Is the statement 'the number of nodal surfaces determines the orbital energy of a given atomic orbital' (...)
more or less, yes
the idea behind the picture was that of a guitar string, let's say the lower E
pick it, and you'll hear a lower E
now, put your finger in the middle of the string (i.e. provoke a node), push, and pull the (remaining) shortended string
the tune you'll hear will be higher, hence of higher frequency, hence of higher energy: E =h*μ
the same was considered to be true for those orbitals electrons were to occupy
however, as as with guitar strings, this is not completely true: if you'd push the E string in a way that it gets a lot of "nodes", the tune resulting thereof might be higher than that of the A sting "unpushed"
(that's why occupying 4s might be of lower energy than 3d , they said)
I don't understand the statement 'occupation of orbitals of higher energy (...)
nor do I anymore
It did make some sense when I started studying chemistry, but by now to me seems nothing but an attempt to "save a doomed modell" ( some old Bohr type ideas, with rotating electrons, spinning all around and moving about)
the original simple scheme for energies - as derived from the model of the H radical - simply doesn't hold up on higher atoms, for starters. Second, there is no such thing as repulsion between electrons
"around" an atom (mostly because there is no such thing as "positively charged particles orbiting the atom's nucleus").
Electrons in the vicinity of a nucleus are - as quantum mechanics teaches us - "standing waves" . the wavelength of these waves is of such a nature , that they can't suffer "disturbing" wave interference (else they would cancel each other out) : that's what all this "repulsion" rubbish is all about.
the solutions of the harmonic equations representing this situation pretty soon become somewhat complex, and don't follow the simple rules for early "occupation schemes" anymore.
Nevertheless, people tried to save their false ideas, declared "exceptions" and "additional principles" (like "inner pair effect"
, "occupation anomalies" and so on), declared differences between their model and reality to be of minor importance ... just like mother nature on a good day was generous, and would pass on making physics rule every once in a while.
now, as you might guess: she is anything but.
fact is : our
model simply is not almighty, hence "false" (or at least incomplete) - and she doesn't care at all
Why is the energy of 4s lower than 3d, (...)
Quantum mechanics - or, if you take relativistic effects into account, too, quantum electrodynamics - seem to provide some mathematical solutions that match reality pretty good.
in case of 4s ./. 3d , this is what the result is there - just like the what mother nature seems to do, too
asides from all I've said above: you'll have to learn it the oldfashioned way in the beginning nevertheless: just "reproduce" their "explanations" , and be done with it
... and don't be too frustrated if things at times simply seem not to add up: chances are, that's because they really
don't add up
you'll learn better in later stages of your studies