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|07.05 Cubic Equations of State||Click here.||73.3333||3||
Virial and Cubic EOS (11:18) (msu.edu)
1. To what region of pressure is the virial EOS limited at a given temperature? Why?
|01.2 Molecular Nature of Temperature, Pressure, and Energy||Click here.||72.6667||30|
|04.02 The Microscopic View of Entropy||Click here.||72||5||
Principles of Probability III, Distributions, Normalizing, Distribution Functions, Moments, Variance. This screencast extends beyond material covered in the textbook, but may be helpful if you study statistical mechanics in another course. (msu.edu, 15min) (Flash)
|12.03 - Scatchard-Hildebrand Theory||Click here.||72||5||
This video walks you through the process of transforming the M1/MAB model into the Scatchard-Hildebrand model using Excel (6min, uakron.edu) It steps systematically through the modifications to the spreadsheet to obtain each new model. You should implement the M1/MAB model before implementing this procedure.
|01.2 Molecular Nature of Temperature, Pressure, and Energy||Click here.||72||15||
Molecular Nature of Energy, Temperature, and Pressure By Etomica(uakron.edu, 17min). We can use a free website (Etomica.org) to visualize the ways that molecules interact, resulting in the average properties that we see at the macroscopic level. The oversimplified nature of the ideal gas model becomes really obvious and the improvement of the hard sphere model is easily understood. Including both attractive and repulsive forces, as in the square well potential model, leads to more surprising behavior. The two effects may cancel and make the Z factor (Z=PV/RT) look like an ideal gas even though it is not. Also, the adiabatic transformation between potential energy and kinetic energy leads to spikes in temperature as molecules enter each other's attractive wells. In certain cases, you might see molecules get stuck in each others' wells. This is effectively "bonding." This bonding is limited at very low density because it requires a third interaction to occur during the collision in order to stay bonded. This requirement lies at the fundamental basis of what is known as "unimolecular reaction," a fairly advanced concept that is easily understood by watching the video. Note: if the etomica applet causes problems with your browser, check the instructions in section 7.10 to download all the apps and run locally. We use the apps for homework in Chapter 7, so it's money in the bank.
|06.2 Derivative Relations||Click here.||72||5||
Exact Differentials and Partial Derivatives (LearnChemE.com, 5min) This math review puts into context the discussion of exact differentials in Section 6.2 of the textbook using an example related to the volume of a cylinder.
1. Given that dU = TdS - PdV, what derivative relation comes from setting ∂2U/(∂S∂P) = ∂2U/(∂P∂S)?
2. Given that dA = -SdT - PdV, what derivative relation comes from setting ∂2A/(∂T∂V) = ∂2A/(∂V∂T)?
3. Given that dG = -SdT + VdP, what derivative relation comes from setting ∂2G/(∂T∂P) = ∂2G/(∂P∂T)?
|01.2 Molecular Nature of Temperature, Pressure, and Energy||Click here.||72||15||
Intermolecular Potential Energy (msu.edu) (7:11)
The intermolecular potential energy is distinct from the gravitational potential energy of the center of mass. Further, understanding of the potential energy relation with intermolecular force is important.
1. Molecules A and B can be represented by the square-well potential. For molecule A, σ = 0.2 nm and ε = 30e-22 J. For molecule B, σ = 0.35 nm and ε = 20e-22 J. Sketch the potential models for the two molecules on the same pair of axes clearly indicating σ's and ε's of each species. Start your x-axis at zero and scale your drawing properly. Make molecule A a solid line and B a dashed line. Which molecule would you expect to have the higher boiling temperature? (Hint: check out Figure 1.2.)
2. The potential, u(r), represents the work of bringing two molecules together from infinite distance to distance r. So, what is the force law between two molecules according to the Lennard-Jones potential model? Hint: W=∫F*dx
3. Sketch the potential and the force between two molecules vs. dimensionless distance, r/σ, according to the Lennard-Jones potential. Considering the value of r/σ when the force is equal to zero, is it greater, equal, or less than unity?
|12.02 - The van Laar Model||Click here.||70||6||
The van Laar Equation (5:54) (msu.edu)
The van Laar equation uses the random mixing rules discussed in Section 12.1 with the internal energy to approximate the excess Gibbs Energy. What we learn is that it is possible to develop models using fundamental principles. Though this model is not used widely in process simulators, it provides a stepping stone to more advanced models.
|06.1 The Fundamental Property Relation||Click here.||70||2||
From the physical world to the realm of mathematics (uakron.edu, 15min) In Unit I, students develop the skills to infer simplified energy and entropy balances for various physical situations. In order to facilitate that approach for applications involving chemicals other than steam and ideal gases, we need to transform that approach into a realm of pure mathematics. In this context it suffices to apply the energy and entropy balance of a very simple system (piston/cylinder) then focus on the state functions that are involved (U,H,S,...). The mathematical realm is relatively abstract, but it is ideally suited for the generalizations required to extend our principles from steam and ideal gases to any chemical.
1. In example 4.16, we noted that the estimated work to compress steam was less when treated with the steam tables than when treated as an ideal gas. Explain why while referring to the molecular perspective.
2. In Chapter 5, we noted that the temperature drops when dropping the pressure across a valve when treating steam or a refrigerant with thermodynamic tables, but the energy balance suggests that the temperature drop for an ideal gas should be zero. Explain how these two apparently contradictory observations can both be true while referring to the molecular perspective.
3. What is the relation of the state variable dU to the state variables S and V according to the fundamental property relation?
4. What is the relation of the state variable dH to the state variables S and P according to the fundamental property relation?
5. What is the significance of writing changes of state variables in terms of changes in other state variables?
6. Why is the compressibility factor (Z=PV/RT) less than one sometimes?
7. Is it possible for Z to be greater than one? Explain.
8. What is the significance of having a relation for P = P(V,T)? How will that help us to solve problems involving chemicals other than steam and ideal gases?
|13.01 - Local Composition Theory||Click here.||68||10||
Local Composition Concepts (6:51) (msu.edu)
The local composition models of chapter 13 share common features covered in this screencasts. An understanding of these principles will make all the algebra in the models less daunting.
1. In the picture of molecules given in the presentation on slide 2, what is the numerical value of the local composition x11?
Molecular Nature of Internal Energy: Thermal Energy
This introduction to "thermal energy" elaborates on the ideal gas definition of temperature, which derives from the way that PV is related to kinetic energy. This PV relation can be easily understood in terms of an ultrasimplified model of ideal gas pressure. (uakron, 6min). Noting empirically from the ideal gas law that PV=nRT, we are led to the derivation of Eqn. 1.1 (uakron, 5min, same as above). This result suggests counter-intuitive implications about the the ways that solid, liquid, and gas molecular velocities must be related. When applying Eqn. 1.1, you must be careful to keep your units straight, as illustrated in this sample calculation of molecular temperature for Xenon (Mw=131g/mol) (uakron, 5min). On a closely related note, we could perform a sample calculation of molecular pressure for Xenon using Eqn. 1.21.
1. If two phases are in equilibrium (e.g. a vapor with a solid), then their temperatures are equal and the rate at which molecules leave the solid equals the rate at which molecules enter the solid. Which molecules are moving faster, solid or vapor? For simplicity, assume that the vapor is xenon and the solid is xenon. Hint: think about the exchange of momentum when the vapor molecules collide with the solid.
2. Compute the average (root mean square) velocity (m/s) of molecules at room temperature and pressure and compare to their speeds of sound. You can search the internet to find the speed of sound.
3. Three xenon atoms are moving with (x,y,z) velocities in m/s of (300,-450,100), (-100,300,-50), (-200,-150,-50). Estimate the temperature (K) of this fluid.
4. Estimate the pressure of the xenon atoms in Q3 above in a vessel that is 4nm3 in size.