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.
Answer
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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