 27 '11 at 7:36
    
@Will Vousden I don't know where to start to do that, or even if it is possible. –  JoeHobbit Jun 27 '11 at 16:55

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It's not easy. However there are attempts to calculate a phase diagram of an element from first principles. For example, in this paper http://prl.aps.org/abstract/PRL/v95/i18/e185701 the solid-liquid transition of diamond is calculated. The calculation of the free energies is done with ab initio molecular dynamics. This means that the carbon nuclei are treated as classical particles, but the electrons are treated quantum mechanically. There are also some other approximations involved in the treatment of the electrons and the electron-nuclei interactions.

Helium is another element for which a phase transition has been studied using quantum mechanics. In that case a different method - path integral Monte Carlo - is used, i.e. the free energy estimation is by Monte Carlo integration. See for example http://prl.aps.org/abstract/PRL/v72/i12/p1854_1.

I think that calculations like these are a step on the way to construct a phase diagram from quantum mechanics, even if we're not at room temperature and pressure yet. Also, the heavier elements are much more challenging.

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Seems to me this can be done in principle using the density operator $\rho$ and driving towards the equation of state by expressing the partition function as

$\rho = \frac{1}{Z_G} exp^{-\beta K}$

where,

$K = H - \mu N$

And write out the Hamiltonian for a system of N atoms. Once you have the partition function written out you can get at the equation of state. Once you have the equation of state you can determine phase of the system at STP. This would be a grand canonical formulation and I'm thinking it's not very easy to do with anything but the simplest of atoms, namely H and He.

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What's the flaw in the logic? I'd like to know why this wouldn't work so I can correct my thinking on this. –  unclejamil Jul 6 '11 at 13:05
    
I don't see any fundamental restrictions for this approach. But! In order to describe something but gases the Hamiltonian should involve interaction of the particles that makes analytical solution impossible. Even if one use supercomputer to deal with it he needs some parameters to describe the particles. These parameters of atoms could be derived from the first principles but this problem is also too hard. –  Maksim Zholudev Dec 16 '11 at 7:23

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