Let us say that I have an equation of state for a non-ideal gas (i.e. the Vanderwaals equation). I would like to calculate the internal energy of the gas U(V,T) without any 'microscopic input'.
1) Is that possible?
2) If so, how should this be done?
@Henk Huinink. Given van der Waals equation, (P+a/V^2)(V−b)=RT
where P is the pressure, V is the molar volume, T is the temperature, R is the ideal gas constant, and a and b are constants specific to each gas that account for intermolecular forces and molecular volume respectively. I suppose you have a and b calculated for your desired gas empirically?. And knowing internal energy U for a gas
dU = TdS -PdV
from the Van der Waals equation P = RT/(V-b) -a/V^2, and dS we could say it is dS = (Cv/T)dT we introduce those terms into dU then we have
dU = CvdT + (a/V^2 -RT/(V-b))dV.
Lets think on an experiment where we fix Temperature so the first term desapear. (we can make several experiments at different fixed temperatures). Then we can use a calorimeter to estimate the heat change when the volume is varied and we plot the data that show the area under the curve. ;)
Thanks Beatriz. What you write is inline with the ideas that I had in mind. In principle you can find the potential energy contribution of an interacting gas by taking the internal energy of an ideal gas as reference point. By calculating the work of compression at constant temperature, you could find the potential energy part. Than I am still left with the heat capacity bit and I was wondering if there is a 'non-microscopic' and a 'non-emperical' argument for finding the heat capacity.
Henk Huinink, we could use Density Functional Theory to computationally determine the total energy and electron density distribution of the offending non-ideal gas (there are geniuses out there mastering DFT, which by the way is very amusing to me). Then, we could link the total energy derived from DFT to the macroscopic variables P, T, and V. Maybe using some ML method could help make sense of these relations?.