NOVEMBEK 5, 1920] 



SCIENCE 



435 



electrons are at tlie ends of their orbits their 

 distance from the center is 0.432 X 10"^ cm. 

 In Bohr's model for the hydrogen molecule 

 the radius of the orbit of the elctrons is 0.953 

 ffl,, or 0.506 X 10"^ cm. In the new model the 

 distance of the nuclei from the center is 

 0.232 X 10-8 (;iji_ .^yjjiig in Bohr's model this 

 distance is 0.292 X 10-^ em. The moment of 

 inertia of the molecule about its center is 

 thus 1.78 X 10-ti g. cm.2 for the new model 

 while Bohr's model gives 2.81 X"*^ S- cm.^ 

 From the theory of band spectra which has 

 recently been developed by Lenz, Heurlinger 

 and others it is possible to calculate the 

 moment of inertia of the hydrogen molecule 

 from certain relationships between lines of the 

 secondary spectrum of hydrogen which were 

 found by Pulcher and Croze. Thus Sommer- 

 feld^ calculates that the moment of inertia 

 of the hydrogen molecule is 1.85 X 10"^^. 

 This value agrees within four per cent, 

 with that calculated from the new model 

 (1.78 X 10-") while Bohr's model gives a 

 value 52 per cent, too high. 



It is of interest to enquire if there are not 

 other simple models for the hydrogen molecule 

 which are consistent with the known chemical 

 facts regarding the remarkable stability of a 

 pair of electrons in molecules. Sommerfeld 

 has modified Bohr's original theory of atomic 

 structure by considering elliptical as well as 

 circular orbits. A two-quantum orbit of an 

 electron in an atom may have both quanta in 

 the form of angular momentmn (circular 

 orbit), or there may be one quantum of 

 angular and one of radical momentum (ellip- 

 tical oi-bit). Both quanta can not be in 

 the form of radial momentum, for the el- 

 lipse would then degenerate into a straight 

 line which would pass through the nucleus 

 and this would lead to infinite velocities for 

 the electron. This reason for the exclusion 

 of orbits having only radial quanta fails for 

 the case of a molecule in which there is no 

 nucleus at the center. We should therefore 

 consider models for the hydrogen molecule in 



1 "Atombau und Speetrallinien," p. 561, 2d edi- 

 tion, soon to be published. 



which the two electrons oscillate in and out 

 along a straight line passing through the 

 center of the molecule, and perpendicular to 

 the line joining the two nuclei. The re- 

 pulsion of the electrons for each other would 

 prevent them from reaching the center. We 

 assume of course that the two electrons are 

 coupled together by some quantimi relation- 

 ship, in such a way that they are always at 

 equal distances from the center. If the elec- 

 trons are at their greatest distance from the 

 center they are more strongly attracted by 

 the two nuclei than they are repelled from 

 each other and they therefore fall in towards 

 the center. When they get close to the center 

 the repulsion increases rapidly and finally 

 causes the electrons to rebound to their 

 original positions. When the electrons are 

 far apart there is a net repulsive force be- 

 tween the nuclei, but when the electrons are 

 close together the attractive force on the 

 nuclei predominates. The length of the path 

 traveled by the electrons must be so related 

 to the distance between the nuclei that the 

 time averages of the repulsive and attractive 

 forces acting on the nuclei must be equal. 



By a series of approximations, based wholly 

 on the classical mechanics, the following 

 results have been calculated. If we take h. 

 the distance between the center of the mole- 

 cule and the nuclei as unity, the maximum 

 distance reached by the electrons (from the 

 center) is 3.710, while the minimum distance 

 within which they approach the center is 

 0.1644. The electrons attain their greatest 

 velocity when they are at a distance of 0.5773 

 from the center, and if they continued to 

 move with this velocity they would travel a 

 distance 8.989 in the time that it actually 

 takes to move from the position of nearest 

 approach to the point in the orbit furthest 

 from the center. The total energy W of the 

 molecule according to this model is 0.8124 

 W^fflg/S where W„ and a„ have the same mean- 

 ings as before. 



In the absence of definite knowledge as to 

 how to apply the quantum theory to this 

 model we may calculate the absolute dimen- 

 sions from the heat of dissociation. Taking 



