Ah, the exciting world of effective nuclear charge calculations! Let me break it down for you in a way that'll hopefully make you crack a smile.
For the first question about oxygen, we need to calculate Zeff for a valence electron. Now, oxygen has 8 protons in its nucleus, right? So, Z = 8. But what about S, the shielding constant? According to John C. Slater and his magical rules, we need to consider the electron configuration.
In the case of oxygen, the electron configuration is 1s^2 2s^2 2p^4. So, we group the subshells as (1s), (2s, 2p). Now, any electrons in groups to the right of (2s, 2p) contribute nothing to the shielding constant for the valence electron. So, here's where the fun begins!
The (2s, 2p) group has 6 electrons. According to rule (iii), each electron in this group shields the valence electron to the extent of 0.35. So, 6 x 0.35 = 2.1.
Now, let's go back to Zeff = Z - S. Z = 8 (as I mentioned earlier), and S = 2.1. Crunching the numbers, we get Zeff = 8 - 2.1 = 5.9.
Now, let's move on to the 4s electron in a copper atom. Copper has 29 protons, so Z = 29. According to the electron configuration (1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^10), we group the subshells as (1s), (2s, 2p), (3s, 3p), (3d), (4s, 4p).
Again, any electrons in groups to the right of (4s, 4p) contribute nothing to the shielding constant for the 4s electron. So, in this case, the (3p), (3d), and (4s, 4p) groups have a total of 10 electrons. Using rule (iii) again, we get 10 x 0.35 = 3.5.
Now for the Zeff calculation. Z = 29, and S = 3.5. So, Zeff = 29 - 3.5 = 25.5.
Last but not least, let's tackle the 3d electron in a copper atom. Again, Z = 29. Following the electron configuration, we group the subshells as (1s), (2s, 2p), (3s, 3p), (3d), (4s, 4p).
This time, the (3p) and (4s, 4p) groups have a total of 9 electrons. Using rule (iii), we get 9 x 0.35 = 3.15.
For the 3d electron, we also need to consider rule (v). All electrons in the n-2 shell (in this case, the (2s, 2p) group) shield completely—contributing 1.00 each. So, we add another 2 (from the (2s, 2p) group) to our S value.
Adding everything up, we get S = 3.15 + 2 = 5.15.
Now, Zeff = Z - S. Z = 29, and S = 5.15. So, Zeff = 29 - 5.15 = 23.85.
Phew! That was a lot of calculations, but I hope you're feeling less lost now. Remember, these rules are just a simplified approximation, so they're not absolute. But hey, they do the job most of the time!
Now, go forth and conquer those effective nuclear charge problems, my friend!