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11.05 - Modified Raoult's Law and Excess Gibbs Energy Click here. 20 1

Extending the M1 derivation of the activity coefficient to multicomponent mixtures  (uakron.edu, 14min) is straightforward but requires careful attention to the meaning of the subscripts and notation. It is a good warmup for derivations of more sophisticated activity models. This presentation begins with a brief review of the M1 model and its relation to the Gibbs excess function, then systematically explains the notation as it extends from the binary case to multiple components.

Comprehension Questions
1. Derive the activity coefficient for the multicomponent M2 model.
2. Derive the activity coefficient for the multicomponent Redlich-Kister model.

08.07 - Implementation of Departure Functions Click here. 20 3

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.

08.07 - Implementation of Departure Functions Click here. 20 2

Helmholtz Example - vdW EOS (uakron.edu, 18min) This video begins with a brief review of the connection of the Helmholtz departure with all other departures then shows four sample derivations assuming that Z is given by the vdW EOS: (1) the Helmholtz departure , (2) the internal energy departure from the Helmholtz departure. (3) the Helmholtz energy from the internal energy (4) the Z factor from the Helmholtz departure. The procedures illustrated here can be applied to any EOS starting with any part (U, A, or Z) as given to derive any other departure: ZUHAGS.
Comprehension Questions: The virial EOS for SW fluids can be written as: Z = 1 + Bρ/RT where B = 4b+[4b(λ^3-1)] [exp(βε)-1], b = πNAσ^3/6.
1. Derive an expression for the Helmholtz departure.
2. Use the result of #1 to derive the internal energy departure.
3. Use the result of #2 to derive the Helmholtz departure. What is the integration constant in this case?

08.07 - Implementation of Departure Functions Click here. 20 4

Helmholtz Example - Modified vdW EOS (uakron.edu, 13min) A sample derivation of the Helmholtz departure implicit in the Gibbs departure given Z = 1 + abρ/(1+)^3 - (9.5aρ/RT)/(1+aρ/RT). Note that the limits of integration matter for this EOS. The audio is inferior for this live video, but it responds to typical questions and confusion from students in the audience. Some students might find it helpful to hear the kinds of questions that students ask. The responses slow the derivation down so that no steps are skipped and key steps are reiterated multiple times. Just turn the volume up!
Comprehension questions:
1. Which part of this EOS is non-zero at the zero density limit of integration?
2. Is there a sign error on one of the terms in this video? Check the derivation independently.
3. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-)2 - (9.5aρ/RT)/{1-a/bRT[1-4bρ+4(bρ)2]}.
4. Derive the Helmholtz departure given Z = 1 + 4bρ/(1-2) - (9.5aρ/RT){1+4/bRT[1-2(bρ)2]}/{1-a/bRT[1-4bρ+4(bρ)2]})/{1-a/bRT[1-4bρ+4(bρ)2]}

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