Using the NIST Webbook for Charts/Tables (uakron.edu, 14min) Shows how to access the NIST fluid properties as needed to design an OVC cycle. Demonstrates the procedure with a problem based on propane at -100F saturated vapor pressure being raised to 10 bars and 180F in an adiabatic compressor by solving for the compressor efficiency and the COP.

Comprehension Questions:

1. Chlorodifluoromethane is used as the working fluid of an OVC cycle at -100F saturated vapor pressure exiting the evaporator and 80F saturated liquid exiting the condenser. Assuming an adiabatic reversible compressor solve for the COP.

Experimental observation of the critical point (LearnChemE.com, 5min) Discusses the background of the critical point and its relation to the 2-phase envelope. Includes a video showing the transition of a 2-phase fluid as it is heated through the critical temperature, then cooled back again.

Comprehension Questions:

1. Based on watching the video, what is different about the behavior of the fluid when it is cooled through the critical point as opposed to being heated from subcritical to supercritical?

Principles of Corresponding States (10:02) (msu.edu)
An overview of use of Tc and Pc and acentric factor to create corresponding states correlation. The relation between acentric factor and deviations from spherical fluids is highlighted.

Comprehension Questions:

1. What is the value of the reduced vapor pressure for Krypton at a reduced temperature of 0.7? How does this help us to characterize the vapor pressure curve?

2. Sketch the graph of vapor pressure vs. temperature as presented in this screencast for the compounds: Krypton and Ethanol. Be sure to label your axes completely and accurately. Draw a vertical line to indicate the condition that defines the acentric factor.

Intro to the vdW EOS. (LearnCheme.com, 5min) Provides a brief overview of the van der Waals (vdW) 1873 equation of state (EOS), which served as a prototype for EOS development for over 100 years. Note: the vdW EOS is just one conjecture of how equations of state for real fluids may be formulated. In reality, each fluid has its own unique EOS. The vdW model conjectures that the pressure is altered relative to the ideal gas by the presence of attractive forces and repulsive forces.

Comprehension Questions:

1. Of the two parameters a and b, which is related to attractive forces and which is related to attractive forces?
2. How are the parameters a and b typically characterized/computed? ie. To what experimental constants are they related in order to compute them?
3. Is the vdW EOS an example of a 2-parameter EOS or 3-parameter EOS?
4. When writing the term (V-b) we subtract b because the molecules occupy volume and when V=b, all the "free volume" is gone. Can you explain the term (P+a/V²) in a similar manner?
5. In the presented example of CO2 at 0.2L and 269K, how does the pressure compare when computed by the ideal gas law vs. the vdW model? (Give both values.)
6. In the presented example of CO2 at 0.0L and 269K, how does the pressure compare when computed by the ideal gas law vs. the vdW model? (Give both values.)

Virial and Cubic EOS (11:18) (msu.edu)
Discusses the strategy of the virial EOS and the cubic EOS and the strategy used to solve as a cubic in Z. Gives formulas for calculating the a and b parameters of both the vdW and Peng-Robinson EOS's, as well as the virial EOS. You might want to watch the video on "Visualizing the vdW EOS" if you want to understand where the equations for a and b come from or how to make quantitative plots of isotherms.

Comprehension Questions:

1. To what region of pressure is the virial EOS limited at a given temperature? Why?
2. Is the Pitzer EOS limited to the same conditions as the virial EOS?
3. Is the virial EOS a 2-parameter or 3-parameter EOS?
4. Is the Peng-Robinson (PR) EOS a 2-parameter or 3-parameter EOS?
5. What is the primary shortcoming of the vdW EOS, as described on slide 4 of this presentation?
6. Is the PR EOS limited to the same conditions as the virial EOS? Explain.
7. How does the "fugacity" help you to identify the stable root of a cubic EOS?
8. When there are 3 real roots to a cubic EOS, what do we do with the center root? Why?

3. Using Preos.xlsx and Interpreting Output (11:38) (msu.edu)
This screencast includes discussion of what we mean by the casual terminology 'three root region' and 'one root region', and how to interpret screen output. Also, the screencast spends time dicussing selection of stable roots using fugacity.

Comprehension Questions:

1. Is it possible to have a 1-root region below the critical temperature?

2. Is it possible to have a 3-root region above the critical temperature?

3. How does fugacity help us to identify the proper root to select?

4. Would argon at 5 MPa be in the 1-root or 3-root region?

1. Peng-Robinson PVT Properties - Excel (3:30) (msu.edu)

Introduction to PVT calculations using the Peng-Robinson workbook Preos.xlsx. Includes hints on changing the fluid and determining stable roots.

Comprehension Questions:

1. At 180K, what value of pressure gives you the minimum value for Z of methane? Hint: don't call solver.

2. At 30 bar, what value of pressure gives Z=0.95 for methane?

3. Compute the molar volume(s) (cm³/mol) for argon at 100K for each of the following?
(a) 3.000 bars (b) 4.000 bars (c) 3.26903 bars.

5. Peng Robinson Using Solver for PVT and Vapor Pressure - Excel (4:42) (msu.edu)

Describes use of the Goal Seek and Solver tools for Peng-Robinson PVT properties and vapor pressure.

Comprehension Questions:

1. Which of the following represents the vapor pressure for argon at 100K?
(a) 3.000 bars (b) 4.000 bars (c) 3.26903 bars.

4. Selecting Stable Roots (1:11) (msu.edu)

Selecting stable roots is often one of the confusing aspects in working with cubic equations of state. This screencasts gives a visual picture of how the roots and stability are related to the vapor pressure and EOS humps at subcritical temperatures.

Using a macro to create an isotherm (Excel) (msu.edu, 14:31) The tabular Excel display is convenient for viewing all the intermediate values, but no so good for building a table such as for an isotherm. This screencast shows how to write/edit a macro to build a table by copying/pasting values. The screencast creates an isotherm on a Z vs. Pr plot over 0.01 < Pr < 10.

Derivative Relations for the Peng-Robinson EOS (3:18) (msu.edu)
The derivatives (∂U/∂V)_T and (∂C_V/∂V)_T are evaluated for the Peng-Robinson EOS and the concept of expressing non-measurable properties in terms of measurable properties is discussed.

Comprehension Questions:

1. What derivative do we need to use when developing formulas for departure functions in Chapter 8?

Nature of Molecular Interactions - Macro To Nano(8min). (uakron.edu) Instead of matching the critical point, we can use experimental data for density vs. temperature from NIST as a means of characterizing the attractive energy and repulsive volume. A plot of compressibility factor vs. reciprocal temperature exhibits fairly linear behavior in the liquid region. Matching the slope and intercept of this line characterizes two parameters. This characterization may be even more useful than using the critical point if you are more interested in liquid densities than the critical point. In a similar manner, you could derive an EOS based on square-well (SW) simulations and use the SW EOS to match the NIST data(11min), as shown in this sample calculation of the ε and σ values for the SW potential. In this lesson, we learn how to characterize the forces between individual atoms, which may seem quite unreal or impractical when you first encounter it. On the other hand, "nanotechnology" is a scientific discipline that explores how the manipulation of nanostructure is now quite real with very significant practical implications. "The world's smallest movie" shows dancing molecules, (IBM, 2min) demonstrating the reality of molecular manipulation, and the accompanying text explains some of the practical implications. Along similar lines, researchers at LLNL and CalTech have developed 3D printers that can display "voxels" (the 3D analog of pixels) of ~1nm3. That's around 10-100 atoms per voxel. Since 2013-14, chemical/materials engineers have been building nanostructures (TEDX, 13min) in the same way that civil engineers build infrastructure.
Comprehension Questions:
1. What does the y-intercept represent in a plot of compressibility factor vs. reciprocal temperature?
2. What parameter does the y-intercept help to characterize, b or ε?
3. What does the x-intercept represent in a plot of compressibility factor vs. reciprocal temperature?
4. What parameter does the x-intercept help to characterize, b or ε?
5. Apply the SW EOS given in the second video to the isochore at 16.1 mol/L. Do you get the same values for ε/k and σ? Explain.

Simple Hard Disk Collisions. (5min) (uakron.edu) Deriving the formula for pressure from the motions of molecules was easy when we were talking about ideal gas molecules and we even got a compact, exact result (aka. the ideal gas law). The problem gets more complicated when the molecules can collide with each other as well as with the walls. This complexity undermines our ability to get an exact solution, but we can obtain a numerical solution by integrating all the collisions with respect to time and computing the average pressure as a result. The process begins with computing collision times of molecules with walls. This computation is simple enough that you should be able to do it even if you can't write a molecular simulation program.

Comprehension questions:

1. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (643, -133) and (133, -643) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule.
2. Two molecules with MW=16 and 0.36nm diameter are traveling in 2D with velocities: (133, -266) and (-133, 266) m/s. Their initial positions in a 5 nm box are (2,4) and (3,1). Estimate the time (ns) and nature of the first collision, whether with a wall or with another molecule.
3. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How far must the molecule travel before it contacts the wall?
4. A molecule with MW=16 and 0.36nm diameter is traveling in 2D towards the east wall of a box. Its velocity is given by (vx, vy) = (123, 345) m/s. It starts off being 2nm from the east wall. How much time(ns) before it contacts the wall?

Nature of Molecular Parking Lots - RDFs(20min, uakron.edu) Molecules occupy space and they move around until they find their equilibrium pressure at a given density and temperature. Cars in a parking lot behave in a similar fashion except the parking lot is in 2D vs. 3D. Despite this exception, we can understand a lot about molecular distributions by thinking about how repulsive and attractive forces affect car parking. For example, one important consideration is that you should not expect to see two cars parked in the same space at the same time! That's entirely analogous for molecular parking. Simple ideas like this lead to an intuitive understanding of the number of molecules distributed at each distance around a central molecule. From there, it is straightforward to multiply the energy at a given distance (ie. u(r) ) by the number of molecules at that distance (aka. g(r) ), and integrate to obtain the total energy. A similar integral over intermolecular forces leads to the pressure. And, voila! we have a new conceptual route to developing engineering equations of state.
Comprehension questions:
1. Sketch u(r)/epsilon and g(r) vs. r/sigma for square well spheres at a very low density. Use a solid line for g(r) and a dashed line for u(r)/epsilon.
2. Sketch u(r)/epsilon and g(r) vs. r/sigma for hard spheres at a high density. Use a solid line for g(r) and a dashed line for u(r)/epsilon.
3. Sketch u(r)/epsilon and g(r) vs. r/sigma for square well spheres at a high density. Use a solid line for g(r) and a dashed line for u(r)/epsilon.

Nature of Molecular Energy - Example Calculation(8min, uakron.edu) Given an estimate for the radial distribution function (RDF) integrate to obtain an estimate of the internal energy. The result provides an alternative to the attractive term of the vdW EOS.

Nature of Molecular Pressure - Example Calculation(11min, uakron.edu) Similar to integrating intermolecular energy to obtain the macroscopic internal energy, integrating the intermolecular force per unit area leads to the macroscopic force per unit area (aka. pressure).