Activity pubs.acs.org/jchemeduc
Exploring the Clapeyron Equation and the Phase Rule Using a Mechanical Drawing Toy Katherine V. Darvesh* Department of Chemistry, Mount Saint Vincent University, Halifax, Nova Scotia B3M 2J6, Canada S Supporting Information *
ABSTRACT: The equilibrium between phases is a key concept from the introductory physical chemistry curriculum. Phase diagrams display which phase is the most stable at a given temperature and pressure. If more than one phase has the lowest Gibbs energy, then those phases are in equilibrium under those conditions. An activity designed to demonstrate the phase rule and to encourage deeper understanding of the Clapeyron equation is presented. This activity uses a mechanical drawing toy (such as an Etch A Sketch) with horizontal and vertical dials that can be adapted to explore mathematical relationships in a graphical situation. This activity is designed primarily for students taking second- or thirdyear physical chemistry, although some aspects of the activity would be appropriate for first-year general chemistry. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Physical Chemistry, Hands-On Learning/Manipulatives, Phases/Phase Transitions/Diagrams and ΔT are the change in pressure and temperature, respectively. What does this equation signify? It is the mathematical expression of just how much and in what direction the pressure must change with a given change in temperature for the two phases to remain in equilibrium. How can an instructor best illustrate this content? The phase diagram itself can be made more familiar by showing examples of substances such as water and carbon dioxide and by reading off the most stable state at different conditions and relating it to our everyday experience. The phase rule and the Clapeyron equation pose more of a challenge. The phase rule is likely to be a new concept, and it would be beneficial to have ways to work with it beyond simply looking at the diagram to see how the phase rule applies to various regions of the diagram. As shown in the Supporting Information, the Clapeyron equation, eq 2, can be derived from
O
ne of the topics introduced in the introductory physical chemistry curriculum at the second- or third-year undergraduate level is that of the equilibrium between phases of a substance. To understand more about which phase or phases of a pure substance is or are the most stable at a given temperature and pressure, and in turn where two or more phases are in equilibrium, we use a phase diagram. We explain that a line on the phase diagram of a pure substance represents a location where two phases of the substance are in equilibrium. A description of the phase diagram also includes introduction of the phase rule. The history of the Gibbs phase rule, and the role of Bakhuis Roozeboom, who through his experiments related to phase-rule studies was able to demonstrate the importance of the phase rule, is described by van Klooster1 and Daub.2 According to the phase rule, for a system at equilibrium: (1)
F=C−P+2
dGm = Vm dp − Sm dT
where F is the number of degrees of freedom, C is the number of components, and P is the number of phases. At or around that same time in the course, we derive the Clapeyron equation, because it tells us about the equilibrium between phases in equation form. In the notation of Atkins and de Paula,3 the Clapeyron equation can be expressed in the following way: Δp =
ΔtrsH × ΔT T ΔtrsV
(3)
where Vm is the molar volume and Sm is the molar entropy of the substance and recognizing that at equilibrium the molar Gibbs energies, Gm, of the two phases must be equal.3 In the introductory physical chemistry course, students should have sufficient mathematical background to work with the Clapeyron equation, but they are still more likely to be comfortable with Δy/Δx than they are with Δp/ΔT, as there are, traditionally, challenges with this “quantity algebra”. Furthermore, what is difficult to illustrate is what this equation actually represents. Students get some sense that the focus is on the phase
(2)
where ΔtrsH is the molar enthalpy of transition, ΔtrsV is the change in molar volume when the transition occurs, and Δp © XXXX American Chemical Society and Division of Chemical Education, Inc.
A
dx.doi.org/10.1021/ed300642v | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Activity
boundary and its location, but how does one emphasize that point? What we are trying to demonstrate with the Clapeyron equation is that a small change in temperature (or pressure) must be accompanied by a predetermined change in pressure (or temperature) in order for the system to remain in the twophase region of the phase diagram. Hands-on activities can be of great assistance with gaining a deeper understanding of concepts in physical chemistry. An example of this is the activity developed by Cloonan et al. involving building blocks to illustrate aspects of chemical reaction kinetics and equilibrium.4 An activity that has been designed to address the challenges of understanding the phase rule and the Clapeyron equation is presented here. A mechanical drawing toy (such as an Etch A Sketch) is employed, because it can be adapted to explore mathematical relationships in a graphical situation. When the left-hand dial of the device is twisted, a dark horizontal line appears on the screen. Twisting the right-hand dial controls the appearance and length of a vertical line. To produce a drawing, both dials are twisted, sometimes simultaneously. The image is erased by shaking the device. In the activity presented, the left-hand (horizontal) dial of the device serves as the T coordinate, whereas the right-hand (vertical) dial serves as the p coordinate. The relationship of the dials of a mechanical drawing toy to coordinates was pointed out by Holdener and Howard in their description of the use of Maple software to create parametric plots.5 When used as described below, the activity presented here can help the student to discover for herself or himself the one possible move of the dial that brings the two phases back to equilibrium. This can lead to a deeper understanding of the Clapeyron Equation. A mechanical drawing toy can also be used to explore the phase rule for various scenarios on the phase diagram. It can even be used to good effect in twocomponent phase diagrams.
Figure 1. Activity setup for exploring the Clapeyron equation and the phase rule.
to derive the Clapeyron equation from eq 3, pointing out that at equilibrium, the molar Gibbs energies of the two phases must be equal. The derivation of the Clapeyron equation, as outlined in Atkins,3 is included in the Supporting Information. Procedure
I start this activity by demonstrating it to one of the students in class, who is then asked to pass the device to the student next to them and to demonstrate the necessary actions to her or his classmate. Although one student is shown the device initially, the instructions are given out loud so the entire class can hear. In this way, every student will get a turn by the end of the lecture. Single-Phase Region
The student starts in the single-phase region, placing the insertion point well within a single-phase region. The student adjusts both dials. They soon see for themselves that they can vary the temperature (left-hand dial) and pressure (right-hand dial), and the insertion point will still remain in the single-phase region. There are, indeed, two “degrees of freedom”, because the student is free to adjust either dial in many different ways.
■
OVERVIEW The purpose of this activity is twofold: first, it allows students to explore the phase rule in an active fashion by exploring the various regions of the phase diagram using the device. Second, it enables students to derive deeper meaning from the Clapeyron equation by relating the move of one dial of the device to a fixed move of the other dial and thereby realizing that a small change in one variable requires a predetermined change in the other for the system to remain in the two-phase region.
Two-Phase Region and the Clapeyron Equation
Next, the student moves the insertion point to one of the lines, for example, the liquid−vapor phase boundary. If the left-hand dial is moved so that the insertion point is now just to the right of the line, this move is ΔT. They are now no longer in the two-phase region. When asked to return to the two-phase region, there is only one possible move of the pressure dial that can be made. This move is Δp. On the boundary line, there is indeed only one degree of freedom. At this point, the Clapeyron equation (eq 2) that was derived in lecture earlier on can be introduced once again, because now Δp has a new significance: it is the only dial change that can be made to maintain equilibrium for the given temperature change ΔT.
■
ACTIVITY DETAILS Detailed instructions on how to carry out this activity are included in the Supporting Information. Materials required are a mechanical drawing toy (available at toy and novelty stores), an overhead transparency or clear plastic sheet, a permanent marker, and clear tape. Before class, the device is prepared by cutting a piece of overhead transparency to fit the screen. With the use of a permanent marker (to avoid smudging), a phase diagram is drawn on the transparency. The transparency is affixed to the screen with clear tape. Anything drawn on the device will be visible through the transparency. A schematic of the setup is illustrated in Figure 1. Before the activity is started, the phase rule (eq 1) is introduced in class. Next, to consider the special circumstance where two phases are in equilibrium, the Clapeyron equation (eq 2) is introduced. If time permits, the instructor may choose
Three-Phase Region
The final demonstration of degrees of freedom can be made by placing the insertion point on the triple point. At the triple point, there are zero degrees of freedom, according to the phase rule. When the student tries adjusting one dial or the other, they see that they are “stuck”; unlike the two-phase region, a change in pressure cannot take the insertion point back to the triple point after the temperature has been adjusted. Neither dial can be moved, so there are indeed zero degrees of freedom at the triple point. B
dx.doi.org/10.1021/ed300642v | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
■
Activity
HAZARDS It is important to abide by all safety guidelines provided by the manufacturer of the device. When used as directed in the classroom setting, a mechanical drawing toy can be used safely; however, if the activity were to be adapted to a distance learning course, a warning should be included about keeping the device away from children under three years of age due to the choking hazard created by small parts.
demonstrating water’s anomalous slope of the liquid−solid phase boundary. It is straightforward to demonstrate using the p dial how an increase in pressure would move the insertion point from the solid to the liquid phase of water at the appropriate temperature. A sample problem to illustrate this has been included in the Supporting Information. The activity can also be extended to phase diagrams involving more than one component. For example, for a two-component system such as two partially miscible liquids, the variables shown could be temperature and composition, with the pressure held constant. The right-hand (vertical) dial could be the “T” dial, and the lefthand (horizontal) dial could be the “X ” or “mass %” dial. Given specified amounts of two partially miscible liquids added together, the two dials would guide the insertion point to the starting composition and temperature. From there, if starting in the two-phase region, the composition dial could lead the insertion point to the two points where the tie line intersects with the saturation curve, and further moves would help with obtaining the composition of the two phases and the relative amounts of the those same phases (via the lever rule). A sample problem for two partially miscible liquids (n-butanol and water6) is included in the Supporting Information. Although this activity is primarily designed for the physical chemistry course, it could be adapted for use in first-year general chemistry, where a mechanical drawing toy could be used to introduce phase diagrams, where the dials can be used to explore the various areas and features of the phase diagram, varying the temperature or pressure and observing the effect on the phases, as described in the Supporting Information example.
■
RESULTS AND DISCUSSION I have used this activity several times over the past decade in my Chemistry 2301 (Chemical Thermodynamics) class. Students take this course in their second or third year of an undergraduate chemistry major or honors program. They have completed a course in general chemistry, and usually they have already studied introductory calculus. I typically introduce this activity in the middle of a 75-min lecture period. Demonstrating the activity out loud to the first student takes only 5 or 10 min, after which time I continue to lecture while each student does the activity and then passes the device on to the neighboring student. Because the class size is small, each student has an opportunity to do the activity by the end of the lecture. There is an initial atmosphere of surprise in the classroom when the device is brought out, but this is soon replaced by a realization that this toy can actually be used to demonstrate effectively the significance of the Clapeyron equation. While I am lecturing, two students are working together on the device; the first student showing the second one which dial moves the second one needs to make, and the second one then performing the necessary tests. They are able to grasp and successfully complete the activity in just a few minutes, and thus, the whole class gets the chance to have a turn in the time available. The reaction as they try out the onephase region is fairly composed, as students expect to be able to adjust either knob any way they want. Trying to move both dials in the two-phase region elicits a reaction of surprise, as the student is no longer free to make the kinds of dial moves that are expected. By the time the student is in the three-phase region, it is not quite as much of a surprise to be restricted in ones’ ability to move about, but it solidifies what was learned of restrictions in the two-phase region. By this point, they are seeing the phase rule in another light, rather than simply as an equation. The students were actively engaged with trying out the dials, and then demonstrating the activity to their neighbor, and did not appear in any way to be bored or annoyed with the change of pace. At no time did they indicate that using a toy for the activity was too juvenile; in contrast, doing something hands-on was appreciated as a welcome respite from the more abstract aspects of the physical chemistry curriculum. The majority of those who attended class and therefore participated in the activity were able to successfully answer an exam question concerning degrees of freedom. With a large class, more than one mechanical drawing toy would be necessary: one per row, perhaps. If finances permit, each student or pair of students could have their own device with which to work. This activity could also form part of a lab session on phase diagrams, the phase rule, and the Clapeyron equation.
■
CONCLUSIONS As phase equilibrium is introduced in the introductory undergraduate course in physical chemistry, a mechanical drawing toy can be a valuable tool for aiding in the understanding of the subtle concepts of the phase rule and the Clapeyron equation. Active learning and a deeper understanding of equilibrium and the Clapeyron equation can be achieved through a variety of in-class activities involving manipulation of the horizontal and vertical dials.
■
ASSOCIATED CONTENT
S Supporting Information *
Detailed instructions on how the instructor can prepare for and carry out this activity in class. The diagrams included in the Supporting Information have been scaled to fit the standard size (18 cm wide × 13 cm high) screen of a mechanical drawing toy. This material is available via the Internet at http://pubs. acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS I would like to thank my students of Chem 2301. I also acknowledge with gratitude the encouragement and inspiration provided by members of the Chemical Education Division of the Chemical Institute of Canada, as well as that of colleagues
■
FURTHER ACTIVITIES Variations of this activity include working with specific phase diagrams, such as those of carbon dioxide and water, C
dx.doi.org/10.1021/ed300642v | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Activity
of Mount Saint Vincent University, and that of the staff of the Mount Saint Vincent University Teaching and Learning Centre. Suzanne Seager and Ardra Cole provided helpful comments and suggestions. I would like to thank the staff of Pearson Education for their help and for their permission to use their figures.
■
REFERENCES
(1) Van Klooster, H. S. J. Chem. Educ. 1954, 31, 594−597. (2) Daub, E. E. J. Chem. Educ. 1976, 53, 747−751. (3) Atkins, P.; de Paula, J. Elements of Physical Chemistry, 5th ed.; W. H. Freeman and Company: New York, 2009; p 111. (4) Cloonan, C. A.; Nichol, C. A.; Hutchinson, J. S. J. Chem. Educ. 2011, 88, 1400−1403. (5) Holdener, J.; Howard, K. Parametric Plots: A Creative Outlet. Journal of Online Mathematics and Its Applications, June, 2004. http:// mathdl.maa.org/mathDL/4/?pa=content&sa= viewDocument&nodeId=323 (accessed Sep 2013). (6) Raff, L. M. Principles of Physical Chemistry; Prentice-Hall, Inc.: Upper Saddle River, NJ, 2001; pp 379−380.
D
dx.doi.org/10.1021/ed300642v | J. Chem. Educ. XXXX, XXX, XXX−XXX
