Category: Departure Functions

P8.1. Develop an expression for the Gibbs energy departure function based on the Redlich-Kwong (1958) equation of state: (ANS. (G – Gig)/RT = –ln(1 – bρ) – aln(1 + bρ)/(bRT3/2) + Z – 1 – lnZ) P8.2. For certain fluids, the equation of state is given by Z = 1 – bρ/Tr. Develop an expression for the enthalpy departure function for fluids of this type. (ANS. –2bρ/Tr) P8.3. In…

Eqns. 8.22–8.30 stand out in this chapter as the starting point for deriving the necessary departure function expressions for any equation of state. It is tempting to use spreadsheets or programs to add up the contributions from departure functions, reference states, and ideal gas temperature effects mindlessly, like a human computer. But keep in mind that…

The study of departure functions often causes students great difficulty. That is understandable since it involves simultaneous application of physics and multivariable calculus. This may be the first instance in which students have applied these subjects in combination to such an extent. On the other hand, it is impressive to see what can be accomplished…

As in the case of the compressibility factor, it is often useful to have a visual idea of how generalized properties behave. Fig. 8.7 on page 324 is analogous to the compressibility factor charts from the previous chapter except that the formula for enthalpy is (H – Hig) = (H – Hig)0 + ω(H – Hig)1. Note that one primary influence in determining the liquid enthalpy…

The tasks that remain are to select a particular equation of state, take the appropriate derivatives, make the substitutions, develop compact expressions, and add up the change in properties. The good news is that many years of engineering research have yielded several preferred equations of state (see Appendix D) which can be applied generally to any…

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…

Formulas for departures at fixed T,P are listed below. These formulas are useful for an equation of state written most simply as Z = f(T,ρ) such as cubic EOSs. For treating cases where an equation of state is written most simply as Z = f (T,P) such as the truncated virial EOS, see Section 8.6. Useful formulas at fixed T,V include:

The remainder of the departure functions may be derived from the first two and the definitions,  The departures for U and S are the building blocks from which the other departures can be written by combining the relations derived in the previous sections. where we have used PVig = RT for the ideal gas in the enthalpy departure. Using H – Hig just derived,

To calculate the entropy departure, adapt Eqn. 8.11, Inserting the integral for the departure at fixed {T, V}, we have (using a Maxwell relation), Since , we may readily integrate the ideal gas integral (note that this is not zero whereas the analogous equation for energy was zero): Recognizing Vig = RT/P, V/Vig = PV/RT = Z, where Eqn. 8.15 has been applied to the relation…

 schematically compares a real gas isotherm and an ideal gas isotherm at identical temperatures. At a given {T,P} the volume of the real fluid is V, and the ideal gas volume is Vig = RT/P. Similarly, the ideal gas pressure is not equal to the true pressure when we specify {T,V}. Note that we may characterize the departure from ideal gas…