Space-Filling P-V-T Models Don B. Hilton University of Lowell. Lowell, MA 01854 Space-filling models help the beginning student visualize the numerical aspects of the empirical gas laws. The pressure-volume-temperature (P-V-T) models are readily made byteachers and students using for patterns the graphical print-outs of isotherms from available computer programs and inexpensive posterboard, binding bars, and glue. When combined with lecture demonstrations and student experiments of the physical properties of gases, the modelbuilding activity reinforces the concepts of the physical principles which are often considered as "hard-to-teach" and "hard-to-learn" topics. These models are often pictured as perspective drawings in textbooks. The tangible model can be used by students to perform calculations-"interactive use", or b;teachers-"demonstration use". Making the Model
Examples of space-filling P-V-T models for an ideal gas and for a van der Wads gas are shown in figures one and two, respectively. The model consists of a three-sided box into which templates of isotherms are inserted. The pattern for the box is cut from a single piece of posterboard of any color (light blue is attractive). The bars are cut to appropriate lengths and glued to the base (V-T plane) of the box using plastic model cement. Spacing is determined by the scale chosen for the tempcrat"re axis. Computer print-outs ( 1 ) of the isotherms ( F \ 'plane), sized appropriately for a model. perhaps hy photoreduction or enlargement, are stiffened by gluing to posterboard and then cut out togive the hyperbolas -1' isotherms. A hindine bar, cut to appropriate of the ~~-~P length, is inserted along the pressure axis of each isotherm. The isotherm assembly is slid into the binding bar, already firmly cemented to the box (V-T plane), until the binding bar along the pressure axis on the isotherm section nearly makes contact with the wall of the box (P-V plane). Cement is applied to this binding bar, which is then firmly seated to the P-V wall of the box. Top and side panels, and labels of the three axes, cut from computer printouts, are glued in place to finish the model. The models made in the workshop measure 15 cm (pressure axis) X 18 cm (temperature axis) X 20 cm (volume axis).
Figure 1. The P-V-T S L ~ B C Bof an dea gas Tne three-way lnlenscton I C i 1 C k 1 0 1 the isocnore (A), isobar 181, an0 sorherm ICI is bm one pa nt on the surlace, any 01 un ch sat's1es tna eqLaloan PV/T = c0nStant.
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Suggestions for Use of the Models The model-building activity enhances a chain of thought from observations of gas behavior-hands-on activities and demonstrations (2) to the formal equations of state. The models enable the predictive value of the relationships to he tested and limitations noted. Use of the interactive software that eenerates the isotherms for these models enables the interested student to further explore more elaborate models that better describe "real" gases and generate print-outs for other models. For example, compressibility factors (Z = PVI nRT) as a function of pressure or temperature display the "goodness" of the various equations of state.
The Ideal Gas Model The ideal gas relationship represents a specific quantity of gas considered as point masses, but without interactions or identity. The P-V-T model clearly illustrates the hyperholas of theP-V (isotherms) relationship and the linear nature of the P-T (isochores) and V-T (isobars) relationships. The 496
Journal of Chemical Education
three-way intersection of lines of these functions represent a surface on which every point satisfies the equation PVIT = constant (Fie. 1).The constant is a composite of the eas constant, R, andthe number of moles, n, oigas. Thread may be used to markoff these lines on themodel's surface. Also, it can be confirmed, using a ruler for extrapolation, that only on the absolute scale (Kelvin or Rankin) do V and T for an ideal gas converge a t T = 0.Values of P V products, each a t some value of temperature, may be taken from the model and plotted vs. the temperature to show extrapolation to -273 O C or 0 K. The van der Waals (VDW) Model Clearlv. - . the nredictive value of the ideal eas model fails a t low temperatures and high pressures-gases do condense. The auestion now becomes, "What can be done to make the idealgas equation better?"~hismay be examined by introducing two simple mechanical arguments-particle volume and attractive forces. Two adjustable parameters, a and b, are introduced. The amount, nb, roughly the volume of n moles of molecules, is subtracted from the total volume, V, to give the "free" volume, V - nb, in which the molecules move. Two molecules entering the same space exert a repulsive force. The parameter, a, characteristic of each kind of gas, is used to account for attractive forces. The relation of this parameter to the pressure is more vague. The measured pressure, P , is proportional to the numbers of molecules (or moles, n) hitting a wall per unit time, and this, in turn, is proportional to the densitvof the collection of particles. nlV. kttiactive fmces on molecules about to hit a wall is pr&ortional to the density of the molecules in the bulk, nlV, which, A
ues of the critical volume (V, = 0.31), the critical temperature (T, = 410K), and thecritical pressure (PC= 33atm), are in good agreement with the observed values for n-pentane; (2) at higher temperatures the repulsive forces exceed the attractive farces [nRTI(V- nb) > a(nlV)z],and the VDW and ideal gas equations predict similar results. There are better, two-parameter equations than the VDW, but this is immaterial here. What is important is the chain of reasoning and the predictive value of the two models that describe eas behavior. Computer graphics as visual aids has progressed to pers ~ e c t i v e("fishnet") P-V-T d o t s of real eas behavior from which instructionai usage dften is evolved from research results (3, 4). The model-building activity described in this paper uses computer programs that may be used by heginning students and inexpensive materials to achieve the 3dimensional effect. These models may be used to introduce topics t o more advanced students. Perhaps students are alieady familiar with the contour lines o n a topographical map, each line depicting a constant elevation. The extent of the spacing of the contour lines indicates degrees of flatness or steepness. Movement on the surface in any direction involves a change of slope. A study of the orthogonal slopes a t the surface offers an opportunity to introduce the physical . . meaning of partial derivatives. Teachers completed one model in a 2-h workshop. Time was available to discuss uses of the models. These models make attractive "corridor" displays when combined with colorful postage stamps that illustrate historical aspects of the kinetic molecular theory (5).The model-building activitv mav " be extended to other X-Y-Z svstems: nressurecompressibility factor-temperature (real gases), speedmass-distribution (illustrating KMT), volume of added base-pH-concentration of acid (titration curves) (6). The author will he happy to share comments and workshop materials with teachers interested in building spacefilling models. ~~
Figure 2. The van der Waais P-V-Tsurface far c-pentane. At the critical point (circle1values of PC, T,, and Vc are in good agreement with those observed. The "dip" of Isotherms at lower temperatures to negative pressures has no physical meaning but is interpreted as the region of gas-liquid phase equiiibrium (region A). in effect, lessens the impact of the wall-hitting molecules. Hence, the pressure is lowered by a term proportional to the square of the density, ~ ( n l V )This ~ . quantity is added to the P to account for the loss in pressure. The combined result of the volume and attraction terms yields the cubic equation,
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[P+a(n/V12][V- nbl = nRT
The surface generated by this equation differs dramatically from that of the ideal gas equation. The VDW parameters, a and b, give identity to the gas. The space-filling model (Fig. 2) depicts the P-V-T relationship for n-pentane. At lower temperatures the isotherms dip to negative pressures-a consequence of the properties nf the cubic equation. This regiun (A, Fig. 2) is interpreted as representing ohase seoaration. The model does not clearlv show the lines hepictin; the saturated liquid state-on thepoint of evaporation-and the saturated vapor state--on the point of condensation. However, the model is in good agreement with two observations of the real substance: (1) . . at the isotherm showine a horizontal tsneent .. (circle. . . Fie. ,, 21. . the, critical region where the rr~ulsiveand attrartiw forces are said tu
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halance [nRT t V
- n b ) approximately equals n ( n / l ? - I , val-
For a complete set of workshop directions please send $3.00 to the author to cover copying costs and postage. included in the kit are all necessary patterns and comDuter printouts of isotherms to make two models (Ideal and VDW).
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Acknowledgment The author would like to thank Judith A. Kelley of the University of Lowell and grant recipient (National Science Foundation through the Science and Mathematics Education Network Program) for her encouragement and financial support that made the workshop possible.
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gen&atpd, by an Apple Ila computer s n d s n Imagewriter I1 printer to make those rnodois. 2. Shakha~hiri.B. 2. Chemical Demonsfrolians: Univ. of Wisconsin: Madison. WI. 1985: 5. ~ ~ 1 .chapter 2 . 3. J o l i ~K. . J.Chem. Edur. 1981.61. 393.
Volume 68
Number 6 June 1991
497
