A Simple Calculation to Illustrate the Meaning of Radial Distribution Functions aud 4 r r r V versus r, When first meeting graphs of students sometimes have difficulty in urtdcrstmding that the maximum probability of finding the electron in, say, a hydrogen 1s orbital is some distance out from the nucleus, although the maximum electron density is at the nncleus. The following tutorial exercise has heeu found useful in clarifying this. Imagine, rings of different radii but of constant (very small) width dr cut from sheets of different densities and the same thickness d. 'The volume of each ring of radius r is 2sr dr X d. Since dr and d are the same for all the rings,.the volume of any ring is proportional to its radins. Thus the weight, w, of each ring is proportional to r and density suppose of thet,he ringring material, material p, i.e. densities w = T X are p. so chosen theNow
e-Irn
r
0,
! ! L 0.2
r+
(For hydrogen atoms, Amis pruport,ional that p o:(e-')S. to e-', and the eleotron densit,y as measwed by vb3 is therefore proportional to cCQ'.) R-sr",. r Then, w .-recZT. A table of the values of ecl' and re@' The srtbsequent for r hetween 0 and 2 is theu drawn up. k graphs of e c " and rcc2' versos r appear as shown in the figures. I t can be seeu that although the density of the ring material dewoases as c'' with increasing r and is a maximum when r = O (corresponding to maximum electrun density a t the nncleus), the maximum ring weight (corresponding to the maximum of the radial distribution 0 I 1 function) occurs some distance out from the origin, a t r = 0.5. With this simple calculation as background, it has been found that st,udents more readily nppreciate the significance of $2 and 4sr2q? itnd may go on to plot aet,,~slfnnctions with more confidence and understanding.
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HARRISCOLLEGE PREBTON, ENGLAND
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b
D. J. MARKS
Volume 45, Number 10, October 1968
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637
