[yiyo]

I have several questions about atomic orbital....Hope you can answer them....

1. Is the statement 'the number of nodal surfaces determines the orbital energy of a given atomic orbital' correct? If yes, why is it correct?

2. I don't understand the statement 'occupation of orbitals of higher energy (eg. 4s) can result in reduction of the repulsions between electrons that would occur if the lower energy orbital (3d) were occupied'. It's because even though electrons will be put into 4s orbital first, but then the extra electrons will be put into 3d orbital, then are there any repulsion?

3. Why is the energy of 4s lower than 3d, in K and Ca? But from Sc to Ga onwards, the energy of 4s is higher than 3d?

[magician4]

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


good luck

Ingo

[yiyo]

but does nodal surface mean angular node? or nodal surface means total number of nodes? And how can we compare the orbital energy in many electrons atom?

[magician4]

but does nodal surface mean angular node? or nodal surface means total number of nodes?

everytime the algebraic sign of an orbital lobe inverts, you've got one type of node or another (point node, node plane..)

for example , consider the sinus function in between -pi and + pi

its negative in the interval - pi to 0, then positive from 0 to  pi

square ( i.e. psi times psi* , to be more precise) of this function (as a 3-dimensional rotation body) will describe the electron density, and result in approx. a p-orbital  (the "real" p-orbital has no left-right "limits" but comes from and goes to infinity), with a node at 0 (i.e. where the nucleus is : in the gravity center of the electron density (though almost all of the density is elsewhere !)[hence, this electron can't "fall further" into the nucleus: it's already there, as much as it can])
... and the former algebraic sign of the sinus function will become the orientation + or - of the lobes.

picture: genesis of the 2p orbital

(from: link )

the functions of the even higher orbitals might be a bit more complicated, but in principle it's the same there, too: everytime the psi-function crosses (or hits) the y=0 line, there's a node in the resulting orbital


regards

Ingo

[Enthalpy]

Question 3: the energy of 4p, not 4s, is higher than 3d. Hence 4p begins filling at gallium.

If you sort 3p < 4s < 3d < 4p you get almost the proper sequence. It works for K, Ca, and Ga. Most transition elements there have 4s², but a few have 4s¹ with one 3d more.

This is for isolated atoms, a rare exception. Properties of normal solid elements (here metals, with electrons shared by all atoms) should not be inferred from lone atoms.

As 3d has nearly the energy of 4s, the shell filling reorganizes according to chemical bonds, so these transition elements have multiple valences, and their properties don't change as radically with the atomic number as for other elements.

[yiyo]

Question 3: the energy of 4p, not 4s, is higher than 3d. Hence 4p begins filling at gallium.

If you sort 3p < 4s < 3d < 4p you get almost the proper sequence. It works for K, Ca, and Ga. Most transition elements there have 4s², but a few have 4s¹ with one 3d more.

This is for isolated atoms, a rare exception. Properties of normal solid elements (here metals, with electrons shared by all atoms) should not be inferred from lone atoms.

As 3d has nearly the energy of 4s, the shell filling reorganizes according to chemical bonds, so these transition elements have multiple valences, and their properties don't change as radically with the atomic number as for other elements.

Is this picture still correct? If I use this picture to answer question 3, I know 3p < 4s < 3d < 4p works for K and Ca, as K and Ca have only one to two electrons, the electrons will only fulfil the 4s orbtial first. But starting from Sc, electrons will occupy 3d orbital first, then 4s orbital and so on.... As the electron configuration of Sc is [Ar]4s²3d¹, therefore, from this configuration, we can see that 3d orbital is in lower energy than that of 4s.
Am I correct?

[magician4]

Is this picture still correct?

even with isolated atoms in the gas phase:
no, it isn't (and hence never had been, as mother nature to the best of my knowledge never has changed her mind about this item): Nb, Mo, Tc, Ru, Rh, Pd, Ag, Ir, Pt, Au, Gd, Cu , Cr , La , all elements from Ac to Np as as Cm are examples of falsification for this rubbish.
how many counter examples does it need, before we are going to drop this ?

regards

Ingo

[Enthalpy]

With the first line of transition elements, the present topic, it works rather well: two exceptions for ten elements - not bad for chemistry rules. Almost, as I said. With shell energies little separated, the filling sequence can't be simple. But is works at Sc: 4s is full, 3d is not.

Electron repulsion is integrated in computer models of atoms, which then make correct predictions, with no added hypothesis.

[magician4]

two exceptions for ten elements - not bad for chemistry rules

with all due respect:
if we'd talk like "general directions", "a rule of thumbs" : I'd agree
but this scheme claims to be derived form basic principles and insights in physics, from a deep understanding of underlying rules.
nevertheless, 2 out of 10 : misses
... and in a well-picked selection at that (take a look at my list)
that's tantamount to nothing less but a complete disaster, in my opinion

Electron repulsion is integrated in computer models of atoms (...)

what type of calculations are we talking about ?
to the best of my knowledge, calculations using quantumelectrodynamic equations incl. relativistic Hamiltonians can be handled by computers since about the late 90th - and those will give you results, I agree.
however, they will not give you those schemes we're talking about: they mostly contradict the inherent "logic" of those schemes.
... and they don't deal with electrons orbiting nucleii in a planet like fashion, along the way repulsing each other every now an then, if memory serves.

of course, there's been models around for longer times that use semi-empirical approaches to one degree or another, plugging Bohr radii "onto" their models, "estimating" repulsive effects or introducing energy differences (mostly from measured spectra), neglecting one part of the equations or another, for simplification purposes: there's a lot of [itex]\approx [/itex] in those equations  (even with SCF : you "guess" φ2(r₂) [i don't know how to put that "vector" ontop of the r here, but I take it you know what I mean ] )

if you mean that by "computing": yeah, you're right here , too. Of course my 'puter will show me repulsive terms in those solutions: after all, I did deliberately introduce those right from the start into the very "structure" to be calculated, didn't I ?


regards

Ingo

[Enthalpy]

To me, the interesting part is that several shells offer nearly the same energy to electrons, so these elements can change their valence easily and have properties that vary not so brutally with the atomic number.

The irregular filling of shells results from the same energy proximity; it's normal that shells won't fill regularly then. Whether 8/10 (first series of transition elements) is good or 2/10 is bad under such conditions, I won't challenge that.

Some computer models of atoms and solids (the ones that solve wavefunctions, not the ones that make weighed sums of standard waveforms) do include the electrostatic repulsion between electrons. This is computed for each combination of positions of the electrons, not jut for pairs of shells.

[magician4]

Some computer models of atoms and solids (the ones that solve wavefunctions, not the ones that make weighed sums of standard waveforms) do include the electrostatic repulsion between electrons. This is computed for each combination of positions of the electrons, not jut for pairs of shells.

can't you see what sort of logical/scientific  hell you're running into, even when only you  try just to write about this ?
how could a wavefunction possibly have a "defined , point-type place", let alone electrostatic "repulsion " inbetween two such "places" and "objects" located at ?

yes, physicists at times do calculations based on those false assumptions (though they know better), claiming that the results are often close enough to reality - but much easier to calculate for with those approaches.
this might be a valid argument, if you're interested in rough numbers instead of understanding underlying principles.
nevertheless, even with the results of those calculations being good enough for many purposes, this still is no proof positive that there are "objects moving about in orbits whilst spinning like planets and all that.." for real.

quantummechanics / quantumelectrodynamics is NOT simple orbital mechanics / kinetics "in disguise": we know better like next to a hundred years by now.
It's about time to include  this knowledge in the education of our students, don't you agree?
... even if knowing about those old ideas, maybe even knowing how to work with them, could be a pathway to a deeper understanding of reality. Knowing about just another collection of rules of thumb which interestingly often are usefull doesn't hurt, I'll give you that.

Just let's not proclaim that they're the holy grail


regards

Ingo

[Enthalpy]

Quantum mechanics has not suppressed electrostatic repulsion. And be reassured: QM has been taught for some time. I learnt it at the engineering school and the University, and meanwhile my beard is getting white, so it's not new.

When you learned the hydrogen atom, remember the electrostatic potential of the electron computed as a function of its distance to the nucleus? This energy is computed for each possible position of the electron, after what a wavefunction solution can be computed - as an algebraic expression in this one case.

Well, people do the same with two electrons and even a few more, and then with a computer. They take all positions of the electron pair (or electron set) and compute the electrostatic energy, plus the effect of the nucleus, and then they solve the wavefunction for the electron pair, which is a single psi function of six coordinates.

It then allows to tell that when one electron is detected in a small volume of the atom (say, by using high energy particles as a probe) then the other electron is probably not near the first one, which is just sensible.

Once this psi function of several electrons is numerically computed, one can reduce the information into the probabilty density of finding an electron around some place in the atom, but this step loses information: the correlation between both electrons.

This is just a reasonable model of what happens in atoms. For instance in singlet oxygen, the energy of the two outmost electrons is 94kJ/mol higher because of electron repulsion and nothing else.
http://en.wikipedia.org/wiki/Singlet_oxygen#Orbital_states
Same orbital as triplet oxygen, but the electrons are closer to another as a mean value.

[magician4]

You keep on telling me that I should consider electrons in the vicinity of a nucleus as point type objects, with, for example, a well defined distance to the nucleus ("(...) computed as a function of its distance to the nucleus (...)")
Additionally, you still seem to understand Heisenberg's uncertainty principle as a kind of  "it's out there somewhere, we just can't detect it precisely" ("(...) to tell that when one electron is detected in a small volume of the atom (...) ")

I have to live with other people holding opinions of their own, might their beards be white or not: I will not give you negative mole snacks for just holding a different opinion and defending your ground

I'm simply holding a completely different opinion, and I am deeply convinced that I have the whole of modern physics behind me to proof it

regards

Ingo

[Enthalpy]

Sorry Ingo...

What I explain is exactly the way orbitals hydrogen are computed with QM. Please read again my explanation, please read again the standard QM theory of hydrogen orbitals, which does compute an electrostatic interaction energy for every possible position of the electron.

I do not mistake with newtonian mechanics nor with planet motion. And I regret that you keep bringing in the discussion assertions that I have not done.

"A small volume" is exactly what is meant by a probabilty density. That an electron occupies some volume still allows to detect it within a smaller volume. This is called a "measure". Again, absolutely standard QM. Please read again the notion of measure in any course about QM.

[Enthalpy]

2. [...] 'occupation of orbitals of higher energy (eg. 4s) can result in reduction of the repulsions between electrons that would occur if the lower energy orbital (3d) were occupied'.

"Higher energy" must refer to orbitals when only ONE electron is in the atom's potential. Based on the model for the hydrogen atom, with 3d having a lower energy than 4s, one would expect electrons to fill 3d first, then 4s.

Though, there is repulsion between electrons. Depending on the shapes of the varied orbitals, it can be that electrons on 3d are strongly repelled by electrons already filling 3p or other orbitals - but less strongly if on 4s. If the difference in repulsion is big enough, it can make 4s more favourable than 3d.

Two electrons on 4s also repel an other, so for the second electron on 4s, the advantage over 3d is smaller. As 3d orbitals fill more and more, they may repel electrons on 4s more than they repel electrons on other 3d, since the 3d are well spaced
http://winter.group.shef.ac.uk/orbitron/AOs/3d/index.html
and then occupying the second 4s place would become less favourable than one place at 3d. Cr has 5 electrons on 5 orbitals 3d, but Mn puts its additional electron on 4s² rather than two electrons on one of the 3d orbitals where they would repel an other.
Well, this is not really accessible to qualitative arguments.

Making it no easier, orbitals adapt their shape to external influences as well as to the filling of other orbitals. This happens in the Zeeman effect for instance. To make this a little bit accessible to computation, people use to express deformed orbitals as weighed sums of undeformed ones, but this is an approximation.