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The fugacity of a solid (uakron, 19min) follows a similar trend to that of a liquid, but there can be unexpected implications. The impact of pressure requires careful consideration. NIST Webbook lists the melting temperature of xenon as 161.45K and the Antoine equation as log10Psat(bars) = 3.80675 - 577.661/(T(K)-13.0), CpV=22.7 J/mol-K, CpL=44.4 J/mol-K, ρL=2.9662 g/cm3. Wikipedia lists the solid density as 3.540 g/cm3 (and the liquid density as 3.084) and the heat of fusion as 2270 J/mol. You may assume CpL=CpS. Use Eq. 7.06 to describe the vapor phase. You may assume ω = 0 for the purpose of these calculations. This screencast shows a sample calculation to solve for: (a) the vapor fugacity at 162 K and 0.085 MPa (b) the liquid fugacity in equilibrium with the same vapor at 162 K and 0.085 MPa (c) the liquid fugacity at 162 K and 8.5 MPa (d) the solid fugacity at 161.45 K and 0.082 MPa (e) the solid fugacity at 162 K and 8.5 MPa. If you are still having trouble understanding the ways that all these fugacities relate, you might like to view the phase diagram implications of VLSE (uakron, 9min).

Comprehension Questions:

1. How much did raising the pressure to 8.5 MPa change the liquid fugacity (bars)?
2. Estimate the fugacity (MPa) of the vapor at 162 K and 1.15 MPa and compare it to the liquid. Which is smaller? Which state do you think best characterizes the fluid (ie. V or L or S)?
3. Estimate the fugacities (MPa) of methane vapor, liquid, and solid at its triple point using PREOS. Compare the vapor pressure from PREOS at the triple point to that from NIST.
4. Assuming VS=VL/1.1, estimate the fugacity of solid methane at 92K and 10 MPa using PREOS for all fluid properties. Consult the NIST Webbook for T and Hfus at the triple point.

Use VLookup and shortcut estimates of Antoine coefficients (see above) to quickly generate the Pxy phase diagram for an ideal solution. (11min, uakron.edu) This video shows a sample calculation for methanol+benzene using Eqn. 10.8. It also shows how to reference the approximate estimates in the cells that will actually be used for later calculations, so you can type in precise estimates when you have them. When no precise values are available, recover the shortcut estimates by simply typing "=" and referencing the cell with the shortcut estimate. The predictive formulas are written in black and should not be edited. The cells for user modification by pasting more accurate values are indicated in blue. We use this approach a lot, and refer to it fondly as the "black and blue" method.

Note: This is a companion file in a series. You may wish to choose your own order for viewing them. For example, you should implement the first three videos before implementing this one. Also, you might like to see how to quickly visualize the Txy analog of the Pxy phase diagram. If you see a phase diagram like the ones in section 11.8, you might want to learn about LLE phase diagrams. The links on the software tutorial present a summary of the techniques to be implemented throughout Unit3 in a quick access format that is more compact than what is presented elsewhere. Some students may find it helpful to refer to this compact list when they find themselves "not being able to find the forest because of all the trees."

Comprehension Questions:

1. Create a Pxy diagram for methanol+benzene at 90C based on the ideal solution SCVP model. Be sure to include all appropriate labels and label your sketch as quantitatively as possible. Compare your model to the data in HW 11.10 by including those points in the plot. Explain similarities and discrepancies.
2. Sketch a Pxy diagram for acetone+acetic acid at 55C based on the ideal solution SCVP model. Be sure to include all appropriate labels and label your sketch as quantitatively as possible. Compare your sketch to Figure 10.9a. Explain similarities and discrepancies.
3. Sketch a Pxy diagram for acetone+chloroform at 35.17C based on the ideal solution SCVP model. Be sure to include all appropriate labels and label your sketch as quantitatively as possible. Compare your sketch to Figure 10.9c. Explain similarities and discrepancies.
4. Sketch a Pxy diagram for 2-propanol+water at 30C based on the ideal solution SCVP model. Be sure to include all appropriate labels and label your sketch as quantitatively as possible. Compare your sketch to Figure 10.8c. Explain similarities and discrepancies.

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-URef).
3. Identify the steps in #2 above as departure function or ideal gas contributions.
4. For propane at 355K and 3MPa, (U-Uig)= -2572 J/mol. We can compute Uig(355K)-Uig(230K)=8000 J/mol. The departure function for liquid propane at 230K, 0.1MPa is (U-Uig)= -16970 J/mol. Compute the value of "U" at 355K and 3MPa relative to liquid propane at 230, and 0.1MPa using this information.
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.)

There are so many activity models, how can you keep them straight? This video shows how MAB, SSCED, and Scatchard-Hildebrand models are all closely related.(9min,uakron.edu) By changing the assumptions, one model can be transformed into the other. So focus on remembering one model very well, then remember the small adjustments to obtain the other models.

Comprehension Questions:
1. Suppose we are trying to find the solvent most compatible with dilute ethanol. Which of the following is most compatible according to the MAB model?  (a) water (b) benzene (c) n-octanol.
2. Suppose we are trying to find the solvent most compatible with dilute ethanol. Which of the following is most compatible according to the ScHil model?  (a) water (b) benzene (c) n-octanol.
3. Suppose we are trying to find the solvent most compatible with dilute ethanol. Which of the following is most compatible according to the SSCED model?  (a) water (b) benzene (c) n-octanol. (Hint: consider the infinite dilution activity coefficient.)

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 Sig(T,V) to Sig(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-) - /(RT1.5).
1.  Use Eqn. 8.19 to solve for (S-Sig)TV/R of the RK EOS.
2.  Use Eqn. 8.27 to solve for (A-Aig)TV/RT of the RK EOS.
3.  Use Eqns. 8.22 and 8.27 to solve for (S-Sig)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.

Stream enthalpies for the DME process (uakron, 7min) can be estimated using the "heat of reaction" pathway (Fig 3.5a) or the "heat of formation" pathway (Fig 3.5b). This presentation is based on Fig 3.5b, which is very similar to Fig 2.6c. The main difference is the inclusion of the heat of formation for each compound relative to its elements. Including the heat of formation puts the reference state for each compound on the same basis of comparison (ie. the elements). If one stream (e.g. "products") possesses more enthalpy than another stream (e.g. "reactants") then the energy difference between the streams (e.g. "heat of reaction") would be accounted for by simply subtracting the two stream enthalpies. Reactions inherently involve multiple components, so including the heats of formation in the stream enthalpies, as well as the other enthalpic contributions represented in Fig 2.6c, is inevitable. These sample calculations are illustrated for all the streams appearing in the DME process. The presentation follows up on the discussion of Fig 2.6c for pure fluids. Once you understand the calculations for each pure fluid, the mixture property simply involves taking the molar average, so: H ≈ ∑(xi*Hfi+CpiigΔT+(qi-1)*Hivap). In this equation, (qi-1)*Hivap accounts crudely for departures from ideal gas behavior. For example, if a stream is a vapor, then q=1 and Hvap doesn't matter. If q=0, then the stream is a liquid and Hvap must be subtracted. We will study more accurate models of ideal gas departures in Unit II.

Comprehension Questions:

1. Compute the enthalpy, H(J/mol), of methanol at 250C and 2 bars relative to its ideal gas standard state elements.

2. Compute the enthalpy, H(J/mol), of DME at 250C and 2 bars relative to its ideal gas standard state elements.

3. Compute the enthalpy, H(J/mol), of water at 250C and 2 bars relative to its ideal gas standard state elements.

4. Compute the enthalpy, H(J/mol), of a stream that is 50% methanol, 25% DME, and 25% water at 250C and 2 bars relative to its ideal gas standard state elements.

This video walks you through the process of transforming the Scatchard-Hildebrand model into the SSCED model using Excel (6min, uakron.edu) It steps systematically through the modifications to the spreadsheet to obtain the new model. You should implement the Scatchard-Hildebrand model before implementing this procedure.

Comprehension Questions:
1. Add Txy capability to this model.
2. Predict the Txy diagram for methanol+benzene by the SSCED model at 2222mmHg. Estimate the phase compositions and phase amounts for the following operating temperatures and feed compositions. (a) 370K and zm = 0.30 (b) 350K and zm = 0.20 (c) 370K and zm = 0.70.
3. Compare your predicted Txy diagram to the predictions by the MAB and Scatchard-Hildebrand models. Describe the differences briefly for each case.
4. Search for experimental data on the system ethanol+toluene. Modify your spreadsheets to plot the experimental data (points) on the same plot with the predictions. Which model provides the most accurate predictions when compared to data?
5. Suppose you set k12=0 in the SSCED model. Does that improve the comparison to experimental data? Other models? Does the combination of k12=0 and k12=k12(alpha,beta) bracket the range of values that fit reasonably?

Closed System Energy Balance: Ideal Gas Expansion (uakron.edu, 9min) An ideal gas is on the left side of a frictionless piston that expands to produce work energy. Beginning with the work energy of expansion and contraction, then contemplating the manners in which other forms of energy could impact this closed system, a checklist is developed for analyzing all the ways that energy can change in the system. This checklist is known as the energy balance, and in this particular case, for a closed system. This system forms the basis for three sample calculations (18min): (1) Adiabatic, reversible expansion from 1000C, 100 bars, and 0.1 L to 0.6L. (2) Isothermal, reversible expansion from 1000C, 100 bars, and 0.1 L to 0.6L. (3) Adiabatic, irreversible expansion from 1000C, 100 bars, and 0.1 L to 0.6L against a perfect vacuum. Calculate the temperature, pressure, work and change in internal energy at the final conditions. The gas can be assumed as pure air. NOTE: Case (1) leads to a very important equation that should be memorized ASAP! Quick answers to common questions (UA, 12min) illustrate easy ideal gas calculations.
Comprehension Questions:
1. Estimate the number of moles in the system.
2. Compute the total work (J) for each case.
3. If all six of the cylinders like Case (1) are firing at the rate of 2500 times per minute, what would be the horsepower of such an engine?