directly. Figure 3 illustrates the input and output of MATHCAD based on the data described above. In contrast t o the application of eq 8 or 9, which are difficult to derive, it is instructive to calculate the pH by setting up and solving simultaneously a set of equations hased on eauilibria. mass balance, and charge balance. This procedure iequires as many independent equations as unknowns. The simultaneous equations can be solved by a variety of commercial software packages, such as Eureka2. Figure 4 shows the input required by Eureka for the specific problem described above. However, because there are four roots to these equations, finding the proper root can be time consuming. his hunting process can be accelerated by inserting as many known constraints as possible and by inserting good initial guesses for the unknowns. The solution is not easily obtained by EUREKA because EUREKA requires tedious manipulations such as specifying the accuracy, iterating, changing the values of the variables, checking for convergence, and starting anew if convergence does not occur. Students may be shocked t o see an initial solution that assiens 9.99999999e999! to a variable. The sikultaneous-equation method, in spite of the userunfriendliness, has the capability to solve a wide variety of complicated chemical equilibrium problems, involving polyprotic acids and bases, solubility, complex formation, etc. using the simultaneous-equation method in the classroom can serve as an excellent forum for discussion and for developing an understanding of the chemical balance in chemical systems. By so doing, students will develop an appreciation for the approximation methods that permeate the textbooks today.
Calculating Percent Boundary Surfaces for Hydrogenlike S Orbitals-An Undergraduate Computer Assignment Lealle J. Schwartr St. John Flsher College Rochester, NY 14618 An important topic in an undergraduate quantum chemistry course is the bydrogenlike wave functions (orbitals). There are many ways to represent these functions (11-15)electron density dot diagrams, contour maps, percent boundary surfaces, radial functions (R.l(r) orR,~(r)~plotted vs. r), radial distribution functions (rZR,l(r)2 plotted vs. r), polar plots, and pseudo-three-dimensional plots-and i t is useful for students to become familiar with them all. Several articles in this Journal (16) have described how to generate some of these lots via computer and have discussed the pedagogical t,enefits that acciue when students run the proerams themselves. This article describes a student programk i n g exercise t o calculate percent boundary surfaces for s orbitals. ~
Method of Solutlon The method is illustrated using the hydrogen atom. The total probability that the electron will be found hetween the nucleus and r = ma0 for an ns orbital is
From tables of radial functions and integrals, the following results are easily calculated:
- (2m2+ Zm + 1)exp(-2m)
n = 1,ls orbital: P,,, =1
+ m2/2 + m + 1)exp(-m)
n = 2,2s orbital: P,,, = 1- (m4/8
- (8mV6561-
n = 3,3s orbital:P,,, =1
-~
~
~
504
Journal of Chemical Education
8m5/729
Results The results are listed in the table. Cross sections of the 1s and 2s surfaces are plotted in Figure 5 as an illustration. The following features are discernible.
Percent Boundary Surfaces for Hydrogen Atom S Orbltals~Tolal Probablllty vs. Dlstance lrom the Nucleus m (distance in units of
1s orbltal
Probability
Bohr radii)
0.10 0.20 0.30 0.40 0.50
0.5511 0.7676 0.9569 1.1425 1.3371
0.60
1.5526
0.70
1.8079 2.1395 2.6612
0.80 0.90
2s orbital
0.10 0.20 0.30 0.40 0.50 0.60
0.70 0.80
.
2 Eureka: The Solver is a trademark of Borland International, Inc. Copyright 1986 Borland International.
(15)
Eauations 14.15. and 16 are transcendental with no aleebraiE solutions. ~ h & are e many search methods that can'be used to numericallv solve for m: the most efficient is bisection (17). A straightforward strategy that also works fine is to simply calculate P, for an increasing series of m values until the desired P,,, is overshot, and then to backtrack to the previous m, reduce the increment in m, and continue the m series until the calculated P,,, is within some specified distance of the desired P,.
~
The A~slsnrnent For a hydrogenlike atom, calculate the radii of spheres centered on the nucleus within which the total probability of findine the is 10%.. 20%.. . . .90% for a 1s. a 29, and a - - ~ ~ ~ electron ~ - ~ 3s orbital. Plot x-y cross sections of the surfaces superimposed on the same graph. Hint: Calculate an expression for the probability that an electron will be found within a distance ma0 (a0 is the Bohr radius) from the nucleus. Set the probability expression equal to 0.10, and solve for m. Repeat the calculation for other values of the probability.
(14)
0.90
3.3701 4.1591 4.7366 5.2660 5.7988 6.3664 7.0239 7.8545 9.1254
mao
(distancein picorneters) 29.16 40.62 50.64 60.46 70.76 82.17 95.67 113.22 140.82 178.34 220.09 250.76 278.77 306.66 337.00 371.69 415.64
482.90
5.3724 284.30 10.3511 547.76 11.6020 613.95 668.56 12.6339 13.6223 720.86 14.6532 775.41 15.8152 836.90 0.80 17.2624 913.49 0.90 19.4392 1028.66 FOI comparison, the 2s node oocurs at m = 2.0000 ~ o h radii r (105.84pml, and the two 3s nodes OOCW at m = 1.9019 Bahr radii (100.64pm) and rn = 7.0981 eohr radii 35 orbital
(375.62 pm).
0.10
0.20 0.30 0.40 0.50 0.60 0.70
Distillation and the Macintosh: PT Nomograph, An "In-Lab" Utility R. Slmon and T. Seneca13
University of Colorado Boulder, CO 80309 With the advent of such programs as Wisconsin's WIMP, ChemIntosh, and Chemdraw, access t o convenient, easy drawing of chemical structures has now become indispensable to the organic chemist. We wish to report the development of a program (desk accessory for the Macintosh) that moves the microcomputer directly into the laboratory: P T Nomograph (v.2.0). General Features of PT Nomograph (v.2.0)
Figure 5. Percent boundary surfaces for hydrogenatom i s and 2s wbite.1~.The surfaces enclose 10%. 20%. . . . .90% of the electron density. Planar cross seclions are shown. The marks the nucleus. The solid line in the upper right b n d corner gives Me scale, and represmts one Bohr radius (53 pm). 1s surfaces -, 29 surfaces 2s node..
+
--
-.
.....
1. The 90% boundary surface for the 1s orbital is contained wholly within the 10% boundary surface of the 2s orbital; the 90% boundary surface of the 2s orbital is contained within the 20% boundary surface of the 3s orbital. Thus, the three orbitals concentrate the electron in very different regions of space, as the shell model of course imnlies. 2. The distances between successive surfaces (i.e., the surface soacines) in the 2s orbital are about 3 times as laree as thosein the-1s orbital, while the spacings in the 3s orbit2 are rouehly 5 times as laree. Further, the spacines follow the s&e pattern in each orbital, first decreasingand then increasing. It is enlightening to discuss these patterns with reference to plots of the radial distribution functions. 3. The 2s node occurs within the 2s 10%boundary surface. The inner 3s node lies verv close to the 2s node and is within the1090 boundary surfaces of both the 2s and the 3s orbitals. The outer 3s node is within the 20% boundarv surface of the 3s orbital. More than 80% of the electron density in the 3s orbital is beyond the second node. I t is very hard to get a feeling for these features from plots of the radial distribution functions alone. 4. Taking the 90% boundary surface to be a measure of atomic size implies that the radius of a hydrogen atom in an excited 2s state is almost 3l/2 times as-large as that of a ground state hydrogen atom, while the radius of a hydrogen atom in an excited 3s state is more than 7 times as large. The 90% 1s boundary surface radius is 141pm, compared to the hydrogen single bond covalent radius of 37.3 pm (18), the hydrogen van der Waals radius of 120 pm (la), and the hydrogen (H-) crystal ionic radius of 154 pm (18). Comment
This assignment has been very successful in generating excitement amone the students and even effort beyond what is strictly required. I credit much of their interestto the fact that the students create the percent boundary surfaces themselves from start to finish-no theory from "outside the scope of the course", no distraction by fancy numerical methods, and no "canned" programs from out of the blue.
,
This Droeram duplicates the eeneric ~ressure-temoerature nomograph, fo&d in a variety of so&es, for the determination of the boiling point of a liauid under reduced pressure (19).No longe;is one requiredto use paper nomographs with mlers and pencils and "eyeing" it; no more "where did I put that Nomograph?" or "Dam, ripped the piece of paper its on.. .". Now this valuable tool for the brganic chemist is a fingertip's reach away. The accuracy of this version of the pressure-temperature nomograph is comparable t o that of the conventional nomograph; however, i t is easier to obtain more precise values. As before, this method should be used as a guideline for the
-0
P I homograph rnrn
Corrected to 760 rnm
Observed
mm)1 Observed Corrected
Pressure
Calculate ... About PT Nomograph ...
Figure 6. PT Nomograph using Method 1: a line is hand drawn with lhe mouse.
determination of the boiling point of a liquid under reduced pressure, not as an absolute measurement. A rough estimation of the obtained boiling point is i 5 "C. P T Nomograph can be used t o calculate either the boiling point a t atmospheric pressure for a compound, the boiling point a t reduced pressure for a compound or the reduced pressure as long as the other twovalues are known. This desk accessory may be used in two ways: Method 1:acompound which boils a t 100 OC a t 0.1 mm Hg can be calculated to boil a t 322 OC a t atmospheric pressur ?. Using this method, one clicks with the mouse on 100 ("C) on the Observed line. Holding the mouse button down, the cursor is dragged to the 0.1 mark on the Pressure line (in mm Hg). The windows a t the bottom of the screen print (Fig. 6) 410 Cody Dr..
Lakewood. CO 80226.
Volume 67
Number 6
June 1990
505
