If you're calculating at constant volume how do you take into account the effects of thermal expansion on the heat capacity? If there's thermal expansion and you hold it at a fixed volume, wouldn't the system be under extreme pressures? If so, wouldn't this pressure increase affect lattice properties and therefore specific heat?
1) For solids, the volume changes very little with temperature, hence I will disregard thermal expansion at first approx.
2) Stricly, yes to get theCv (constant volume heath capacity) you should keep the solid at fixed volume, and get enormous pressure.
The Debye model is very simple, doesn't care about pressure or lattice structure.
3) If you are at for ex. at Ambient Pressure, the solid when expanding does a little work against external pressure. You should add this work to Debye's Cv to get your actual Cp (heath capacity at constant pressure) that is its actual capacity in "normal" conditions. This pressure work will be little.
In gas, Cp and Cv are very different. In solids, very close.
Note that Debye Cv does not even depend on Volume; only particle number!
So is the argument here that at Constant Volume, lattice properties are fixed so thermal expansion shouldn't affect specific heat? It follows then that the extra P*dV energy that you exclude at constant volume would give you the Cp. For solids this is small , so you assume Cp is about Cv?
Yes, it should work well for solids Cp~Cv
Lattice properties however can change. See for example the phase diagram of water ice, that has different crystals depending on pressure.
The lattice properties will affect the specific heat somehow; specially when you have a phase transition from a crystal to another. But that is not included in the Debye model. This Debye model is a rough estimation, it is not good like the formulas for perfect gases.
If you're under very high pressure, maybe the thermal expansion can do some significant P*dV work?
in general for solid state materials we consider Cp approximately equal to Cv. Of course this does not hold at all conditions. It depends also on the temperature and the range of temperature of your measurements. If this range within few hundred degrees then you have to take precautions.
Condensation (partial) differential functionals make relationships amongst micro-environments within the condensate by imposing boundary circumstances for the most likely anisotropy that has already formed there.The resulting asymmetry brings about some on-the-spot Deltas in various epi-phenomenological levels of "C" according to our calculations here in our Centre.
When microfluid currents cannot be described as undergoing any multi-dependency in coordinations of anhomological nature, then the quantity of continuous ( micro-)changes are mostly calculated by our team as parameters of chaotic phase transitions with their in-common universality within quasi-linear vectorial behavior. This makes the order of the said changes to critically shift as best measured with Cv _ [ ƛ-transition] echoing the same critical vicinity of a wide spectrum of phase spaces.
Such space shuttles not only act as infinitely context-less differences, but also give us the capacity to order condensation parameters even partly according to Cv differentials.
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