Now, let's look closer at these electron orbitals and their shapes. Remember, we used a two-dimensional plot of the wave function versus x to visualize the standing wave of an electron trapped in one dimension. To visualize the standing waves (or orbitals) of electrons bound to a positively charged nucleus in three dimensions, we will need a four-dimensional plot of the wave function vs. x, y, and z. This can be a bit tricky since our visual perception is limited to three spatial dimensions. So we will need a few tricks to help us visualize the four-dimensional standing waves of the electron in 3 dimension.

The lowest energy orbital of the hydrogen atom.

The second harmonic state.

Remember that at the node, the probability of finding the electron is zero. In general, an orbital with high n (principal quantum number) (e.g. n = 2, 3, 4...) means that the electron will extend out from the nucleus further, and so will be held less tightly than a 1s electron.

When n = 2, we have 2 possible values for ℓ. The first is ℓ = 0, or 2s orbital, which we just discussed above. The second possibility is ℓ = 1 or the 2 p orbital. For a given value of ℓ there are 2 ℓ + 1 possible m_{l} values. So for ℓ = 1, we have m_{ℓ}= -1, 0, +1. These three values of m_{ℓ} correspond to three different p-orbitals.

P-orbitals look like dumbbells along each axis. Instead of a radial node, we have an angular node, which lies along the plane perpendicular to the axis in which the orbital lies. Since the energy, E, of each orbital is a function of only n, then all the n = 2 orbitals (2s, 2p_{x}, 2p_{y}, 2p_{z}) have the same energy.

For the n=3 orbitals the possible quantum numbers are:

n=3 | ℓ=0 | m_{ℓ}=0 | 3s orbital |

n=3 | ℓ=1 | m_{ℓ}=-1, 0, +1 | 3p orbitals |

n=3 | ℓ=2 | m_{ℓ}=-2, -1, 0, +1, +2 | 3d orbitals |

For much nicer three-dimensional renderings of all the atomic orbitals visit Mark Winter's Orbitron site .

Chemisty, The Central Science, 10th Ed.

6.49, 6.51, 6.53, 6.55, 6.57

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