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Estimate the molar heat capacity of gadolinium at room temperature, given it's Debye temperature $155 \ K$.

$C=2.4\pi^4 N_A k \left(\frac{T}{T_D} \right)^3\approx13000\ \frac{J}{molK}$.

I didn't think this would work because this isn't supposed to apply to solid metals. Molar heat capacity of gadolinium is about $37\ \frac{J}{molK}$, according to Wikipedia.

Can Debye's model be used here. Appreciate any help.

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    $\begingroup$ Why do you claim that it "isn't supposed to apply to solid metals"? $\endgroup$
    – daydreamer
    Commented Oct 12, 2020 at 19:37
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    $\begingroup$ I'm just learning and I made a reckless assumption. The question comes from an exam paper. Is it possible that a mistake was made composing this particular problem? $\endgroup$
    – Some Student
    Commented Oct 12, 2020 at 21:12
  • $\begingroup$ Probably, Peter... Keep up with the studies! Good luck $\endgroup$
    – daydreamer
    Commented Oct 14, 2020 at 11:03

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Gadolinium belongs to the rare earths class of materials. Their physics is highly dominated by localized states (please, check this nice simple overview). Electrons in metals tend to be highly non-localized (Bloch states), so yeah, I wouldn't expect Debye's theory to work for these guys (because even though Debye is concerned about lattice dynamics, electron localization spoils the fun most of the time).

You get to the right answer (namely, that Debye fails) via wrong reasons: the Debye model actually was designed to address low-temperature physics of all solids, and does it quite successfully for some materials. For intermediate temperatures, however, you can but cheer for it and see the failure.

Of course, what actually are intermediate/high temperatures heavily depend on your hamiltonian and its parameters, i.e., what material is being studied.

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    $\begingroup$ OP's answer was way off. Heat capacity at ordinary temperatures is normally dominated by the lattice, but in the case of gadolinium there is the magnetic phase transition around room temperature. (This can be used for magnetic refrigeration.) $\endgroup$
    – user137289
    Commented Oct 12, 2020 at 20:40
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    $\begingroup$ Yeah, probably Motts-like?, idk the technical details. But what bothered me was his affirmation that he expected Debye to fail for metals. $\endgroup$
    – daydreamer
    Commented Oct 12, 2020 at 20:44
  • $\begingroup$ But I did not understand why you posted that comment in my answer lol. No passive-agressiveness here, just curious if I got something wrong or the like...? $\endgroup$
    – daydreamer
    Commented Oct 12, 2020 at 20:45
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    $\begingroup$ Just a comment that I expect most rare earths to be very ordinary with a $c_v$ according to Dulong & Petit. Gadolinium is different because of this magnetic phase transition. $\endgroup$
    – user137289
    Commented Oct 12, 2020 at 20:48
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    $\begingroup$ The Curie temperature of gadolinium is at 20 °C, so its room temperature $c_v$ is affected. It stands out among the rare earths in this list: en.wikipedia.org/wiki/… $\endgroup$
    – user137289
    Commented Oct 12, 2020 at 21:03

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