Demystifying The Departure Function (11min) (uakron.edu)
...a peek inside the magician's hat. The connection from the real fluid to the ideal gas is not really magic. You can look at it as transforming the interaction potential (ie. u(r)) from 0 (ideal gas) to a Lennard-Jones model (real fluid). Alternatively, you can view it as the difference between the real fluid and ideal gas at each density, summed from zero density (where both exhibit ideal gas behavior) to the density of interest. This latter approach is most convenient and makes good use of our new expertise in derivative manipulations and equations of state.

Comprehension Questions:

1. In the diagram of (A-Ac)/RTc, which state represents the closest point to an ideal gas: A, B, C, or D?
2. Write out the departure function pathway in its various steps to compute "U" = (U-U^_Ref).
3. Identify the steps in #2 above as departure function or ideal gas contributions.
4. For propane at 355K and 3MPa, (U-U^ig)= -2572 J/mol. We can compute U^ig(355K)-U^ig(230K)=8000 J/mol. The departure function for liquid propane at 230K, 0.1MPa is (U-U^ig)= -16970 J/mol. Compute the value of "U" at 355K and 3MPa relative to liquid propane at 230, and 0.1MPa using this information.
5. Compare your answer to the value given by PREOS.xlsx.
6. Compare your answer to the value given by the pathway of Figure 2.6c. (Hint: use Eqn. 2.47 to decide whether 355K,3MPa corresponds to a vapor or liquid.)

The Entropy Departure Function  (11:22) (uakron.edu)
Deriving the general formula for the entropy departure function is analogous to the derivation for the internal energy formula. There are two points of interest however: (1) The entropy formula for an ideal gas depends on volume (or pressure) as well as temperature, necessitating a contribution of lnZ to correct from S^ig(T,V) to S^ig(T,P). (2) When all is said and done, combining S with U (derived in 08.02) gives A (=U-TS) and A gives G (=A+PV), implying that other departure functions can be obtained by simple arithmetic applied to U and S.

Comprehension Questions: The RK EOS can be written as: Z = 1/(1-bρ) - aρ/(RT^1.5).
1.  Use Eqn. 8.19 to solve for (S-S^ig)_TV/R of the RK EOS.
2.  Use Eqn. 8.27 to solve for (A-A^ig)_TV/RT of the RK EOS.
3.  Use Eqns. 8.22 and 8.27 to solve for (S-S^ig)_TV/R of the RK EOS.

Helmholtz Energy - Mother of All Departure Functions. (uakron.edu, 10min) This screencast begins with a brief perspective on energy and free energy as they relate to concepts from Chapter 1 and through to the end of the course. Then it focuses on how the Helmholtz departure function is one of the most powerful due to the relations that can be developed from it. The Helmholtz departure is relatively easy to develop from a density integral of the compressibility factor. Then the internal energy departure can be derived from a temperature derivative. Alternatively, if the internal energy departure is given, the Helmholtz energy can be inferred by integration, and the compressibility factor can be derived from a density derivative. 
Comprehension Questions: (Hint: some of the following may be answered in later videos below.)
1. Write an equation that takes you from the Helmholtz energy departure function to Z.
2. Write an equation that takes you from the Helmholtz energy departure function to (U-Uig)/RT.
3. Derive the internal energy departure function for the vdW EOS using Eqn. 8.22.
4. Derive the Helmholtz energy departure function for the vdW EOS using Eqn. 8.25.
5. Use the result of #4 to derive the internal energy departure function for the vdW EOS.

Helmholtz Example - vdW EOS (uakron.edu, 18min) This video begins with a brief review of the connection of the Helmholtz departure with all other departures then shows four sample derivations assuming that Z is given by the vdW EOS: (1) the Helmholtz departure , (2) the internal energy departure from the Helmholtz departure. (3) the Helmholtz energy from the internal energy (4) the Z factor from the Helmholtz departure. The procedures illustrated here can be applied to any EOS starting with any part (U, A, or Z) as given to derive any other departure: ZUHAGS.
Comprehension Questions: The virial EOS for SW fluids can be written as: Z = 1 + Bρ/RT where B = 4b+[4b(λ^3-1)] [exp(βε)-1], b = πNAσ^3/6.
1. Derive an expression for the Helmholtz departure.
2. Use the result of #1 to derive the internal energy departure.
3. Use the result of #2 to derive the Helmholtz departure. What is the integration constant in this case?

Helmholtz Example - Modified vdW EOS (uakron.edu, 13min) A sample derivation of the Helmholtz departure implicit in the Gibbs departure given Z = 1 + abρ/(1+bρ)^3 - (9.5aρ/RT)/(1+aρ/RT). Note that the limits of integration matter for this EOS. The audio is inferior for this live video, but it responds to typical questions and confusion from students in the audience. Some students might find it helpful to hear the kinds of questions that students ask. The responses slow the derivation down so that no steps are skipped and key steps are reiterated multiple times. Just turn the volume up!
Comprehension questions:
1. Which part of this EOS is non-zero at the zero density limit of integration?
2. Is there a sign error on one of the terms in this video? Check the derivation independently.
3. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-bρ)2 - (9.5aρ/RT)/{1-a/bRT[1-4bρ+4(bρ)2]}.
4. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-2bρ) - (9.5aρ/RT){1+4aρ/bRT[1-2(bρ)2]}/{1-a/bRT[1-4bρ+4(bρ)2]})/{1-a/bRT[1-4bρ+4(bρ)2]}

Helmholtz Example - Scott+TPT EOS. (uakron.edu) A sample derivation (8min) for the compressibility factor given that (A-Aig)TV/RT = -2ln(1-2ηP) - 18.7ηPβε/[1+0.36βεexp(-5ηP)]. This equation of state is a little complicated, but the derivation is no problem if you just go slow and steady. The remainder of this screencast shows a sample calculation (21min) to solve the resulting equation of state at a given value of pressure and temperature following the methodology of "visualizing the vdW EOS." This problem was adapted from an actual test problem. This screencast is live so the audio is inferior, but it gives insight into questions that real students have. 
Comprehension Questions:
1. Derive an expression for the internal energy departure function of this EOS.
2. Is there a sign error on one of terms in this video? Check the derivation independently.
3. Derive the Z factor given (A-Aig)TV/RT = -2ln(1-2bρ) - (9.5aρ/RT)/{1-a/bRT[1-4bρ+4(bρ)2]}.

Helmholtz Departure - PR EOS (uakron.edu, 11min) This lesson focuses first and foremost on deriving the Helmholtz departure function. It illustrates the application of integral tables from Apx. B and the importance of applying the limits of integration. It is the essential starting point for deriving properties involving entropy (S,A,G) of the PREOS, and it is a convenient starting point for deriving energetic properties (U,H).

Internal Energy Departure - PR EOS starting from Helmholtz Departure (uakron.edu,9min) This sample derivation supplements what is in the textbook by starting from the Helmholtz departure function. It also includes a few intermediate steps to help clarify how the formal equations in the textbook were developed. Hopefully, seeing this content from slightly different perspectives will make it a little easier to comprehend. See also the derivation for (U-Uig).

Comprehension Questions: Starting from the Helmholtz Departure function and referring to the above results...

1. Derive the internal energy departure function for the "modified vdW" EOS.
2. Derive the entropy departure function for the "modified vdW" EOS. (Hint: A=U-TS)
3. Derive the internal energy departure function for the "Scott+TPT" EOS.

Derive the internal energy departure function (uakron.edu, 20min) for the following EOS:
P = (RT(1+V^1.5)/V^1.5)*(1+sqrt(V)) - a/(V^{^2}T^{^1.3})/(1+sqrt(V)) This sample derivation is more complicated than average, but the usual procedure still works. We begin by rearranging to obtain an expression for Z and finding the Helmholtz departure, then differentiating to get the internal energy.

Comprehension: Given (A-A^ig)_TV/RT = -2ln(1-η_P) - 16.49η_Pβε/[1-βε(1-2η_P)/(1+2η_P)^2 ]

1. Derive the internal energy departure function.

2. Derive the expression for the compressibility factor.

3. Solve the EOS for Zc.

Heat Pump Design (20min) (uakron.edu) This sample calculation was adapted from a 1987 practice test. It walks carefully through the coefficient of performance (COP) determination for a heat pump that might be applied to a home in Ohio, and the practicality of a heat pump compared to a furnace. The problem assumes a desired indoor temperature of 70F and an average outdoor temperature of 45F, with a 10F approach temperature on both sides. Care is needed to adapt the COP to heat pump application because the quantity of interest is QH, not QC. This video applies Freon-12, which has since been outlawed because it contributed to the hole in the ozone layer that comprised what may be the first example of anthropogenic catastrophe on a global scale. The good news is that the ozone hole has begun to heal itself since the regulation of Freon-12. Nevertheless, Freon-12 did possess remarkable properties as a refrigerant, highlighting the motivation to contiuously search for its replacement. As an exercise, it is suggested that you redesign this heat pump using HFO1234yf, a new refrigerant with a global warming potential of 4, compared to 3800 for HFC134a and virtually infinity for Freon-12.

Comprehension Questions: Assume a reference state of the saturated liquid at 1 bar. Use Eqn. 2.47 (SCVP eq) to estimate saturation conditions.
1. Compute the enthalpy (J/mol) of saturated vapor HFO1234yf at 80F.
2. Compute the enthalpy (J/mol) of saturated liquid HFO1234yf at 35F
3. Compute the enthalpy (J/mol) of HFO1234yf exiting an adiabatic reversible compressor being fed saturated vapor at 80F.
4. Compute the COP for this OVC cycle.

Real Gas Expansion (LearnChemE.com, 5min) determines the final state of a real gas that expands adiabatically into a vacuum by an energy balance. Real Gas Expansion Part 2: Excel Solver (LearnChemE.com, 5min) uses the Peng-Robinson equation of state to compute the necessary properties. This two-part series shows the solution of a fairly challenging problem. Nevertheless, the solution appears to be easy when using the right tool.

Comprehension Questions:

1. This problem involves the use of just the energy balance. Can you think of a similar problem that would use both the energy and entropy balance?

Thermodynamic pathways of EOS's for arbitrary reference states (uakron.edu, 20min) The development of a thermodynamic pathway from an arbitrary reference state to a given state condition is independent of the thermodynamic model. It depends only on (1a) identifying the condition of the reference state (e.g. ideal gas, real vapor, or liquid) (1b) transforming from the reference state to the ideal gas, if necessary (2) transforming from the ideal gas at the condition of the reference state to the ideal gas at the given state condition (3a) identifying the condition at the given state (3b) transforming from the ideal gas at the given state to the real fluid at the given state. The methodology is illustrated for two thermodynamic models: the P^sat/H_^vap model of Figure 2.6c,Eqs 2.45,47 vs. the PR EOS. The screencast is a bit long, but it covers 16 sample calculations (8 for H and 8 for S) and comparisons between PREOS vs P^sat/H_^vap. You might like to refer back to Sections 2.10 and 3.6 to review the P^sat/H_^vap model and the elemental reference state. Push pause before each sample calculation and check whether you can predict the next answer.

Comprehension Questions:

1. Compute "H" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cp^ig/R = 8.85 and the PR EOS. You may use PREOS.xlsx to compute H-Hig, but you must show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
2. Compute "S" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cp^ig/R = 8.85 and the PR EOS. You may use PREOS.xlsx to compute S-Sig, but you must show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
3. Compute "H" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cp^ig/R = 8.85 and the P^sat/H_^vap model. Show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
4. Compute "S" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cp^ig/R = 8.85 and the P^sat/H_^vap model. Show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.

This sample calculation shows how to compute the liquefaction in the Linde process for methane as the operating fluid. (uakron, 8min) The Linde process is a slight variation on the OVC cycle wherein the liquefied fraction exiting the throttle is captured as product and removed from the process. There is also heat integration in the sense that the cold vapor is used to precool the feed to the throttle.

FYI: Since natural gas is mostly methane, this process could be easily adapted to the production of liquefied natural gas (LNG) or liquified petroleum gas (LPG, mostly propane). Liquefied gases may seem impractical when you first encounter them, but they are more efficient for transport because they are so much more dense than the gases. Keeping them as liquids is basically a reflection of the effectiveness of the insulation. If any gas leaks from the relief valve (~1.1 bar), then liquid must evaporate to fill the space. The requisite heat of vaporization in that case cools the remaining below the boiling temperature. No heat = no vaporization.