# Calculate the enthalpy of a pure compound with a cubic equation of state

I would like to calculate the enthalpy of a pure compound with a cubic equation of state (the ones deriving from the Van der Waals equation of state).

I have an analytic expression of the heat capacity of the pure fluid in the perfect gas state as a function of temperature, allowing me to estimate the enthalpy of the pure fluid in the ideal gas state. Thus the cubic equation of state shall be used to estimate the deviation between the enthalpy of the pure fluid in the perfect gas state and the real state. And then come the questions:

• How to estimate this enthalpy deviation with a cubic equation of state?
• How to choose a wise enthalpy reference state (as no absolute enthalpy can be estimated I need to choose a reference enthalpy that might simplify some calculations)?
• How to treat the cases when wether I have (i) a single phase system or (ii) a two phase system (vapor-liquid equilibrium, i.e. how to take account of the phases proportions)?

I have read many references that give partial answers but none of them seem to tackle these questions in a single, clear and educationnal way. I hope someone is brave enough to try to answer all these questions.

## 1 Answer

For the reference state of zero enthalpy, you choose a convenient temperature in the ideal gas region. Then you integrate the heat capacity at constant pressure from the reference temperature to the temperature T of the system, still in the ideal gas region (basically, zero pressure). Then, at the desired temperature, you calculate the change in enthalpy between zero pressure and the pressure of interest by integrating the equation $$\left(\frac{\partial H}{\partial P}\right)_T=V-T\left(\frac{\partial V}{\partial T}\right)_P$$

To decide what you want to do about the two phase region, you need to plot P as the ordinate as a function of H as the abscissa at constant T. For a real gas, this will be flat in the two-phase region. For your substance, it will exhibit a maximum and a minimum. You need to decide upon a criterion for judging where the ends of the two phase region lie along the line (pure liquid and pure gas).

ADDENDUM

An alternate is to first determine $\Delta U$ and then add $\Delta (PV)$. The change in U in the ideal gas region is obtained by integrating $C_v^{IG}dT$. Then the correction for specific volume (not being infinite) is obtained by integrating $$\left(\frac{\partial U}{\partial v}\right)_T=P-T\left(\frac{\partial P}{\partial T}\right)_v$$ from v to infinity.

Many books have approaches for establishing the two end points of a two-phase line. See Smith and Van Ness, Introduction to Chemical Engineering Thermodynamics.

• The proposed integral cannot be calculated with a cubic equation of state which is a function of $T$ and $v$ but not $P$ and $v$. Thus derivative at fixed $P$ cannot be estimated. Regarding the two phase problem, my concern is more about how to estimate the enthalpy of the system when it is in vapor-liquid equilibrium and that we know the enthalpy of both phases. – yolegu Jan 10 '18 at 9:51
• Which cubic equation of state are you using? Are you willing to express H as a function of v and T? – Chet Miller Jan 10 '18 at 13:15
• The possible equations are the Redlich-Kwong or the Peng-Robinson ones. And yes, I would like to calculate $H$ as a function of $T$ and $v$. – yolegu Jan 10 '18 at 14:13
• See my addendum. – Chet Miller Jan 10 '18 at 15:07