Question #3b3d4

1 Answer
Mar 12, 2017

The spin-only magnetic moment is most applicable to light-enough transition metals with minimal orbital contribution to muμ 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+).