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Is there a scientific explanation to why p orbitals are shaped like two balloons, etc. I think it has got to do with electron repulsions. Wikipedia says they are 'characterised by unique values of quantum numbers', of which I don't understand.

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The pictures that we typically use to represent orbitals are really just graphs showing the 3-dimensional probability of finding an electron (that occupies the orbital being examined) in space. $\ce{\Psi}$ is the wavefunction that describes the electron with a set of specific quantum numbers. $\ce{\Psi^2}$, gives rise to the electron density distribution (the shape of an orbital) plotted in the graph.

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    $\begingroup$ I'm sorry to be a grave digger, but why does the $\psi^2$ have such unsymmetrical graphs for $l\ge1$? And if possible, could you explain the non mathematical aspect of it? Because i am not yet acquainted with wave equation's advanced math (i have only seen the raw form). From Resnick, I know that the three orbitals (in case of p orbitals)combined together are symmetrical, but why are they indivudally non spherical? $\endgroup$ – FreezingFire Jun 11 '16 at 17:02
  • $\begingroup$ @FreezingFire sounds different from the question above and the answer will require more than a sentence or two. It might be best if you post this as a new question. $\endgroup$ – ron Jun 11 '16 at 18:08
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The orbitals are solutions (also called wave functions) to the (time-independent) Schrödinger equation. The solutions are related to Bessel functions and Legendre polynomials.

The drawing of orbitals as you often find in text books is in a sense the Copenhagen interpretation of quantum mechanics. In this interpretation, the wave functions (actually, the square of the wave function) are seen as probability distributions of where to find the electrons with 90 % probability (the cutoff might be different from 90 %).

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I like to think of an atom as a 3-dimensional version of a vibrating string. If you pluck a guitar string while lightly touching the exact center point, it will vibrate in the first harmonic - make a sound one octave higher (twice the frequency). That vibrational mode is kind of like a 2p orbital in that there is zero amplitude at the exact center. You get the same behavior in 2 dimensions--and more complex vibrational modes--with a vibrating drum head. Put some iron filings on a drum head and then use different frequencies of sound to stimulate vibration. Some of the vibrational modes look a lot like pictures of the orbitals.

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