Question #3b3d4

1 Answer
Truong-Son N.
Mar 12, 2017

The spin-only magnetic moment is most applicable to light-enough transition metals with minimal orbital contribution to #mu# that it accurately corresponds to the electronic structure. Then, we would have something like:

#bb(mu_S = 2.00023sqrt(S(S+1)))#

where #S# is the total spin in the system. The electron configurations of these ions are:

#[Ar]3d^9#, #"Cu"^(2+)#
#[Ar]3d^8#, #"Ni"^(2+)#
#[Ar]3d^6#, #"Co"^(3+)#
#[Ar]3d^6#, #"Fe"^(2+)#

Write out the orbital diagrams.

#3d^9#:

#ul(uarr darr)" "ul(uarr darr)" "ul(uarr darr)" "ul(uarr darr)" "ul(uarr color(white)(darr))#

#3d^8#:

#ul(uarr darr)" "ul(uarr darr)" "ul(uarr darr)" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))#

#3d^6#:

#ul(uarr darr)" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))" "ul(uarr color(white)(darr))#

One can clearly see the increasing number of unpaired electrons going from #"Cu"^(2+)# to #"Fe"^(2+)#. #mu_S# for each of these configurations is:

#color(green)(mu_(S,3d^9)) = mu_(S,"Cu"^(2+)) = 2.00023sqrt(1/2(1/2+1)) = color(green)(1.732)#

#color(green)(mu_(S,3d^8)) = mu_(S,"Ni"^(2+)) = 2.00023sqrt(2/2(2/2+1)) = color(green)(2.829)#

#color(green)(mu_(S,3d^6)) = mu_(S,"Co"^(3+),"Fe"^(2+)) = 2.00023sqrt(4/2(4/2+1)) = color(green)(4.900)#

Alternatively, #mu_(S+L)#, the spin-and-orbital magnetic moment, is:

#bb(mu_(S+L) = 2.00023sqrt(S(S+1) + 1/4L(L+1)))#

where #L# is the total orbital quantum number for each unpaired electron. For each of the above configurations, we would have the same increasing pattern with the number of unpaired electrons:

#color(green)(mu_(S+l,"Cu"^(2+))) = 2.00023sqrt(1/2(1/2+1) + 1/4*2(2+1)) = color(green)(3.000)#

#color(green)(mu_(S+l,"Ni"^(2+))) = 2.00023sqrt(2/2(2/2+1) + 1/4*3(3+1)) = color(green)(4.473)#

#color(green)(mu_(S+l,"Co"^(3+),"Fe"^(2+))) = 2.00023sqrt(4/2(4/2+1) + 1/4*2(2+1)) = color(green)(5.478)#

For instance, #L# for #3d^6# is #2+1+0+(-1) = 2#, and for #3d^8# is #2+1 = 3#.

Do note that the observed magnetic moments for each of these cations are #1.7-2.2#, #2.8-4.0#, #~5.4#, and #5.1-5.5#, respectively (Inorganic Chemistry, Miessler et al., pg. 360).

So, #mu_S# has the better agreement for #"Cu"^(2+)# and #"Ni"^(2+)#, but #mu_(S+L)# has better agreement for #"Co"^(3+)# and #"Fe"^(2+)#.

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