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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.
01.2 Molecular Nature of Temperature, Pressure, and Energy Click here. 63.3333 12

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?
10.03 - Binary VLE using Raoult's Law Click here. 60 2

Raoult's Law (5:39) (msu.edu)
What type of components make an ideal solution that follows Raoult's Law? What does a diagram look like for a system that follows Raoult's Law? Can you identify the regions? What is the K-ratio for Raoult's Law? What simple principles must be followed for the K-ratios of the components in a binary mixture?
08.08 - Reference States Click here. 60 2

Thermodynamic pathways of EOS's for arbitrary reference states (uakron.edu, 20min) The development of a thermodynamic pathway from an arbitrary reference state to a given state condition is independent of the thermodynamic model. It depends only on (1a) identifying the condition of the reference state (e.g. ideal gas, real vapor, or liquid) (1b) transforming from the reference state to the ideal gas, if necessary (2) transforming from the ideal gas at the condition of the reference state to the ideal gas at the given state condition (3a) identifying the condition at the given state (3b) transforming from the ideal gas at the given state to the real fluid at the given state. The methodology is illustrated for two thermodynamic models: the Psat/Hvap model of Figure 2.6c,Eqs 2.45,47 vs. the PR EOS. The screencast is a bit long, but it covers 16 sample calculations (8 for H and 8 for S) and comparisons between PREOS vs Psat/Hvap. You might like to refer back to Sections 2.10 and 3.6 to review the Psat/Hvap model and the elemental reference state. Push pause before each sample calculation and check whether you can predict the next answer.

Comprehension Questions:

1. Compute "H" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cpig/R = 8.85 and the PR EOS. You may use PREOS.xlsx to compute H-Hig, but you must show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
2. Compute "S" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cpig/R = 8.85 and the PR EOS. You may use PREOS.xlsx to compute S-Sig, but you must show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
3. Compute "H" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cpig/R = 8.85 and the Psat/Hvap model. Show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
4. Compute "S" by hand for propane at 80C and 3 MPa relative to a reference at 230K and 1bar, assuming Cpig/R = 8.85 and the Psat/Hvap model. Show your hand calculations for each step (1a-3b). Compare your answer to the result tabulated in PREOS.xlsx.
09.10 - Saturation Conditions from an Equation of State Click here. 60 2

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?
09.10 - Saturation Conditions from an Equation of State Click here. 60 1

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?)
14.04 LLE Using Activities Click here. 60 2

Txy Phase Diagram Showing LLE and VLE Simultaneously (9min,uakron.edu)

The binary Txy phase diagram of methanol+benzene is visualized with sample calculations of the SSCED model with several values of the nonideality (kij) parameter. The calculations show the liquid-liquid equilibrium (LLE) phase boundary as well as the vapor-liquid equilibrium (VLE) boundary. As the estimated nonideality (kij) increases, the LLE boundary crashes into the VLE. It is so exciting that it makes a thermo nerd wax poetic about the "valley of Gibbs."

Comprehension Questions:

1. The LLE phase boundary moves up as the nonideality increases. Which way does the VLE contribution move? Explain how this relates to the molecules' escaping tendencies.
2. How would this phase diagram change if the pressure was increased to, say, 10 bars?
3. What value of kij is required to make the LLE binodal barely touch the VLE at 1 bar?
4. What value of kij is required to make the LLE binodal barely touch the VLE at 10 bars?
03.6 - Energy Balance for Reacting Systems Click here. 60 1

Heat Removal from a Chemical Reactor (uakron, 8min) determines heat removal so that a chemical reactor is isothermal following the pathway of Figure 3.5b using the pathway of Figure 2.6c if a heat of vaporization is involved. The reaction is: N2 + 3H2 = 2NH3 at 350C and 1 bar. The pathway to go from products to the reference condition is to correct for any liquid formation at the conditions of the product stream then cool/heat the products to 25C (the reference temperature), then "unreact" them back to their elements of formation. Summing up the enthalpy changes over these steps gives the overall enthalpy of the reactor outlet stream. The same procedure applied to the reactor inlet gives the overall enthalpy of reactor inlet stream. Then the heat duty of the reactor is simply the difference between the two stream enthalpies.

Comprehension Questions:
1. Use this approach to compute the heat of reaction for 2 CH3OH = CH3OCH3 + H2O at 250C and 1 bar. Compare to your answer when using the pathway of Figure 3.5a. 
2. Methanol is a liquid at 45C and 2bars. Compute the enthalpy of a stream that is 100 mol/h of pure methanol at 45 C and 2 bars according to the method of Figure 3.5b. Hint: this is different from the pathway of Figure 2.6c because it includes the heat of formation.
07.08 Matching The Critical Point Click here. 60 2

Visualizing the vdW EOS (uakron.edu, 16min) Building on solving for density, describes plotting dimensionless isotherms of the vdW EOS for methane at 5 temperatures, two subcritical, two supercritical, and one at the critical condition. From these isotherms in dimensionless form, it is possible to identify the critical point as the location of the inflection point where the temperature first exits the 3-root region. This method can be adapted to any equation of state, whether it is cubic or not. The illustration was adapted from a sample test problem. This screencast also addresses the meaning of the region where the pressure goes negative, with a (possibly disturbing) story about a blood-sucking octopus.

Comprehension Questions:

1. What are the dimensions of the quantity (bP/RT)?
2. Starting with the expression for Z(ρ,T), rewrite the vdW EOS to solve for the quantity (bP/RT) in terms of (bρ) and (a/bRT).
3. Consider the following EOS: Z = 1 + 2bρ/(1-2bρ) - (a/bRT) /(1-bρ)2. Estimate the value of bPc/(RTc) for this EOS.
4. Consider the following EOS: Z = 1 + 2bρ/(1-2bρ) - (a/bRT) /(1-bρ)2. Estimate the value of (a/bRTc) for this EOS.
5. Compute the values of a(J-cm3/mol2) and b(cm3/mol) for methane according to this new EOS.

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