Question #964eb

I got #Z_(eff) = 13.8# for the #3s# electron of manganese. If you want the #4s#, you'll have to go through the calculation again and include the #3s,3p,3d# electrons.

That changes the whole calculation. We then look at shielding via the #1s,2s,2p,3s,3p,3d# electrons, and via the one other #4s# electron.

#S(4s) = 2 cdot overbrace(1.00)^(1s) + 2 cdot overbrace(1.00)^(2s) + 6 cdot overbrace(1.00)^(2p) + 2 cdot overbrace(0.85)^(3s) + 6 cdot overbrace(0.85)^(3p) + 5 cdot overbrace(0.85)^(3d) + 1 cdot overbrace(0.35)^(4s) = 21.4#

And from that,

#color(blue)(Z_(eff)) = 25 - 21.4 = color(blue)(3.6)#

For this we refer to Slater's rules for calculating #Z_(eff)#:

#Z_(eff) = Z - S#

where #Z# is the atomic number and #S# is the shielding constant. It's in #S# where all the little 'rules' come in.

We partition the orbitals as follows, in the so-called "Slater ordering of atomic orbitals":

#(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(5f)(6s,6p)(cdots)#

Their contributions to #S# are as follows for a particular electron. Based on the value of #n# for the orbital that holds the shielding electrons:

s and p electron

#S = 0.35# if the electron is in an orbital of the same #n#. For a #1s# electron shielding another one, it is #0.30#.
#S = 0.85# if the electron is in an outer core orbital with #n-1#.
#S = 1.00# if the electron is in a deep core orbital (#n - 2# or below).

d and f electron

#S = 0.35# if the electron is in a #d# or #f# orbital of the same #n#.
#S = 1.00# if the electron is an #s# or #p# orbital of the same #n#.
#S = 1.00# if the electron is in a deep core orbital (#n - 1# or below).

We have a #3s# electron in #"Mn"# of electron configuration of #[Ar] 3d^5 4s^2#, so its contributions are from the #1s, 2s, 2p,3s,3p# electrons.

I don't know why you would want the #3s# electron, but here you go...

#S = 2 cdot overbrace(1.00)^(1s) + 2 cdot overbrace(0.85)^(2s) + 6 cdot overbrace(0.85)^(2p) + 1 cdot overbrace(0.35)^(3s) + 6 cdot overbrace(0.35)^(3p)#

#= 11.25#

And so, for the #3s# electron...

#color(blue)(Z_(eff)) = 25 - 11.25 = color(blue)(13.8)#