Using Quantum Mechanics to describe electrons

Spherical coordinates are more convenient than Cartesian coordinates when describing an atom.

If spherical coordinates are used, the equations that describe the electron distribution around the nucleus can be separated into two parts, a radial distribution function and an angular distribution function.

The angular distribution function:
The square of the angular distribution function describes the probability of finding the electron at angles q and f.
In other words, picture yourself standing on the nucleus, and the electrons look like fog around you. The square of the angular distribution function descibes how thick fog is when you look in different directions.
The Angular distribution function describes the basic shape of the orbital, or the number of lobes in an orbital.
The angular distribution functions depend only on the quantum numbers l and ml. That is, the angular distribution functions of all electrons with the same l and ml values are the same regardless of n or ms. Another way of saying this is that all s orbitals have the same basic shape. For example The 2s orbital (n = 2, l = 0, ml = 0), the 3s (n = 3, l = 0, ml = 0) and the 4s (n = , l = 0, ml = 0) have the same basic shape (spherical)
The radial distribution function:
The square of the radial distribution function describes the probability of finding an a electron a given distance from the nucleus.
In other words, it tells you how far a way the fog starts to tapper off, and it tell you whether or not there are pockets of clear sky in the fog.
The radial distribution function describes how far away from the nucleus the orbital extends and the number of nodes the orbital has.
The radial distribution functions depend on both n and l. This means that the number of nodes an orbital has and how far that orbital extends from the nucleus depends on the principle quantum number or energy level of the orbital (the 1 in 1s, the 2 in 2s, the 3 in 3s, etc.) and the type of orbital (s vs. p vs. d).

Although distribution functions allow us to determine the probability of finding an electron in a certain region of space they do not give us a good picture.To generate pictures of s, p, d, or f orbitals we need to make a limitation. The limitation is that the surface depicted represents the volume in which we should find an electron 90% of the time. (Why?)

 

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