Edited bv
textbook forum The Rotational Levels and Rotational Partition Function for 1602
GEORGE WlGER California state university Carson. CA SO747
"Centrifugal" Force and the Bohr Model of the Hydrogen Atom Blanca L. Haendler Lafayene Colleue ~astonP . A 180i2
University of Zimbabwe P.O. Box MP 167 Mount Pleasant. Salisbury. Zimbabwe
The influence of nuclear spin on the rotational levels of '602 is treated in a numher of spectroscopic and statistical thermodynamics texts. The purpose of this note is to correct a common error in the treatment which has arisen by authors treating the 2; ground state as if it were 2;: This results in the erroneous statement that only those rotat~onallevels with even values for the nuclear rotation quantum numher can exist. ' can he To a good approximation the total wavefunctiou P written as
where the subscripts refer to translation, nuclear spin, molecular rotation, vibration, electronic orbital, and electronic spin, respectively. Because the nuclear spin of 160is zero it is a hoson, i.e., the total wavefunction must be symmetric to interchange of the nuclei. For all diatomic molecules the are symmetric and can he igwavefunctions $t, $, and nored. The ground electronic state of O2 is gerade and therefore is symmetric to interchange of nuclei. For the overall wavefunction to be symmetric it is therefore necessary for the product $,+, to he symmetric. Because the nuclear spin is zero there is only one nuclear spin wavefunction and it is necessarily symmetric. Therefore the rotational wavefunction must be symmetric to interchange of identical nuclei. Reference to Figure 114(h) of Herzherg1 shows that for a state it is the odd rotational levels which are symmetric; therefore only the odd rotational levels can exist. If the Oz molecule can be treated as a rigid rotor and the coupling between nuclear rotation and electron spin can he ignored, then the rotational energy can be written as
+,
+,
+;
where N is the quantum numher characterizing the nuclear (end over end) rotation. The rotational partition function in this case is
Almost every freshman chemistry text and many physical chemistrv texts contain some treatment of the Bohr model of the hydrogen atom. Some books simply present the postulates of the model and assert that the formulas for the energy and radius of the orbits may he derived from them, hut many texts include a derivation. A large numher of these use misleadine terminology in dealing with the forces operating in the system. The usual treatment is to assert that there is an electrostatic (Coulomb) force of attraction between electron and nucleus which oulls the electron toward the nucleus. This is. of course. correct. It is then stated that this force is exactly bdlanced b; a centripetal force (or centrifugal, the term used varies with the text) which pushes the electron away from the nucleus. This is misleadine, because it imolies that there are two forces acting in the system, and that these forces are in balance. In fact, the only force acting is the electrostatic force, and this force produces a centripetal acceleration. The correct derivations may he found in Bohr's original oaoer'and in standard texts on p h y ~ i c s ~TO , ~say . that there are two forces in balance may cause the reader to conclude that there is no net force on the electron. If this were true, the electron would have no acceleration and would have to move away from the nucleus in a straieht line. hed difficulty discussed here is an example of the problems associated with presenting the difficult concepts of centripetal and centrifugal force which constantly plague teachers of ohvsics. The use of more orecise ohraseolom oresentine the -. in . ~ ~model h of r hydrogenmay help to alleviate this confision for students of chemistry. ~
~
~
~~
~
'
Herzberg. G., "Spectra of Diatomic Molecules." 2nd ed.. D. Van Nostrand Co. Inc.. Princeton, 1950, p. 2 3 8 . 2 5 1 . 'Ballik, E. A., and Ramsay, D. A,, J. Chem. Phys., 31, 1 1 2 8 ,
~~~~~~~~
' Bohr, N., Phil. Mag., 26, l ( 1 9 1 3 ) .
Symon, K. R.. "Mechanics." Addison-Wesley, Reading, Mass., 2nd ed., 1960, p. 9 0 . Resnick, R. and Haliiday. D., "Physics." John Wiley, New York. N.Y., 3rd ed.. 1977. pp. 59-63. This feature is the direct descendent of the "Textbook Errors" column which appeared for many years in T H ~ JounhlaL S under the editorship of N Ifam tnernara! A aecteils? in me number of rcpurtca errors coupleo a m a c o n l l n ~ng nee0 fur u scuss on 01 nlvdern tcxlooors has F O ..F 10 ~rwadonm c SCODP 01 lne Errors co Jmn me lone of lnts formal rnanae is Seen in the n$w title. Within the "Forum" we intend to address a broad range of textbook related topics. Errors will remain as an integral part of the "Forum". In this regard, authors wishing to address text errors should follow the usual guidelines. The error should appear in at least two fndependent textbooks. A list of the texts containino the error should accombany the submission, but the article itself s h o i d be tree of direct reference* to l r w ollcnd ng oouns ! s 1.rthcr s.ggeslad mat a-lnors refer to tlw P ~ F I ' U U S I ~0.0 Snca lnUlCes I J C ~ E M kO .C 44. 156 19671 an0 57. 129.19801.. .~ r i o lro o r ~ o n rnoandsl.nmnnaaman.srt~ot As ntheoasl. the iditor Is more t h i n Allins to discuss ideas for papers relatinu to'textDODL errors r m polenlmi 3blnors ilebono W O l F , r c mleno 10 mco-ruge d WJSF on over R Droad rmqe of w x n m u relaled lop cr Mntlerr s ~ c n dr Ine formal conlenl, aooronc I ECOoe and role uf mc lexfDOor are ooten.. tial t o p i c s . ' an ~ ~ initial stage thi; expansion, the editor welcom& suggestions regarding topics readers would like to present or have presented. Full manuscripts in areas pertinent to the textbook are likewise welcomed. The only restraint is that. in deference totextbook authors, manus c r i p t ~should c o n s ~ i e n t i o ~ s avoid ly presenting direct criticisms or reviews of panicular texts. Let us hope to establish continuing and productive exchange of our ideas about the textbook. ~
The ground state of Oz is 32; and the coupling between electron soin and nuclear rotation cannot he ienored in a DroDer treatment of the rotational energy levels:~f people wish' to treat an examole of a homonuclear diatomic molecule in which altt~rnalpIf~vvIsnrr mis>ing, t h e n poj~ihly;l bettw 1.x~mplr i< ('> w h e r e i t I> n m kntran. t h a t t h e zrnund eIectron~c.htate
~~~
~
~-~
.
(1959).
Volume 58
Number 9
September 1981
719
