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12.01 - The van der Waals Perspective for Mixtures Click here. 70 4

Mixing Rules (7:23) (msu.edu)

How should energy depend on composition? Should it be linear or non-linear? What does the van der Waals approach tell us about composition dependence? This screencasts shows that the mixing rule for 'a' in a random mixture should be quadratic. A linear mixing rule is usually used for the van der Waals size parameter.

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.

12.03 - Scatchard-Hildebrand Theory Click here. 68.8889 9

Scatchard-Hildebrand Theory (6:53) (msu.edu)

Have you ever heard 'Like dissolves like'? Here we see that numerically. The Scatchard-Hildebrand model builds on the van Laar equation by using pure component information. Scatchard and Hildebrand replaced the energy departure with the experimental energy of vaporization. Because this is related to the 'a' parameter in the van Laar theory, they developed a parameter called the 'solubility parameter', but based it on the energy of vaporization. Interestingly, the model reduces to the one parameter Margules equation when the molar volumes are the same.

Comprehension Questions:

1. Based on the Scatchard-Hildebrand  model, arrange the following mixtures from  most compatible to least compatible.  (a) Pentane+hexane,   (b) decane+decalin,  (c) 1-hexene+dodecanol,   (d) pyridine+methanol,
Most compatible                                                                     Least compatible

 _____                          ______                             ______                          ______

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.

Comprehension Questions:

1. In the picture of molecules given in the presentation on slide 2, what is the numerical value of the local composition x11?
2. In the same picture, what is overall composition x1?
3. What value of Ω21 can you infer from 1 and 2 above and the equations on slide 3?

08.01 - The Departure Function Pathway Click here. 68 5

Departure Function Overview (11:22) (msu.edu)
The philosophy and overall approach for using departure functions.

13.02 - Wilson's Equation Click here. 66.6667 6

Wilson's model concepts (2:44) (msu.edu)

The background on the assumptions and development of Wilson's activity coefficient model.

Comprehension Questions:

1. What value is assumed by Wilson's model for the coordination number (z)?
2. What are the values of Λ21 and Λ12 at infinite temperature, according to Wilson's equation?
3. Solve for x1+x2Λ12 in terms of volume fraction (Φ1) and mole fraction (x1) at infinite temperature.
4. What type of phase behavior is impossible to represent by Wilson's equation?

04.02 The Microscopic View of Entropy Click here. 65 4

Principles of Probability I, General Concepts, Correlated and Conditional Events. (msu.edu, 17min) (Flash)
Comprehension Questions:
1. Estimate the probability of pulling an king from a randomly shuffled deck of 52 cards.
2. A coin is flipped 5 times. Estimate the probability that heads is observed three of the 5 times.
3. A die (singular of dice) is a cube with the numbers 1-6 inscribed on its 6 faces. If you roll the die 7 times, what is the probability that 5 will be observed on all 7 rolls?

04.02 The Microscopic View of Entropy Click here. 65 4

Principles of Probability II, Counting Events, Permutations and Combinations. This part discusses the binomial and multinomial coefficients for putting particles in boxes. The binomial and multinomial coefficient are used in section 4.2 to quantify configurational entropy. (msu.edu, 16min) (Flash) You might like to check out the sample calculations below before attempting the comprehension questions.
Comprehension Questions:
1. Write the formulas for the binomial coefficient, the multinomial coefficient, and the multinomial with repetition.
2. Ten particles are distributed between two boxes. Compute the number of possible ways of achieving 7 particles in Box A and 3 particles in Box B.
3. Note that the binomial distribution is a special case of the multinomial distribution where the number of categories is 2. Also note that the total number of events for a multinomial distribution is given by M^N where M is the number of categories (aka. outcomes, e.g. boxes) and N is the number of objects (aka. trials, e.g. particles). The probability of a particular observation is given by the number of combinations divided by the total number of events. Compute the probability of observing 7 particles in Box A and 3 Particles in Box B.
4. Ten particles are distributed between three boxes. Compute the probability of observing 7 particles in Box A, 3 particles in Box B, and zero particles in Box C.
5. Ten particles are distributed between three boxes. Compute the probability of observing 3 particles in Box A, 3 particles in Box B, and 4 particles in Box C.

11.02 - Calculations with Activity Coefficients Click here. 65 4

This example shows how to incorporate activity calculations into Excel for solutions that follow the Margules 1-parameter (M1) model.(9min, uakron.edu)

You should be able to adapt this procedure along with the procedure for the multicomponent ideal solutions to create a multicomponent M1 model. If you are having trouble, the video for the multicomponent SSCED model illustrates a very similar procedure. You can check your answers by putting in the same component twice. For example, instead of an equimolar binary mixture, input a quaternary mixture with 0.25 moles of methanol, 0.25 methanol (ie. type it as if it was another component), 0.25 of benzene and 0.25 of benzene. If you don't get the same results as for the binary equimolar system, check your calculations.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: Assume the SCVP model (Eq. 2.47).
1. Develop a Pxy diagram for the IPA+water system like Figure 10.8c, guessing values of A12 until you match the maximum pressure (azeotrope). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)
2. Develop a Pxy diagram for the acetone+chloroform system like Figure 10.9c, guessing values of A12 until you match the minimum pressure (azeotrope). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)
3. Develop a Pxy diagram for the acetone+acetic acid system like Figure 10.9a, guessing values of A12 until you match the pressure at x1=0.5 (305mmHg). What value of A12 did you find? (Hint: A12 is not the same as A12*RT.)

01.2 Molecular Nature of Temperature, Pressure, and Energy Click here. 62 10

Molecular Nature of Internal Energy: Configurational Energy. (uakron.edu, 19min) Making the connection between "u" and "U" requires the concept configuring the molecules such that their potentials overlap. Then it is a simple matter (conceptually) to count the number of overlaps that occur and multiply by the energy of the overlap to get the "configurational energy." Adding the configurational energy to the translational (and vibrational) energy (Uig, discussed above), gives the total "U."

Comprehension Questions:

For 1-4, assume 100 molecules are held in a cylinder with solid walls. A piston in the cylinder can move to adjust the density.
1. Suppose the range of the potential (λ) was increased. Would the configurational energy increase, decrease, or stay the same?
2. Suppose the density was decreased. Would the configurational energy increase, decrease, or stay the same?
3. Suppose the temperature was increased at constant density. Would the configurational energy increase, decrease, or stay the same?
4. Suppose the temperature was increased at constant density. Would the configurational energy characterized by (U-Uig)/RT  increase, decrease, or stay the same?
5. 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.)
6. 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?

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