 | CASE TEACHING NOTES
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| CASE TEACHING NOTES
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Chemical separations play an important role in many scientific endeavors and can aid in the identification, isolation, purification, and/or quantitation of individual chemical constituents from complex mixtures. Therefore, it is important for students to gain an appreciation of the many factors that influence the efficiency of a chromatographic separation. The introduction of key concepts, including capacity factor, relative retention (or selectivity factor), and height equivalent of a theoretical plate (or more simply, plate height H), often culminate in the presentation of the van Deemter equation, which to students may appear to be no more than a collection of three mathematical terms with little obvious correlation to peak shape or chromatographic response.
Exposure to chromatographic theory and the van Deemter equation usually takes place during a junior- or senior-level instrumental analysis class. At this level, class sizes are often small and so active-learning exercises can be more readily incorporated, which may aid student understanding of the important but challenging concepts that underpin chromatographic theory. However, in large institutions (such as the University of Alberta where this case study was first developed and implemented), class sizes even at the junior/senior level can appear prohibitively large for substantial student involvement. This case study, to be presented by the students themselves as an informal theatrical performance, was designed specifically for large classes (75 to 125 students) to provide a memorable and clear visual presentation of the practical effects of each of the three terms of the van Deemter equation on the efficiency of a chromatographic separation.
By "experiencing" the effects of column packing, longitudinal diffusion, and the kinetics of partitioning, students develop an understanding of the relative impact of these terms on plate height as a function of the average linear velocity of the mobile phase, and hence, have the tools necessary for the design of better separations.
The purpose of this case is to provide junior/senior-level chemistry majors with a firm understanding of the factors that contribute to separation efficiency and/or peak broadening in chromatography. Specifically, these include:
These three factors constitute the three contributions to plate height, as modeled by the van Deemter equation, and are (i) not, (ii) inversely, and (iii) directly dependent upon the average linear velocity of the mobile phase. The van Deemter equation is most simply written as
where H is the plate height, vm is the average linear velocity of the mobile phase, and A, B, and C are (positive) constants determined by various physical properties of the mobile and stationary phases. By understanding these three major contributions to plate height, along with the role of mobile phase velocity, the students should be better equipped to vary separation conditions as necessary for optimization of separation efficiency.
This is a short, factually-driven case that does not involve substantial decision making or debate on the part of the students. Instead, it engages the students, in three large groups, as actors taking the roles of solute molecules, stationary phase particles, or mobile phase molecules. By requiring students to identify their roles and plan their "action" under specified chromatographic conditions, it is believed that students will develop an in-depth understanding of the factors affecting chromatographic efficiency. This case effectively augments the usual lecture-based presentation of the van Deemter equation, and should be preceeded by at least a cursory introduction to the basic tenets of chromatography, including partition coefficients, selectivity coefficients, and definition of plate height (or number of theoretical plates). The instructor may or may not choose to present the van Deemter equation prior to conducting this case study. This decision will not affect the success of the case, but will alter how discussion should be led following the case (see "Directing the Discussion: Expected Outcomes").
The case, presented by the students as a three-act play, is designed to be completed in a single 60-minute lecture period for a large class (75 to 125 students). Suggestions aimed at making the presentation viable for smaller classes are provided at the end of the next section, although the impact of the presentation will not likely be as great under such circumstances.
This case study uses the three acts of a play to present the three terms of the van Deemter equation, and as such, requires the class to be divided into three groups of roughly equal size. This can be accomplished most quickly by simply grouping the students according to their seating locations in the lecture hall. If the class is already split according to lab sections, you may wish to use these to form the three necessary groups. After assembling into a working unit, each group should be assigned to perform one act of the play, and should be provided with some guidelines for casting and action. Copies of each act of the Case Study itself, along with corresponding Stage Directions, would serve this purpose. If the instructor has chosen to present the details of the van Deemter equation prior to this case, then the groups could be asked to enact each term of the equation without the provision of any casting or action guidelines. However, this option would require more than one 60-minute lecture period to complete.
After allowing sufficient time for group members to read the handout pertaining to their act, groups should be encouraged to collectively discuss the study questions prior to "rehearsing" their scene. Laboratory teaching assistants could be asked to sit in on the class to help moderate the discussion in each group, although groups can be left entirely unsupervised at this point. There is no need to assign a formal leader in each group, nor to assign roles to individual "cast members," since these will naturally evolve as the groups work to plan their scene. Advise groups that they may or may not choose to use dialogue during their scene, and that simple props or other means of identification of characters can be used. For example, simple hats or flags made out of different colors of paper are effective visual aids that may help a group to better convey the identities of its members. During this preparation and discussion time, some groups will choose to physically rehearse their scene, while others will simply discuss their expected actions. Regardless, about 15 to 20 minutes should be allotted for this stage of the case study.
To physically stage the play, the instructor will adopt the role of narrator and will announce each act in succession. When their act is announced, group members will move from their seats in the lecture hall down to the front of the room. This poses an inevitable disturbance and breaks the "flow" between acts, but unfortunately can't be avoided in a large class setting. Each act typically requires no more than five minutes to perform, with several minutes between acts for students to get organized (for a total of about 20 to 25 minutes for this stage of the case study). Class members should be encouraged to clap loudly after the completion of each act!
Following the presentation of all three acts, students should return to their group locations in the lecture seats and should discuss (again, in their acting groups) the "After the Scene" questions. Alternatively, the instructor may wish to use these questions to direct an all-class discussion of these key points, or to ask each group to lead the all-class discussion of the "After the Scene" questions pertaining to their own scene. In any case, this discussion will likely consume the time remaining in the class period. Students can be sent home with individual follow-up questions if desired, or, such questions could be posed by way of an on-line survey or on-line quiz on a class website prior to the next class meeting. The following class meeting should begin with a summary of the key points from the play, presented more formally as the van Deemter equation. This will help the students to make the connection between the essential chromatographic theory and the related action they participated in/observed.
Many of the organizational issues presented above are independent of class size. However, smaller classes will not have sufficient students to be able to fully "cast" the roles suggested in the one-page handouts associated with this case. In Act I (Eddy Diffusion), when student numbers are small, it would be possible to use chairs or desks (instead of students) randomly positioned across the front of the classroom to represent stationary phase particles in the column. In Act II (Longitudinal Diffusion), one single group of students, rather than three different subgroups, could repeatedly travel across the front of the room at different rates on subsequent trips. In Act III (Rate of Mass Transfer), chairs could again be used to represent stationary phase particles, and students who act as solute molecules could simply be seated for varying lengths of time to represent interactions with the stationary phase. In small classes, however, students might need to participate in the action in each scene in order to have a sufficiently large cast. This would detract from the overall impact of the exercise, because there would be no substantial audience for the presentations, and because it would preclude the students from viewing any of the scenes from the perspective of an audience member.
The "A" term in the van Deemter equation accounts for the effect of inhomogeneities in the packing of a chromatographic column on the travel of solute molecules through the column. If stationary phase particles are not highly regular and are not carefully packed to form a homogeneous bed, then larger channels or pockets may exist in the column along with the normal pores. These channels and pockets present various different paths to the solute molecules. As a consequence, spreading of a sample zone will occur as identical solute molecules within that zone travel different effective distances through the column along the various paths available to them. This process is called eddy diffusion, and is effectively independent of the mobile phase velocity. In order for students participating in this scene to understand this concept fully, they should address the following questions prior to the scene:
Act I-1. Why is it important for the three solute molecules in each zone to be identical (even though the solute molecules between zones may differ)?
While acting out the process of eddy diffusion, it must be made clear that the three molecules in each zone arrive at the end of the chromatographic column at different times not because they are different molecules, and hence, interact differently with the stationary phase molecules, but rather, because they travel different paths through the stationary phase. That is, students must realize that eddy diffusion results in different travels (and, consequently, zone broadening or increased plate height) for identical molecules within any given zone.
Act I-2. What physical aspect of a chromatographic column can lead to the existence of different routes or paths through that column?
Most obviously, the column must be packed with a stationary phase, and that packing must contain inhomogeneities in order for different paths to exist. Wall-coated, capillary chromatographic columns have no "A" term contribution to plate height since they contain no packing and hence present only one path—the capillary itself—to the solute molecules. Other factors allowing the magnitude of the "A" term to be diminished will be considered in the "After the Scene" questions.
The "B" term in the van Deemter equation accounts for longitudinal diffusion along the column axis. Students must understand that longitudinal diffusion is different from eddy diffusion: the latter (as described above) involves the motion of solute molecules along different paths, whereas the former is driven by the random motion of solute molecules, which tend to move from regions of high concentration to regions of low concentration. Diffusion is governed by Fick's first law, but there is no need to introduce this law in order to adequately describe this familiar phenomenon. Such diffusion occurring within a solute zone on a chromatography column is not unlike the familiar observance of, say, a drop of food coloring placed in a glass of water. Initially, the drop appears highly colored and concentrated in one area, while over time and even without stirring the drop spreads out and eventually the entire glass of water will be faintly colored with no single region of high food dye concentration. Solute molecules in a sample zone injected onto a column likewise spread out from their initial uniform region of high concentration, broadening the zone and increasing the plate height. The more time the zone spends in the column, the more opportunity there is for such longitudinal diffusion to occur. Hence, this "B" term contribution to plate height is inversely dependent upon the velocity of the mobile phase. These concepts should be reinforced by having students who will participate in Act II consider the following question prior to their performance:
Act II-1. Why is it important for the three subgroups to contain the same number of solute molecules (and for the three groups to represent the same type of solute) before beginning their travels through the chromatographic column?
Solute zones containing different numbers of solute molecules have effectively different concentrations, and, hence, will experience different extents of diffusional broadening. More diffusional broadening will occur in highly concentrated zones relative to zones with lower starting concentrations. To ensure that the "broadening" presented in this act is not affected by differences in starting concentration, the three subgroups or zones of solute molecules must begin with the same number of molecules (or actors). Furthermore, if one of these subgroups or zones contained different types of solute molecules, then they would interact with the stationary phase to different extents. In this scene, the only factor that should be illustrated is zone broadening due to longitudinal diffusion rather than broadening or separation caused by other factors.
The "C" term in the van Deemter equation accounts for non-instantaneous equilibration time or resistance to mass transfer between the mobile and stationary phases. Mass transfer or partitioning depends on the equilibrium constant or partition coefficient, the diffusion coefficient of the solute in the stationary phase, and the diffusion coefficient of the solute in the mobile phase to be transported to the stationary phase. Because of these multiple contributions, more in-depth discussions of the van Deemter equation present the "C" term as the sum of two parts, (CM + CS ), to differentiate between the contributions of diffusion of solute across the flow direction in the mobile phase (towards the stationary phase), and the reaction rate of partitioning of the solute in the stationary phase, respectively. The combined effects become important when the mobile phase velocity or flow rate is too high for equilibrium between the two phases to be obtained and, hence, the "C" term is directly dependent upon the velocity of the mobile phase. Again, to ensure that students participating in this scene are aware of these concepts, they should address the following questions prior to their act:
Act III-1. Why is it important for the solute molecules to be identical (i.e., how would the "separation" be affected if they were different)?
It is the purpose of this act to demonstrate the effects of resistance to mass transfer rather than inherent differences in the ability of different solute molecules to interact with a particular stationary phase material. If solute molecules are all of the same type, then differences in their travel through the column cannot be attributed to differences in the nature of their interaction with the stationary phase but, rather, can be attributed to the resistance to mass transfer offered by their diffusion into and out of the stationary phase (as well as their diffusion to the stationary phase).
Act III-2. What determines the nature of interaction between a solute molecule and a stationary phase particle, or between a solute molecule and the mobile phase?
The association of a solute molecule with either the stationary or mobile phase can be dictated by a wide variety of properties, such as molecular charge, size, mass, polarity, chirality, and so on. Differences in these properties between solutes results in differential partitioning and hence, separation as the solute travels through the chromatographic column.
Following the performance of all three acts, it is instructive to recall and interpret the action as it relates to the van Deemter equation. As mentioned previously, such follow-up can be conducted as a discussion within individual acting groups or as a whole-class discussion led either by the instructor or by the groups themselves. The following questions can be distributed to students either before or after the play, and will serve to focus the discussion and highlight key concepts:
1. In Act I: Eddy Diffusion, why did the three molecules in each "zone" or solute group travel at slightly different rates through the column, even though the molecules themselves were identical? How could this effect be minimized or eliminated?
The focus here is on the existence of different paths through the column packing. This effect can be minimized by using highly regular stationary phase particles of uniform shape and size. The use of smaller stationary phase particles also helps to minimize this effect, since smaller particles provide less opportunity for large channels to form. In addition, a more narrow column (smaller diameter bed) means that any irregular channels that might happen to form will have less chance of being isolated and hence will have a less significant effect on solute particle travel routes. Finally this effect can be eliminated altogether by using wall-coated capillary columns, since these have no packing.
2. Would the phenomenon that you described in answer to #1 be affected by the velocity of the mobile phase and, if so, how?
The existence of multiple paths through a packed column is not significantly affected by mobile phase velocity and, hence, this contribution to peak broadening is effectively independent of mobile phase velocity.
3. In Act II: Longitudinal Diffusion, each group or zone consisted of the same number of identical solute molecules (representing the same initial concentration). What aspect of their travel through the column resulted in different extents of spreading (or longitudinal diffusion) for each of the three solute zones? Consequently, how does the velocity of the mobile phase affect the zone broadening in this case? How could broadening be minimized in this case?
Solute molecules within each zone naturally spread from their initial region of high concentration (within the zone) to regions of lower concentration (at the zone edges) as they traveled through the column. This is a natural process of diffusion that affects all solutes in all systems. The solute zone being carried through the column by the slowest moving mobile phase had the most time to undergo diffusion and, hence, exited the column having experienced a greater extent of spreading than the solute zone being carried through the column by the fastest moving mobile phase. Zone broadening due to longitudinal diffusion can be minimized by increasing flow rates through the column and/or by injecting zones with lower concentrations to begin with.
4. Even with mobile phase molecules traveling at a constant speed via regular paths through the chromatographic column in Act III: Resistance to Mass Transfer, did the six identical solute molecules arrive at the far end of the column at the same time? If not, why not?
The zone of identical solute molecules broadened as it traveled through the column, apparently due to solute molecules spending different lengths of time associated with the stationary phase particles. However, since the solute molecules were all chemically identical, the nature of their interactions with the stationary phase would not differ from molecule to molecule. Thus, the apparently different associations with the stationary phase particles must have been kinetic in nature due to differences in mass transfer from one particle to the next (or differences in the extent of diffusion, for example, of the solute molecules in the stationary phase). This effect could be minimized by decreasing the thickness of the stationary phase coating on the solid support particles, so that the equilibration time of the solute between this stationary phase and the mobile phase would be decreased. Likewise, the use of smaller, less porous stationary phase particles would reduce the extent of broadening due to mass transfer, since equilibrium can be reached more quickly under these conditions.
5. Would the velocity of the mobile phase have any impact on the phenomenon that you described in answer to #4? If so, what would that effect be?
The slower the mobile phase velocity, the greater the opportunity for solute molecules to reach equilibrium with the stationary phase before being carried further along the column. Conversely, high mobile phase velocities do not permit sufficient time for the solute molecules to reach equilibrium in their partitioning and, hence, the solute zone would be broadened as it travels through the column. Zone broadening due to mass transfer can be minimized by reducing the velocity of the mobile phase.
It is effective to end this case study with a restatement of the van Deemter equation. By relating the "A" term of the equation to Act I, the "B" term to Act II, and the "C" term to Act III, students will be able to make better sense of the three contributions to plate height H and band broadening. If, however, the instructor chooses not to expose the student to the van Deemter equation prior to conducting this case study, then it can be left as a take-home exercise to the student to "construct" a mathematical equation, consisting of three terms, relating plate height or band broadening to mobile phase velocity. Discussion of the student-constructed equations could begin the following class session with formal introduction of the van Deemter equation at that point. Construction of a graph of plate height versus mobile phase velocity, showing each of the three terms along with the net relationship dictated by the van Deemter equation is also a good follow-up exercise to be conducted in class.
In any case, by having staged the equation as a three-act play, the student will have a much more vivid understanding and longer-term recollection of the physical contributions to band broadening that take place during a chromatographic separation. As such, the student should be well equipped to make judgments about variable selection and control for separation design and optimization. For completeness, a presentation of the experimental determination of plate height (based on the equation H = (w2L)/(16 tr2), where w represents peak width, L represents column length, and tr represents retention time) should follow this presentation of plate height theory and the van Deemter equation, so that students get a feel for assessing the separation efficiency of experimental data, after any given separation design has been implemented.
Acknowledgements: Development of this case study was made possible with support from The Pew Charitable Trusts.
Date Posted: 3/13/02 nas
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