# How to calculate the volumetric energy density of a molecule?

I am trying to store hydrogen in a hydrocarbon. The example can be taken as cyclohexane ($$\ce{C6H12}$$) which is in a hydrogen-rich state. Its hydrogen-lean state is Benzene.

I want to calculate the volumetric energy density of cyclohexane.

The reported value of density on Wikipedia is 0.7739 g/mL for the liquid and 0.996 g/mL for the solid, along with a note:

"Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa)."

The molar mass of cyclohexane is 84.16 g/mol.

I could not understand how to calculate the density value as given on the above mention Wikipedia page.

• That density is probably obtained by an experimental measurement. Why do you want to "calculate" it? Apr 20, 2021 at 21:03

The values that you quoted for the densities in the liquid and solid phases, were most likely obtained from experiments, in the following very simple way:

• (1) take exactly 1 liter of the substance
• (2) measure it's mass in grams
• (3) the density in units of g/L is simply the result of the measurement in step (2)
• (4) divide the result in step (3) by 1000 to get the density in units of g/mL.

You mentioned the molar mass of the substance as being 84.16 g/mol, which is the mass if you have exactly 6.02214076 × 1023 copies of the molecule.

It's not very easy to "calculate" the density of a substance knowing only the chemical formula and the molar mass, because you would have to know how much space a certain amount of the substance occupies.

The best way to find out how much space a certain amount of a substance takes up, would probably be to do a molecular dynamics simulation. Here you would start with thousands (for example) of copies of the molecule, in some initial configuration, which you can get using the software PACKMOL. Then you would set the temperature and pressure (perhaps to 25°C and 100 kPa in this case, according to the quote in your question, but I suspect you might actually have to find out where those density values were obtained and choose the values of temperature and pressure based on what they were when those numbers were measured, since it's unlikely that the substance is both liquid and solid at the same temperature+pressure).

Then you would use a software such as LAMMPS or GROMACS (just to name a couple examples) with some choice of forcefield, to see the effect of Newton's 2nd law on the system in that forcefield (i.e. how the molecules will push and pull each other until they get comfortable with the amount of personal space they have). Eventually when the system reaches equilibrium, you will have some idea about how much space that number of molecules of your substance takes up at that temperature and pressure. Then you can do the same simulation with more and more molecules in your simulation, until you converge on a value for the number of molecules/volume. This final number of molecules/volume will be quite accurate under two main assumptions: the forces between the molecules are given by the forcefield that you chose, and the dynamics of the system follows Newton's second law rather than the time-dependent Schroedinger equation or something more accurate but more complicated that you would get from a more advanced quantum field theory).

Once you have the number of molecules/volume, you can finally use that molar mass of 84.16 g/mol that you correctly looked up, and get the mass/volume!

You will probably have to do this procedure twice: once for the temperature+pressure for the liquid phase and once for the temperature+pressure for the solid phase.

It would be very nice if there was a theory that could allow you to estimate the volume of a substance based on the chemical formula, so that you wouldn't have to run this type of "brute force simulation" known as "molecular dynamics" to get an estimate of the volume of a substance.

• Thank you for the detailed response. We can calcu the density using comptox.epa.gov/dashboard/predictions/index . This will be in gm/cm^3 which can then be converted to gm/liter. Can we do this? I don't know how accurate and true will be the results. Apr 22, 2021 at 3:01
• 1 Litre is a cubic decimeter, so yes, you can convert gm/cm^3 to gm/L. Oct 10, 2021 at 21:52