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For a lot of years, I had been believing that sphere was the most stable 3-dimensional shape. But after coming across the p,d and f-orbitals, I am unable to comprehend the fact that these orbitals have such crude shapes. Can we prove that these shapes(dumb bell and crossed dumb bell) are the regions in which the electrons would be most stable (No wave functions please. I am only in high school)?

P.S.: An intuitive answer would be much appreciated than a mathematical one.

P.S.S: This is not a duplicate question. I don't want to know how the shapes are like that. I know it is due to the probability functions, I want to know if it is just the rule of nature, or can we show it is because of stability.

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marked as duplicate by M.A.R., bon, Todd Minehardt, Jan, Martin - マーチン Nov 1 '15 at 5:50

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    $\begingroup$ I know you said no wavefunctions, so I really hate to say this, but it has to be said. You absolutely cannot rationalise the shapes of the orbitals without quantum mechanics. In fact, you cannot even rationalise the existence of orbitals without QM, because when you say that electrons can only go into the 1s, 2s, 2p, 3s.... orbitals, you are effectively saying they are only allowed to adopt a specific set of energies. And only QM can predict, and explain, this quantisation of energy. Any classical model of the atom (i.e. using the laws of electromagnetism) is doomed to failure. $\endgroup$ – orthocresol Oct 31 '15 at 17:48
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    $\begingroup$ @ShankRam Stop thinking about the orbitals, or you will invent some wild misconception and instill it in your head forever. I am serious. You are talking of orbitals as if they were real, which is not quite so. You are talking of their shapes as if they had shapes, which is not quite so. You are talking of them being stable, but in fact, the very word "stable" is absolutely inapplicable here. You may have called them "green" or "smiling" as well; that would make just as much sense. $\endgroup$ – Ivan Neretin Oct 31 '15 at 17:56
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    $\begingroup$ Well, the shapes of s, p, d, etc. orbital come about because atoms are spherically symmetric. The spherical symmetry let's you split the wavefunction/orbital into 2 parts, a radial one (which basically just determines how many nodes your orbital has) and an angular one (which determines the shape), which are then multiplied with each other. The angular functions (called spherical harmonics), loosely speaking, correspond to different angular momenta and indeed the spherical shape is the one with the lowest angular momentum and in that sense "the most stable one" (don't take that too literal)... $\endgroup$ – Philipp Oct 31 '15 at 18:05
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    $\begingroup$ ... But, as has already been mentioned by others, quantum mechanics dictates that each energy level/orbital can only be occupied by 2 electrons of different spin (Pauli Exclusion Principle). Thus the electrons of each shell can't all go into the "most stable" spherical s orbital. Only 2 are allowed and the rest has to occupy the other differently-shaped orbitals. The same principle applies to the different shells (those are linked to the above-mentioned radial part of the wavefunction): not all electrons can go into the first and "most stable" shell. $\endgroup$ – Philipp Oct 31 '15 at 18:09
  • $\begingroup$ @orthocresol, I do understand that the orbitals are just plots of the square of probability functions in 3--dimensional space, but I don't get how the electrons are in such an arrangement. I am a little bit familiar with QM (like Arthur-Beiser modern physics book level), but the mathematical analysis is too much for a high school student. If you could interpret the same in an elegant way, I will surely get it. Btw, this is not a duplicate question. I don't want to know how the shapes are like that. I want to know if it is just the rule of nature, or can we show it is because of stability. $\endgroup$ – ShankRam Nov 1 '15 at 7:47
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I'm guess that you have read about orbital shapes in the wikipedia article, or done a google search on the term.

In general, such "orbitals" shown are typically calculated for a lone electron, not any "real" multi-electron configuration.

Think of an orbital like a single loop of cotton fiber wound into a cotton ball. The planetary model of the electron-nucleus pair indicates that the electron is a solid ball following the fiber. The quantum model of the electron-nucleus pair indicates that the electron has no fixed position, but that it is essentially at all places on the loop at the same time.

So how big is the cotton ball in diameter? It has no limiting diameter, it depends on how hard you squeeze! So if I catch a Hydrogen atom with my magic tweezers, there is a finite probability according to the Schrödinger wave equation that its electron can be in orbit even further away than the dwarf planet Pluto, or inside the nucleus! So we we talk of an the "size" of an ion it is somewhat of an artificial abstraction, and the Schrödinger wave equation can't be "the gospel truth." (I don't mean to badly disparage the Schrödinger wave equation for it is very useful.)

Not to leaving you totally confused, the size of an ions does have some real physical significance. For instance consider table salt, $\ce{NaCl}$. This is really $\ce{NaCl}$. Using x-ray diffraction we can measure how close the $\ce{NaCl}$ are, so we know their "size."

Starting with the Aufbau principle for the atoms, chemists can predict the electron structure of an atom, say carbon. Given the structure of bonds in the atoms, say carbon and hydrogen, then chemists can predict how carbon and hydrogen will bond to form molecules. The fact that chemists can make predictions about molecular structure is the "proof" that the models work.

Howver weird things do have with real electron orbitals. For instance if there was a single "normal electron configuration" for chromium, then there would only be one chromium chloride. However chemists have synthesized three.

  • Chromium(II) chloride, also known as chromous chloride.
  • Chromium(III) chloride, also known as chromic chloride or chromium trichloride
  • Chromium(IV) chloride

So there must be weird configurations that are stabilized by some sort of "hybridization." As another example all the C-H bonds in methane, $\ce{CH4}$, are the same because the four carbon orbitals (one 2s, and three 2p) orbitals of the carbon atom hybridize into four $sp^3$ orbitals that are equivalent.


To put such orbital shapes into perspective, a chemist thinks a unicorn is simply a hybrid of a horse and a rhinoceros. So a model helps making predictions, but you can't have a fixated belief that the model is the whole truth.


Another example would be showing you a picture of a blueberry pie. Would just such a picture "explain" a blueberry pie? In order for the picture to have meaning you must have some underlying knowledge.

  • You know in general what a pie is.
  • You have know that pies contain fruit pieces, so this kind is made from blueberries.
  • You know that pies are sweet.
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