menu with the choices Titrate. Save. End. and Help. Selecting Titrate s t a r t s the d a t a col'lecti&~routine and prompts the user to enter the initial volume reading of the buret. Once a volume is entered, the system reads the meter and ulaces the DH or potential datum in the cell adiacent to the volume-reading, moves the cursor to the next cell, and prompts for the next buret reading. At the end of the titration, signaled by entering 999 when prompted for a buret-reading entry, the menu reappears. Selecting Save automatically extracts the data to the user's floppy disk; the macro code is not saved on the data disk, thus keeping data file size to a minimum. Data files may be extracted in Lotus 1-2-3 or Quattro Pro format, a s specified by the user. Choosing Help presents brief information screens, which are also accessible during data collection by typing 911. Selecting End returns manual spreadsheet control, allowing the user to access all of the usual spreadsheet features for data reduction and graphing. The skills required to carry out a n accurate and precise wet chemical analvsis are still stressed. and students use the sprt~adnhectthemselves to carry out all data redurt i m and plotting, thus beromingproficicnt spreadsheet 11sers. Collection of Calibration-Curve Data
The macro routine for the collection of potentiometric calibration-curve data presents the menu choices Collect Data, Save, End, and Help. The user is prompted to enter the numbers of standards, samples, and replicates to be measured. A macro subroutine automatically prepares a random sampling sequence and instructs the user& insert the electrodes into the proper solution. Once the electrodes are inserted, the user preises Enter, the system pauses for 30 s to allow electrode equilibration, reads the potential, and enters it beside the solution description in the spreadsheet. After the last data point is collected the system reorders the samplelstandard list and presents the user with a menu similar to the one a t the heginning of the program. Essential error-checking routines are built into the data acauisition macros so that command echoes from the mete; will be rejected and not affect the data. A complex cleaning routine removes unwanted characters from the innut strines . . and ~ converts the strine..data from the instmment to numerical values w ~ t hthe appropriate numbers of sienificanl firmres. As a safcmx~rd.s c l e r t l n ~ T ~ t r a t eor Collect Data r e h s in a caution message warning that any existine data in the sureadsheet will be overwritten and gives the user the option of extracting the data into a separate file before continuing. We have chosen not to fully automate the analyses because we believe it to be more instructive for students to handle their own calculations and graphmg with the spreadsheet. The use of these uroerams makes the sureadsheet-controlled . appnrach UI irlsuument interfxinga simple and inexpensive means of introducinp automated data rollection into the Ink) in a manner that isreadily accessible to students. The programs have been used successfully in quantitative analysis, instrumental analysis, and biochemistry laboratory courses, as well as in oceanographic research. The macro routines have been developed and tested on a personal computer running Quattro Pro 4.0 under DOS 5.0, interfaced to a n Orion model 520ApWmV meter using the RS232-C uorts available on each device. Before loading the spreadsheet itself, the DOS MODE command is used t; confieure the comuuter's serial port. No add-in uromams or adbitional hardware other t h i n a cable are required if the instrument to be interfaced has bidirectional RS232 capability. Copies of the spreadsheet-controlled data acquisition macros described in this paper are available (with brief ~~~
.~~~~ -
.
documentation) by sending a blank disk and postage-paid mailer to J. Mullin. Acknowledgment The authors wish to thank the University of New England College of Arts and Sciences Dean's Office for partial funding of this project, and Gene Yonuschot of the UNE College of Osteopathic Medicine for the loan of a computer and printer.
Where the Electrons Are Roger Earth West Chester University West Chester, PA19383, E-mail:
[email protected] Locating electrons and modeling their behavior is a t the heart of much of today's chemistry. Acursory glance a t any organic chemistry textbook shows the central role played by electrons in our view of chemical reactions. Chemists need to have a n approximate pictorial understanding of where the electrons are. For students of chemistry, an understanding of the behavior of the electron begins with atomic orbitals, in particular, with the s, p, and d orbitals of hydrogen. To most chemists, a n orbital is used as a picture, not a mathematical function. Our concern as educators should be that the correct picture be implanted in the minds of chemistry students. Methods of Representation Adesirable picture would be one that is closely related to where the electrons are. Ease of drawing and remembering would also be helpful. The best picture, in my view, is a contour surface of the probability density (26). A related, and also effective picture, is the dot-cloud diagram (27). The use of surface plots of the wavefunction has been advocated (281, but I think that students may find them less helpful in the central concern: learning where the electrons are. I n any event, the same methods used for the contour plots can be made to give surface plots. Kikuchi and Suzuki (29) have made a lucid comparison of various representations of orbitals. Methods and programs for preparing all of these representations have apneared in this Journal. Baughman's proposal, that s t u d e k s draw the contours (in twodimensions) as an exercise (301,should be given very serious consideration by teachers and textbook authors. I will describe a method for preparing two-dimensional contours using a commonly avaiiable general-purpose mathematics-engineering program. I will also demonstrate that this same method Fan be used to prepare the matrices that serve as the basis for students'own hand-drawn diagrams, a s recommended by Baughman. The Shaded Hourglass
The usual representation of a p orbital shown in general chemistry textbooks differs in nearly every detail from the contour plot of a 2p, orbital shown in Figure 2. The most popular textbook drawing is what I call a shaded hourglass, a two-dimensional version of which is shown in Figure 3. As an indication of where the electrons are, Figure 3 is not realistic. I t leaves the impression that there is significant electronic charge density a t the nucleus. The shaded hourglass is actually a graph, in spherical coordinates, of the square of the angular part of the wavefunction (y2).The distance from the origin represents the magnitude of Y'. Afew texts represent the p orbitals with a pair of tangent spheres (Fig. 4),which is a graph of the angular part (Y). It is unthinkable that anyone teaching general chemistry Volume 72 Number 5 May 1995
401
Fiaure 2. Contours ofthe probability density ofthe
Figure 3. Two-dimensionaldraw- Figure 4. Tangent circles: occaing of a shaded hourglass: the spnally used to represent a P orusual representation of a p or- bml. would want these graphs to be the students' internal picture of where the electrons are. Many of the texts improp erly identify these graphs as probability contours or as 90 or 99% boundary surfaces. These errors are inexcusable in a first-year general-chemistry text. That the same errors are also found in some physical chemistry textbooks can onlv be described as embarrassine. -
Figure 5.Two views of the suriace plot of the probability density fora 2p orbital. Plotted by Mathcad. perience replace those from the text. It is now possible, with inexpensive math-engineering programs, for students to display and print accurate contour plots of a twodimensional slice of an orbital. The characteristic of these programs is that they can accept mathematical functions
0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.8 0.8 0.8 0.7 0.5 0.4 0.2 0.1 0.1 0.0
Unrealistic Diagrams
-
Of seventeenrecent eeneralchemistrvtextswhich ~ ~ Iexamined,on?vthreemadeconsiste~useofrealisticdiagramsofpanddorbitals~Oftheremaining fourteen, seven showed the shaded hourglass, incorrectlvident~f~eitasag00r99%boundarvsurfaee ~~
~~~
shadedhou%lasslaterinthebook(honecase,in
0.1 0.1 0.2 0.4 0.6 0.9 1.2 1.4 1.5 1.4 1.2 0.9 0.6 0.4 0.2 0.1 0.1 0.1 0.2 0.3 0.6 0.9 1.4 2.0 2.5 2.7 2.5 2.0 1.4 0.9 0.6 0.3 0.2 0.1 0.1 0.2 0.4 0.7 1.3 2.2 3.2 4.1 4.5 4.1 3.2 22 1.3 0.7 0.4 0.2 0.1 0.1 0.7. 0.4 0.8 1.6 2.9 4.6 6.2 6.9 6.2 4.6 2.9 1.6 0.8 0.4 0.2 0.1 0.1 0.2 0.4 0.8 1.6 3.1 5.5 81 9.3 8.1 5.5 3.1
0.0 0.0 0.0 0.0 atendencytopresentthe~Zplotintwodimensions, 0.0 0.0 whichistheoriginoftheshadedhourglass,usuall~ 0.0 0.1 but a few improperlyidentify it 0.1 0.2 as a probability contour. The shaded hourglass 0.1 0.2 seemstodominatethepresentationsinno,,,,,ajors~ 0.1 0.2 texts. o h n without any identification.
the diagrams in the chapter problem set) or had errors in the diagrams. Physical chemistry texts have
Contour Plots of the Probability Densities
Oneremedyforthisunfo~tesituationisto
1.6 0.8 0.4 0.2 0.1
0.1 0.2 0.4 0.9 2.0 4.1 5.9 4.1 2.0 0.9 0.4 0.2 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 00 0.0 0.0 0.0 0.0 00 0.1 0.2 0.4 0.9 2.0 J I 5.9 4.1 2.0 0.9 0.4 0.2 0.1 0.0 0.0 0.2 0.4 0.4 0.4
0.5 0.8 0.8 0.7
1.1 1.6 1.6 1.3
2.4 4.8 8.1 9.9 8.1 4.8 2.4 1.1 1.6 2.9 4.6 6.2 6.9 6.2 4.6 2.9 1.6 2.2 3.2 4.1 4.5 4.1 3.2 2.2 1.3 3.1 5.5 8.1 9.3 8.1 5.5 3.1
0.5 0.8 0.8 0.7
0.2 0.4 0.4 0.4
0.1 0.2 0.2 0.2
0.0 0.1 0.1 0.1
0.1 0.2 0.3 0.6 0.9 1.4 2.0 2.5 2.7 2.5 2.0 1.4 0.9 0.6 0.3 0.2 0.1 0.1 0.1 0.2 0.4 0.6 0.9 1.2 1.4 1.5 1.4 1.2 0.9 0.6 0.4 0.2 0.1 0.1 0.0 0.1 0.1 0.2 0.4 0.5 0.7 0.8 0.8 0.8 0.7 0.5 0.4 0.2 0.1 0.1 0.0
have students prepare their own contour plots of the probability densities (yz)for various orbitals. 'rhe Figure 6. Scaled values of the 2p, probability density for student preparation of a objective is to have the images fmm hands.on ex. contour plot by hand. Printed from Mathcad. 402
Journal of Chemical Education
Table 1. Contour Plots of Hydrogen Orbitals
"Some constants' 1 9.10953x
lo3'
+
1 1.67265x 10.~
so = 8.854188 x 1 0-l2
I
"Reduced mass of H 'Vacuum permittivity"
Surface and Contour Plots of Atomic Orbitals
hsr = 1.05459x lo4 Oe= 1.60219~ lo-''
"Elementary charge"
Yzp(C0) =-
"Normalized wavefunction SI units"
"Mapping Cartesian space into circular coordinates"
"Number of data points: must be even" k = O...N
A variation on this approach is to print out a small matrix (about 15 x 15) and have the students connect points of equal probability density a s advocated by Baughman. I n this case, the values within the matrix should be scaled to fall within the range of 0-9.9. One decimal place should be specified, and trailing zeros should be retained so that all the numbers will occupy the same amount of space.
I=0... N
"Number Bohrs origin to edge" "Origin is (M2, N12)"
A a 1 --
2
The Mathcad (version 5.0 for Windows) document presented in Table 1can he used to prepare surface and contour plots of atomic orbitals. Explanations are set off in quotation marks. The constants are eiven in SI units. The wavefundion gwcn is that for tht: 2p, orhtal. The pararneterNsets thesize of the matrix M rN + 1 k.V T 11. whch wdl contain the ~ n h ability density. Larger values of N give smoother cokour plots but require more calculation. Surface plots with high values ofN have too many lines and are dimcult to interpret. The parameterA is the number of Bohr radii from the origin to the edge of the plot. The subscripted variablesx and Yare the Cartesian coordinates assigned to the matrix locations corresponding to the subscripts. The matrix has 0,O in the upper left comer, so the origin of the Cartesian coordinates must be offset to the center. The atomic orbitals are given in spherical coordinates, so the Cartesian coordinates X and Y are converted t o r and 8.The probability density is then calculated for each point on the matrix. This matrix can be displayed as a contour plot (Fig. 1, N = 80) or as a surface plot (Fig. 4 , N = 16).Matrix G is a scaled version of matrix M with the maximum value of the probability density held to 9.9. This scaled matrix can he printed out as shown in Figure 6 to allow students to prepare their own contour plots. In this case, N should be about 14 to 20. Mathcad and similar mathematics-engineering packages can be valuable tools in allowing students to manipulate atomic orbitals. The same method can be used for hybrid or molecular orbitals. The vivid images and the possibilities for hands-on manipulation can help overcome misconceptions, perpetuated by textbooks, about where the electrons are.
"Special rdes for x axis and origin" 1[ %---I
3.") %n$es are
eXhl+l
counterclockwisefrom y axis." "Normalized Probability Density"
Transformed Probability Density values from 0 to 9.9
and carry out mathematical operations upon them, including integration, solving for variables, finding roots, and preparing various sorts of plots. One example is a program that has been used for a variety of chemistry and chemistry-education applications: Mathcad (Math Soft) (3132).Mathcad can prepare contour plots and surface plots of data presented in a rectangular matrix. The position within the matrix gives the x and y Cartesian coordinates; the value a t that position provides the z coordinate. What is needed is a means of mapping the locations within the matrix to the polar coordin a t e s t h a t a r e n a t u r a l to wavefunctions. Then any wavefundion can be typed into the document, and with some adjustment to the scaling, either a contour plot (Fig. 2) or a surface plot (Fig. 5) of the value of the probability density in the xy plane can be produced.
Chemical Education via MOLGEN C. Benecke, R. Grund, A. Kerber, R. Laue, and T. Wieland Depaltment of Mathematics University of Bayreuth D-95440Bayreuth. Germany A main topic a t certain school classes, university lectures, or seminars is the very large variety of hydrocarbon molecules, such a s t h e well-known benzene. Therefore, i t i s important to show students t h a t the molecular formula CsH6 corresponds to more t h a n just t h e famous benzene ring; there are many isomers of t h a t particular formula. Few students know of the existence of the 217 structural formulas, and they will be astonished to hear t h a t about 50 of them have been observed i n n a t u r e o r laboratory. Ideally t h i s fact could be demonstrated with a n example on the screen of a computer monitor. MOLGEN provides a solution to t h a t problem. I t was awarded the German-Austrian University Software Prize 1993 for excellent educational software in chemistry. Several schools, universities, and chemist r y companies in Germany use this program, which i s the result of a research project supported by the Deutsche Forschungsgemeinschaft for several years. We will describe MOLGEN by presenting short examples taken from chemical education and research. Volume 72 Number 5
May 1995
403
