Phase equilibrium in a pure fluid (uakron, 11min) can be contemplated in terms of the following question: Suppose propane exists at a set temperature in an uninsulated piston/cylinder with half the volume as vapor and half as liquid. What is the final pressure when the piston is pressed down. A proper thermodynamic answer leads to the consideration of the Gibbs energy, with implications that open up an entire new world of problems to be solved related to equilibrium partitioning for pure fluids and mixtures.

Comprehension Questions:

1. Write dG for the total piston/cylinder system in terms of the individual phases.
2. What is the criterion for equilibrium in a pure fluid?
3. What is the stable state (L,V,L=V) when G^L > G^V ?
4. For the vdW fluid at 62C, 0.35 MPa, the following roots were obtained: Z^L = 0.02598,
Z^V = 0.92718, A=0.08608, B=0.01820. What is the stable state (L,V,L=V)?
5. For the vdW fluid at 62C, 0.25 MPa, the following roots were obtained: Z^L = 0.01859,
Z^V = 0.94910, A=0.061487, B=0.013000. What is the stable state (L,V,L=V)?

Hint: (G-G^ig)/RT = -ln(Z-B)-A/Z + Z - 1 - ln(Z) where A=a*P/(R²T²); B=bP/RT; b=0.125*RT_c/P_c

Estimating the Heat of Vaporization from Antoine's Equation (3min, learncheme.com) The Clausius-Clapeyron equation shows that the heat of vaporization is slope of the vapor pressure curve. So you just need to differentiate the Antoine equation to estimate the heat of vaporization. This sample calculation shows how to compute the heat of vaporization of benzene given Antoine coefficients.

Comprehension Questions:

1. Estimate the vapor pressure (torr) of benzene at 55C using the equation given in the screencast.
2. Estimate (or report) the heat of vaporization (J/mol) of benzene according to the screencast.
3. Estimate the vapor pressure (torr) of benzene at 55C using the SCVP equation (2.47).
4. Estimate the heat of vaporization (J/mol) of benzene using the SCVP equation.
5. Estimate the heat of vaporization (J/mol) of benzene using Eq. 2.45.

Shortcut estimation of thermodynamic properties (sample calculation) can be very quick and sometimes reasonably accurate.(6min, uakron.edu) As a follow-up exercise, it is suggested to adapt the shortcut vapor pressure equation in combination with Eqn. 2.45 and the pathway of Fig. 2.6c to rapidly estimate stream properties. Briefly, all you need is an "IF" statement that checks whether the T is less than T^sat at the given P. If so, then H=H^ref+CpΔT+H_vap. If not, then H=H^ref+CpΔT. This can be a quick and convenient method to estimate stream attributes of a process flow diagram. One equation per cell and you're done. This sample calculation illustrates the process for the heat duty of a butane vaporizer and compares the PREOS to the methods of Chapter 2 (ie. Eq. 2.45 etc.)

Comprehension Questions: Suppose you want to tabulate the entropy (S) of your stream attributes by this approach.
1. How would you compute the S^ig(T,P)-S^ig(T^ref,P^ref) contribution?
2. How would you compute ΔS_vap?
3. Compute "S" for propane at 355K and 3MPa relative to the liquid at 230K and 0.1MPa by this approach.
4. Compute "H" for propane at 355K and 3MPa relative to the liquid at 230K and 0.1MPa by this approach.
5. Compute H and S for the same conditions/reference using the PREOS.
6. Explain the discrepancies between the two approaches. e.g. compare the H_vap values and the (H^V-H^ig) values, where H^V represents the enthalpy of the vapor phase, not the heat of vaporization (H_vap).

Gibbs Energy - Nuts to Soup. (learncheme.com, 8min) It is straightforward to start from the definition of Gibbs Energy and derive all the changes in Gibbs energy. These can be graphed for H2O to see how familiar quantities from the steam tables relate to changes in this unfamiliar property.

What is fugacity? (10min) (learncheme.com) Defines fugacity in terms of Gibbs Energy and describes the need for defining this new property as a generalization of how pressure affects ideal gases.
Comprehension Questions
1. The phases in this video start with concentrations 0.0007kg/L and 1.0 kg/L, when not at equilibrium. What are the equilibrium concentrations?
2. Why is concentration an unreliable indicator for the direction of mass transfer?
3. Name two indicators for the direction of mass transfer that are superior to concentration.  

In a contest for "the most hated word in Chemical Engineering," fugacity won by a landslide. This video (15min, uakron.edu) reviews how the term was developed and why it's not really as bad as all that. In fact, it's a nice word that sets the stage for all of phase and reaction equilibrium with a straightforward extension of the same conceptual basis to mixtures. On second thought, perhaps the power of that conceptual basis and all that it implies is what really intimidates new students. Many perspectives have been offered to help overcome the frustration that students feel toward fugacity. If you like a comic book perspective, even that is available.

Comprehension Questions:

1.What is the fugacity of a vapor phase component in a mixture according to Raoult's law?
2.What is the fugacity of a liquid phase component in a mixture according to Raoult's law?
3. What word is modern usage is closely related to the latin root "fuga-"?
4. Water is in VLE at 0.7 bars in a fixed volume vessel. Five cm³ of air are injected into the vessel and the temperature is allowed to return to its original value. Does the water in the vapor phase increase, decrease, or remain the same? (Learncheme.com, 2min) (Hint: you may assume that air does not dissolve in the liquid water and the pressure is sufficiently low that the vapor can be assumed to behave as an ideal gas.)

When liquid is added to an evacuated tank of fixed volume, equilibrium is established between the vapor and liquid. (3min,learncheme.com) The fugacity criterion characterizes this equilibrium as occurring when the escaping tendency from each phase is equal.

We occasionally require the fugacity in the vapor phase by an EOS other than the PR EOS. (learncheme, 3min) This becomes especially common in Unit 3 when we extend our methods to mixtures. Another skill demonstrated in this screencast is a sample derivation using the pressure dependent formulas. Note that there is a typo in the initial problem statement. The equation of state should be: PV = (1-0.05 P)RT.

Comprehension Questions:

1. Rearrange the given EOS to solve for Z and apply Eq. 9.23 to solve for the change in fugacity. Compare your answer to that given in the screencast. Which method seems easier to you?
2. Use Eq. 7.5 with Eq. 9.23 to derive an expression for the fugacity.
3. Apply the result of #2 to evaluate the fugacity of n-pentane at 398 K and 1 MPa.
4. Does this condition for pentane satisfy Eq. 7.10? Explain.

Liquid fugacity relative to vapor fugacity. (LearnChemE, 5 min) This screencast shows a sample derivation and sample calculation for the vapor equation of state given by: Z = 1-0.01P, solve for: (a) the vapor fugacity at 500K and 30 bar (b) the liquid fugacity in equilibrium with the same vapor at 500K and 30bar (c) the liquid fugacity at 500K and 60 bar. Data: V^L = 25 cm³/mol.

Comprehension Questions:

1. How much did raising the pressure to 60 bar change the liquid fugacity (bars) (+/- 1%)?
2. Estimate the fugacity (bars) of the vapor at 500 K and 60 bar and compare it to the liquid. Which is smaller? Which state do you think best characterizes the fluid (ie. V or L)?
3. Estimate the fugacity (bars) of n-pentane vapor at 30 bar and 460 K by Eqn. 7.5.
4. Assuming V^L=229cm³/mol, estimate the fugacity of liquid n-pentane at 460K and 600bar.
5. Compare your answers for 3 and 4 to the PREOS.

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 log₁₀P^sat(bars) = 3.80675 - 577.661/(T(K)-13.0), Cp^V=22.7 J/mol-K, Cp^L=44.4 J/mol-K, ρ^L=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 Cp^L=Cp^S. 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 V^S=V^L/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 H_fus at the triple point.

Solving for the saturation pressure using PREOS.xls simply involves setting the temperature and guessing pressure until the fugacities in vapor and liquid are equal. (5min, learncheme.com) It is not shown, but it would also be easy to set the pressure and guess temperature until the fugacities were equal in order to solve for saturation temperature. One added suggestion would be to type in the shortcut vapor pressure (SCVP) equation to give an initial estimate of the pressure. Rearranging the SCVP can also give an initial guess for Tsat when given P. This presentation illustrates a sample calculation for toluene to explore when the vapor is the stable, when the liquid is the stable phase, and when the phases are roughly in equilibrium.

Comprehension Questions:

1. Estimate the vapor pressure (MPa) of n-pentane at 450K according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
2. Estimate the saturation temperature (K) of n-pentane at 3.3 MPa according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
3. Estimate the vapor pressure (MPa) of n-pentane at 223K according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?
4. Estimate the saturation temperature (K) of n-pentane at 3.3 kPa according to the PREOS. Compare your result to the value from Eq. 2.47 (SCVP) and to the Antoine equation using the coefficients given in Appendix E. What do you think explains the observations that you make?

We can combine the definition of fugacity in terms of the Gibbs Energy Departure Function with the procedure of visualizing an equation of state to visualize the fugacity as characterized by the PR EOS. (21min, uakron.edu) This amounts to plotting Z vs. density, similar to visualizing the vdW EOS. Then we simply type in the departure function formula. Since the PR EOS describes both vapors and liquids, we can calculate fugacity for both gases and liquids. Taking the reciprocal of the dimensionless density ( V/b=1/(bρ) ) gives a dimensionless volume. When the dimensionless pressure (bP/RT) is plotted vs. the dimensionless volume, the equal area rule indicates the pressure where equilibrium occurs and this can be checked by comparing the ln(f/P) values for the liquid and vapor roots. When the pressure is not exactly saturated, we may still be in the 3-root region. Then you need to check the fugacity to determine which phase is stable.

Concept Questions:

1. What equation can we use to estimate the fugacity of a compressed liquid relative to its saturation value?
2. How accurate is that equation relative to the change in pressure when we are close to saturation?
3. The video shows a graph of ln(f/P) vs. P. Which phase gives the lower value of fugacity when you are to the right of the intersection point? (ie. vapor or liquid?)