Question #3b3d4

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+)`.