Modified Raoult's Law and Excess Gibbs Energy (6:27) (msu.edu)

What are 'postive deviations' and 'negative deviations'? What are the 'rules of the game' for working with deviations from Raoult's law?

This screencast show the three main stages of modeling deviations from Raoult's law: 1) obtaining the activity coefficient from experiment; 2) fitting the activity coefficient to an excess Gibbs energy model; 3) using the fitted model to perform bubble, dew, flash calculations. These three stages are often jumbled up when first learning about activity coefficients, so explicit explanation of the strategy may be helpful.

Fitting One-Parameter Margules Equation (4:01) (msu.edu)

This screencast show application of the Stage I and Stage II calculations using experimental data and the one-parameter Margules equation. It is helpful to follow this screencast with the application of Stage III calculations described in the screencasts for Section 11.2.

Dew Pressure (7:41) (msu.edu)

The culmination of the activity coefficient method is application of the fitted activity coefficients to extrapolate from limited experiments in a Stage III calculation. The recommended order of study is 1) Bubble Pressure; 2) Bubble Temperature; 3) Dew Pressure; 4) Dew Temperature. Note that an entire Pxy diagram can be generated with bubble pressure calculations; no dew calculations are required. However, many applications require dew calculations, so they cannot be avoided. The dew calculations are more complicated than bubble calculations, because the liquid activity coefficients are not known until the unknown liquid mole fractions are found. This screencast describes the procedure and how to implement the method in Matlab or Excel.

Activity Coefficient Calculations in Matlab (6:12) (msu.edu)

An overview of the strategy of placing the activity coefficient models in a single folder, how the gammaModels .m files are used with scalars and vectors, and how to use the Matlab 'addpath' command to run the code from any folder on your computer.

Bubble Pressure (5:25) (msu.edu)

The culmination of the activity coefficient method is application of the fitted activity coefficients to extrapolate from limited experiments in a Stage III calculation. As the easiest routine to apply, the bubble pressure method should be studied first. The recommended order of study is 1) Bubble Pressure; 2) Bubble Temperature; 3) Dew Pressure; 4) Dew Temperature. Note that an entire Pxy diagram can be generated with bubble pressure calculations; no dew calculations are required.

Dew Temperature (7:57) (msu.edu)

The culmination of the activity coefficient method is application of the fitted activity coefficients to extrapolate from limited experiments in a Stage III calculation. The recommended order of study is 1) Bubble Pressure; 2) Bubble Temperature; 3) Dew Pressure; 4) Dew Temperature. Note that an entire Txy diagram can be generated with bubble temperature calculations; no dew calculations are required. However, many applications require dew calculations, so they cannot be avoided. The dew calculations are more complicated than bubble calculations, because the liquid activity coefficients are not known until the unknown liquid mole fractions are found. This screencast describes the procedure and how to implement the method in Matlab or Excel.

Bubble Temperature (2:43) (msu.edu)

The culmination of the activity coefficient method is application of the fitted activity coefficients to extrapolate from limited experiments in a Stage III calculation. The bubble temperature is the easiest after bubble pressure. The recommended order of study is 1) Bubble Pressure; 2) Bubble Temperature; 3) Dew Pressure; 4) Dew Temperature. Note that an entire Txy diagram can be generated with bubble temperature calculations; no dew calculations are required.

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.

M1/MAB Extension of the Multicomponent Flash Spreadsheet (19min, uakron.edu) adapted from Ideal Solutions (cf Section 10.4)

Binary VLE Flash Calculations Using the Lever Rule (uakron.edu, 6min)

When you want to perform flash calculations with one activity model and many components, you should use the methods of Section 10. 4 or Section 12.7. When you want to perform flash calculations with two components and many activity models, this video shows the best method. Starting with a Txy or Pxy binary phase diagram, the procedure of Section 10.1 is easily adapted. Since binary Pxy and Txy diagrams are the first thing you do for any activity model, you can simply apply this procedure any time for any activity model. This example shows how to interpret the phase diagram for 2-propanol+water at 30C, similar to Figure 11.5.

Comprehension Questions: Use the SCVP model of vapor pressures and the M2 activity model for the following. (Hint: you might want to watch the videos below before answering these.)
1. Sketch the Pxy diagram for methanol+benzene at 373K with A12=1.5 and A21=2.0.
At P = 2683mmHg and zM = 0.4, the mole fraction methanol in the liquid phase is closest to:
(a) 0.25 (b) 0.33 (c) 0.66 (d) 0.75.
2. Sketch the Pxy diagram for methanol+benzene at 373K with A12=1.5 and A21=2.0.
At P = 2683mmHg and zM = 0.4, the molar amount of the liquid phase is closest to:
(a) 0.25 (b) 0.33 (c) 0.66 (d) 0.75.
3. Sketch the Txy diagram for mtbe+ethanol at 760mmHg with A12=1.5 and A21=1.2.
At T = 333K and zM = 0.3, the mole fraction mtbe in the liquid phase is closest to:
(a) 0.25 (b) 0.33 (c) 0.66 (d) 0.75.
4. Sketch the Txy diagram for mtbe+ethanol at 760mmHg with A12=1.5 and A21=1.2.
At T = 333K and zM = 0.3, the molar amount of the liquid phase is closest to:
(a) 0.25 (b) 0.33 (c) 0.66 (d) 0.75.

Fitting the M2 model parameters using Excel. (uakron.edu, 6min) You can type Eqns. 11.2 and 11.38 directly into Excel then it is a simple matter to compute the A12 and A21 values from a single data point and compute the γ's, P's, and y's at all compositions assuming constant values for A12 and A21. This generates a phase diagram in short order. The procedure is illustrated in this presentation for the 2-propanol+water system at 30C, similar to Examples 11.1 and 11.5 in the textbook. It would be a simple matter to adapt this spreadsheet to fit the experimental data in Example 11.8 by computing the deviations at each composition. With this tool readily available, you should be able to apply the M2 model to any binary mixture in short order.

Comprehension Questions: You may assume the SCVP model for purposes of the calculations below (but you should use more accurate vapor pressure estimates for more professional purposes).

1. At 760 mm Hg the system acetone(1)+hexane(2) exhibits an azeotrope at 68 mole percent acetone with a boiling point of 49.8°C. 
a. Estimate the A12 and A21 parameters.
b. Estimate the bubble point pressure and vapor composition of 10 mole percent acetone at this temperature.
2. Acetonitrile+water forms an atmospheric pressure azeotrope at 70 mole% acetonitrile, 76°C.
a. Estimate the A12 and A21 parameters.
b. Estimate the bubble point pressure and vapor composition of 80 mole percent acetonitrile at this temperature.

The extension to the multicomponent M2 flash spreadsheet (uakron.edu, 10min) adapts the multicomponent M1 spreadsheet by recognizing that summing rows of a matrix times mole fractions involves a simple matrix multiplication. (Matrix operations involve highlighting the cells of interest, typing the MMULT function, and hitting ctrl+shift+enter.) The column multiplication simply applies the sumproduct function. In this way, we just need to insert one more column relative to the multicomponent M1 spreadsheet, then change the expression for gi, and we are done.

Comprehension Questions:
1. Find the bubble and dew temperatures of an equimolar mixture of 2-propanol, water, and methanol at 2 bars using the M2 model. Then compute V/F x and y at the temperature that is halfway between dew and bubble. Is it what you expected?
2. Find the bubble and dew temperatures of an equimolar mixture of 2-propanol, water, and methanol at 3 bars using the M2 model. Then compute V/F x and y at the temperature that is halfway between dew and bubble. Is it what you expected?
3. Find the bubble and dew temperatures of an equimolar mixture of 2-propanol, water, and methanol at 4 bars using the M2 model. Then compute V/F x and y at the temperature that is halfway between dew and bubble. Is it what you expected?
4. Find the bubble and dew temperatures of an equimolar mixture of 2-propanol, water, and methanol at 5 bars using the M2 model. Then compute V/F x and y at the temperature that is halfway between dew and bubble. Is it what you expected?

Fitting the M2 model parameters using Excel. (uakron.edu, 6min)  Computing the A12 and A21 values from a azeotropic data is just like fitting at a single data point. The procedure is illustrated in this presentation for the benzene+ethanol system at 68.24C where the azeotropic composition is xE=0.448, like Example 11.6 in the textbook. Also following that example, the application of accurate Antoine constants and bubble temperature computation is illustrated. As another problem, you might be given infinite dilution activity coefficients. For example, Lazzaroni et al. list the ginfM=2.03 and ginfB=2.10 at 313K for the 1-butanol+methylethylketone system. Taking the limits of Eqn. 11.37 shows that A12=ln(ginf1) and A21=ln(ginf2). Predict whether this system is expected to exhibit an azeotrope at 760 mmHg.

Comprehension Questions: You may assume the SCVP model for purposes of the calculations below (but you should use more accurate vapor pressure estimates for more professional purposes).

1. At 760 mm Hg the system acetone(1)+hexane(2) exhibits an azeotrope at 68 mole percent acetone with a boiling point of 49.8°C. 
a. Estimate the A12 and A21 parameters.
b. Estimate the bubble point pressure and vapor composition of 10 mole percent acetone at this temperature.
2. Acetonitrile+water forms an atmospheric pressure azeotrope at 70 mole% acetonitrile, 76°C.
a. Estimate the A12 and A21 parameters.
b. Estimate the bubble point pressure and vapor composition of 80 mole percent acetonitrile at this temperature.

Fitting Activity Models to Multiple Data Points (6:40) (msu.edu)

In early sections of chapter 11, we discussed fitting a single point. This technique is good pedagogically, but using a single point can lead to spurious results. Fitting of multiple data is preferred. Various options are discussed, as well as the bubble line method used in the textbook.

Fitting Pxy data using Excel (9:00) (msu.edu)

An illustration of using Excel for fitting Pxy data of the IPA+water system using the M2 model and some suggestions for working with the GammaFit.xls file, showing that the sum of squared deviations is 14 mmHg^2. Dividing by the number of points (18 including x1=0 and x1=1), and taking the square root gives a root mean square deviation (rmsd) of 0.89 mmHg. Noting that the pressure ranges from roughly 30-60 mmHg, this corresponds to roughly 2%rmsd. This effectively corresponds to sample validation of the M2 model for the IPA+water system since the deviation of 2% is quite small. We could argue that a model is valid as long as the rmsd is less than 10%, but you need to report the %rmsd and show the plot in order to be clear. For example, if the plot shows that there is systematic deviation from the experimental data, then a better model probably exists and should be sought. If there is no systematic deviation and the data are simply very scattered, then the model is probably as good as can be expected.

Comprehension Questions:

1. If experimental data for vapor pressures are included for a particular data set, should you use the values from the data set or the values calculated from Antoine's equation?
2. What is the objective function applied by the GammaFit spreadsheet?
3. Apply the procedure illustrated here optimize the M2 model for the ethanol+benzene data given in HW 10.2. What values do you obtain for A12 and A21 in that case? What is the value of the rmsd that you obtain?
4. Explain how you would modify this spreadsheet to apply to the M1 model.
5. Explain how you would modify this spreadsheet to apply to data obtained at constant pressure instead of constant temperature.

Introduction to Henry's Law (10:16) (msu.edu)

Fugacities are calculated relative to standard state values, and the relations developed earlier in the chapter use a pure fluid standard state. What if the pure fluid does not exist as a liquid when pure? One choice is to use Henry's law.

Osmotic Pressure (7:23) (Learncheme.com)

A derivation of the relation for osmotic pressure, and an explanation of why the pressures are different on each side of the semi-permeable membrane.

MW of protein by osmotic pressure - (8:23) (learncheme.com)

An application of osmotic pressure measurement to determine MW of a protein.