Occasionally, our equation of state is difficult to integrate to obtain departure functions using the formulas from Section 8.5. This is because the equation of state is more easily arranged and integrated in the form Z = f (T,P), such as the truncated virial EOS. For treating cases where an equation of state is written most simply as Z = f(T,ρ) such as a cubic EOS, see Section 8.5. We adapt the procedures given earlier in Section 8.2.
1. Write the derivative of the property with respect to pressure at constant T. Convert to derivatives of measurable properties using methods from Chapter 6.
2. Write the difference between the derivative real fluid and the derivative ideal gas.
3. Insert integral over dP and limits from P = 0 (where the real fluid and the ideal gas are the same) to the system pressure P.
4. Transform derivatives to derivatives of Z. Evaluate the derivatives symbolically using the equation of state and integrate analytically.
5. Rearrange in terms of density and compressibility factor to make it more compact.
We omit derivations and leave them as a homework problem. The two most important departure functions at fixed T,P are
The other departure functions can be derived from these using Eqns. 8.20 and 8.21. Note the mathematical similarity between P in the pressure-dependent formulas and ρ in the density-dependent formulas.
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