Question

In my previous question Enthalpy of a Van der Waals gas, I got the expression of the enthalpy generalised, but I am still having issues finishing it since it might have some quite hard calculus...

For my Van der Waals gas with constant temperature we got to the following expression by the already mentioned question:
dH = TdS + VdP = T (−∂P / ∂V dP ) + VdP = ( V − ∂P / ∂V T ) dP
while the Van der Waals equation is
( P + an² / V²) ( V − nb ) = P ( V − nb ) + an² / V − an³b / V² = n R T
and solving for P to get the ∂P / ∂V we get
P = nRT − an²V − an³b / V² / V − nb = nRTV² − an³V − an³b / (V−nb)V²
and so we can find the partial derivative term
∂P / ∂V = (2nRTV−ab³) (V² (V − nb)) − (nRTV² − an³V − an³b) (3V² − 2nbV) / (V³ − nbV²)²
which seems already extremely large, am I on the right path or is there an easier way?

Answer 1

The complex partial derivative of enthalpy for a Van der Waals gas is on the right track. Simplification is generally recommended, and under conditions where the gas behaves ideally, the equation simplifies greatly, demonstrating negligible influence from Van der Waals corrections.

Step-by-step explanation:

When addressing the enthalpy change for a Van der Waals gas, the complex equation you've derived involving the partial derivative ∂P/∂V seems to be on the correct path. However, solving the enthalpy for a Van der Waals gas indeed involves challenging calculus. Given that a Van der Waals gas corrects the ideal gas law by considering molecular volume and intermolecular forces, your expansion of the differential dH looks correct. The next steps would typically involve simplifying the derivative expression as much as possible and potentially substituting any known variables or constants that could simplify the resulting equations.

If the conditions are such that the gas behaves nearly ideally, where 'a' and 'b' are relatively small, the Van der Waals equation reduces to the ideal gas law, PV = nRT. Under these conditions, the deviations due to intermolecular attractions and finite molecular volume are negligible, meaning the calculus and resulting expressions will be much simpler. The extreme complexity of your derived partial derivative might be indicative that simplifications can be made, especially if the conditions approach ideality in the limits of large volume and low pressure.