Compound with pressure sensitive melting point at moderate temperature and pressure

I'm in need for a compound with ideally all of the following properties:

• A melting point of 5 °C or less at 1 atmosphere (or slightly lower pressure)
• A melting point of 12 °C or more at less than 20 atmospheres.
• A significant enthalpy of fusion

(long term stability, a boiling point > 50 °C, and less than a 3 digit \\$ cost per kg would also be nice).

Seeking such a compound, a melting point range this wide appears to be unusual at such small pressure differences. So far, the best I have is cyclohexane, but the melting point is a bit too high at 6.5 °C at normal pressure, and it needs over 100 atmospheres before the melting point gets above 12 °C, which is impractical.

• I am afraid the required parameters are not realistic. Feb 14 '21 at 20:29
• @Poutnik A negative would also be helpful as an answer, as I can then conclude that I would have to look for other solutions. Feb 15 '21 at 15:51
• It would help ( unless it is a research secret ), if you reveal the background, what is the intended application of this material. As it is possible others could suggest alternative ways to reach the goal, which is unknown for now. Feb 15 '21 at 17:36
• @Poutnik It's as mundane as a cooling requirement for an application where providing electricity is troublesome. If this compound existed, it would be convenient. If not, there are already many slightly less convenient solutions. (gas-liquid phase transitions being the most on-topic). It's a solved problem, I was just hoping an even simpler solution existed. Feb 15 '21 at 22:24

I'm not going to do your research for you (at least not for now—perhaps if the mood takes me I'll do the Mathematica programming at some later point), but I will give you guidance that will help you find the answer:

1. It's hard to find data on melting points at different pressures for a wide range of compounds (and you want to search through a wide range of compounds). What is much more comprehensively available are data that can be used to calculate such pressure dependency, namely the specific densities of substances in their solid and liquid phases, and their enthalpies of fusion. So:

According the Clapeyron equation,

$$\frac{dT}{dp}=\frac{\Delta V_m}{\Delta S_m}$$

where $$T$$ is the phase transition temperature, $$\Delta S_m$$ is the molar entropy change for the phase transition, and $$\Delta V_m$$ is the molar volume change for the phase transition.

For a phase transition, $$\Delta S_m$$ = $$\frac{\Delta H_m}{T}$$, where $$T$$ is in Kelvin. And you can determine $$V_m$$ (in $$\pu{\frac{{cm}^3}{mol}}$$) from the molecular mass, $$M_r$$ (in $$\pu{\frac{g}{mol}}$$), divided by the specific density, $$\rho$$ (in $$\pu{\frac{g}{cm^3}}$$):

$$V_m = \frac{M_r}{\rho}\equiv \frac{\pu{\frac{g}{mol}}}{\pu{\frac{g}{cm^3}}} = \pu{\frac{{cm}^3}{mol}}$$

Thus:

$$\Delta V_{m, fus} = \frac{M_{r, liq}}{\rho_{liq}} - \frac{M_{r, solid}}{\rho_{solid}}$$

The subscript "fus" means "fusion", indicating this is a solid-liquid phase transition.

Substituting, we obtain:

$$\frac{\Delta T_{fus}}{\Delta p} \approx \frac{dT_{fus}}{dp}=\frac{\Delta V_{m, fus}}{\Delta S_{m, fus}} =\frac{\frac{M_{r, liq}}{\rho_{liq}} - \frac{M_{r, solid}}{\rho_{solid}}}{\Delta S_{m, fus}}=T_{fus}\frac{\frac{M_{r, liq}}{\rho_{liq}} - \frac{M_{r, solid}}{\rho_{solid}}}{\Delta H_{m,fus}}$$

What you want is to identify the substances that have the largest values for the above expression, and then filter according to your other criteria.

Note that this expression only gives an exact value for $$\frac{\Delta T_{fus}}{\Delta p}$$ if you integrate with respect to $$T_{fus}$$ using $$\Delta H_{fus} (T), \rho_{liq}(T) \text{ and } \rho_{solid}(T)$$—i.e., if you account for the temperature-dependencies of each of these. However, since you are determining $$\frac{\Delta T_{fus}}{\Delta p}$$ over a range not far from $$T_{fus}$$ at 1 atm, and since the densities for the phases that are stable at room temperature are probably at $$25 ^\circ C = 298 K$$, which is close to your desired $$T_{fus}$$, and since they will not change much over your pressure range, this should give you a very good approximation. Once you narrow it down to the actual substance, you can apply corrections to get a more precise estimate.

1. You could do this search manually by plugging in values into an Excel spreadsheet, or semi-manually by finding downloadable data tables with these values and importing them into Excel, but the best approach would be to use a program like Mathematica*, which has access to large chemical databases, and ask it to search for all substances for which it has density data in the solid and liquid phases, and enthalpies of fusion, calcualate the values for the above expression, and sort by magnitude. You can then apply additional filters (e.g., melting point and enthalpy of fusion), as desired.

[*This of course requires knowing how to use Mathematica, or learning to use it. It's possible someone on MathematicaSE could help you write such a query. Alternately, it's possible the online CRC is also searchable in this way (certainly seems it should be), though I don't know myself. One way to get help might be to check the reference desk of your local library. They might have the capability to query chemical databases in this way.]

Once you have this, you can then identify the best candidate compounds, and further filter by cost and stability.

1. Note: It may not be possible to find a compound whose melting point changes enough (a comprehensive chemical database search would help you determine this). Finding a large pressure-induced change in melting point is challenging, because substances don't change volume that much in going from a solid to a liquid. Where you do get a large pressure-induced change in a phase transition temperature is when going from liquid to gas because, under normal conditions (room temp and 1 atm), gases have about 1000x the molar volume of liquids. That's why water boils at a notably lower temperature when you go into the mountains, even though the change in pressure there is a mere fraction of an atmosphere. [By "notably lower" I mean enough of a change to affect cooking times.]

Plus you make your search even more challenging by requiring a high $$\Delta H_{fus}$$, since a high $$\Delta H_{fus}$$ lowers $$\frac{\Delta T_{fus}}{\Delta p}$$ (everything else being equal).

• Will report back if this yields results. Feb 14 '21 at 13:39
• @SE-stopfiringthegoodguys When I gave my initial answer I forgot to include the entropy term! I've amended my answer accordingly. Feb 14 '21 at 21:28
• After trawling through large tables, there's indeed nothing of the sort to be found. Mar 4 '21 at 12:39