Chapter 8, Harder Questions, Answers
Heavy Atoms and Transition Metals
- Is the result of the calculation in question 5 above trustworthy? Why or why not?
The calculation referred to in question 5 is:
Use the simple semiclassical Bohr equation for the velocityvof an electron in an
atom (Eq. 4.12 to calculate a value ofvforZ¼100 and energy leveln¼1:
v¼Ze^2
2 e 0 nh(4.12)
e¼1.602% 10 #^19 C,e 0 ¼8.854% 10 #^12 C^2 N#^1 m#^2 ,h¼6.626% 10 #^34 Js
What fraction of the speed of lightc¼3.0% 108 ms#^1 ) is this value ofv?
Using the “Einstein factor”pffi
(1#v^2 /c^2 ), calculate the mass increase factor that
this corresponds to.
The calculation yieldsv¼2.19% 108 ms#^1. The value ofvis correct for
hydrogenlike atoms (one electron), because for these the Bohr atom is a correct
model, at least mathematically if not conceptually. It should be approximately
right for atoms with more than one electron, because we are consideringn¼1,
an s electron, and the effect of outer-shell electrons on the first shell is not large.
This velocity is 2.19% 108 /3.00% 108 ¼0.73 of the speed of light.
Asvapproaches c, the mass increase factor approaches infinity. Thus the factor
we seek is 1/pffi
(1#v^2 /c^2 )¼1/pffi
(1#0.73^2 )¼1.47. The mass increases by 47%.- Should relativistic effects be stronger for d or for f electrons?
For d electrons. This may seem like a trick question because of the quirky filling
of d and f shells, but there is no reason to doubt that the effect of the nuclear
potential on electron shells increases in the order f, d, p, s. Thus the speed at
which the “orbiting” electrons move increases in that order. - Why are the transition elements all metals?
First, note that by the point in the periodic table where the transition elements are
reached (i.e. byZ¼22, titanium), there still lie several nonmetals beyond:
germanium–krypton (Z¼32–36), tellurium-xenon (Z¼52–54), and astatine
and radon (Z¼85 and 86), thus ten at least (there are a few elements of
ambiguous metallicity which could be included here or omitted; this has no
effect on the argument). So it is not simply that with the first transition element
we have reached the end of the nonmetals, noting that beyond radon all the
elements are essentially metallic. The reasons for this lie more in the realm of
solid-state physics than in conventional “single-atom/single/molecule” chemis-
try, for metallicity is a bulk property: characteristics like electrical conductivity,
lustrousness and malleability are not properties of single atoms or molecules.
Without going into solid-state physics, we content ourselves with the suggestion
that beyond aboutZ¼86, the outer electrons of the atoms in the bulk solid
are not held strongly enough to abstain from merging into a common pool.
652 Answers