# Estimation of specific heat of a solid

I'm investigating a material, that under high temperature-pressure conditions undergoes a phase transition. In order to correctly calculate the temperature, or even estimate a reasonable value, I need to know the specific heat of the material post phase transition. Unfortunately these values haven't been determined for the materials I'm interested in, and I'm not in a position to easily determine them experimentally.

My question then is, do reasonably straightforward techniques exist for the estimation of specific heat of fairly arbitrary materials? I suspect I could look at molecular dynamics or atomistic techniques, but these feel as involved as doing the work experimentally.

Edit with some extra information

The phase change is solid-solid, and compression to the high PT state occurs sufficently quickly that isothermal conditions, or temperature control are not credible. I can't say which material I'm interested in exactly due to commerical confidence, but sucrose is a not unreasonable analogue.

• Difficult to answer on a completely general basis. Knowing which material and if the solid phase you are referring to is a crystalline or not could help. It is strange that in an experiment there is no direct control of temperature. Jan 30 '19 at 23:42
• Both phases are solid, and are crystaline. The compression occurs sufficently rapidly that isothermal, or even temperture controlled, conditions are not possible unfortunately. Jan 31 '19 at 14:07
• ok, now I understand the difficulty to get the temperature. Still a realistic estimate about its order of magnitude may help. A few hundreds K ? thousand ? Jan 31 '19 at 14:49
• With no specific heat data post phase change its hard to pin down, but up the the phase change its an increase of a few hundred Kelvin over ambient conditions, so around 500 - 600 K. Jan 31 '19 at 14:58

On the basis of the available information on the system, I would consider unsafe to use results from models in conditions where it is highly probable that strong anharmonicity and/or potentially unknown structural changes take place.

For a system not too different from sucrose, ab-initio simulations would be the method of choice to get reliable values for specific heat (which would not be a too difficult quantity to evaluate).

I agree that performing an ab-initio simulation requires some specific expertise, but here we are moving out of the scientific ground. Maybe it could not be too difficult to find a person with the right expertise interested in working on the problem.

If your material is a pure substance and your final phase is a solid or gas, you could estimate the specific heat using the equipartition theorem as

$$c_v = \frac{f}{2} \frac{R_\text{u}}{M}$$

where

• $$f$$ is the number of internal degrees of freedom per particle, which is $$6$$ for a solid, $$3$$ for a monoatomic gas, and $$5$$ (at room temperature) or $$7$$ (at very high temperature) for a diatomic gas, but can also be evaluated for larger molecules,
• $$R_\text{u}$$ is the universal gas constant, and
• $$M$$ is the molar mass of the substance.
• It's a solid-solid phase transition. I'll add a note to the original question to that affect. Jan 31 '19 at 14:05
• Equipartition provides the specific heat of a solid only under the assumption that the system behaves like an ideal harmonic solid. Reality is often not as much ideal. Defects and anharmonicity play an important role. Jan 31 '19 at 15:26
• Fair point. I mention it because it is a very straightforward way of getting a rough estimate, which is what the question was seeking. Feb 1 '19 at 0:52