Lecture 15sf

# Lecture 15sf - The energy levels of the hydrogen atom...

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The energy levels of the hydrogen atom depend on the principal quantum number n, and not on any other quantum numbers. E = 2 18 n 10 x 178 . 2 - - J Each level has one or more orbitals associated with it. The n th energy level has n types of orbitals, for a total of n 2 orbital. The first level has one orbital: 1s. The second level has four orbitals: 2s, + 3 orientations of 2p. The third level has nine orbitals: 3s, + 3 orientations of 3p + 5 orientations of 3d. The fourth level has sixteen orbitals: 4s, + 3 orientations of 4p, + 5 orientations of 4d, + 7 orientations of 4f. The energy levels for hydrogen are pictured below:

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With some modifications, it is possible to use the hydrogen orbitals to describe the electrons in other atoms.
The energy level picture for multi-electron atoms is shown below

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There are some differences between multi-electron atoms and hydrogen. The simple formula: E = 2 18 n 10 x 178 . 2 - - J applies only to hydrogen. There is no comparable formula for other atoms. Also, as can be seen in the diagram, the energy level now depends on the type of orbital within a level. There is a splitting of energy between 2s, 2p, and between 3s, 3p,3d. For a given level: Energies: s < p < d < f Electron Configurations An electron configuration is a notation which describes the orbitals occupied by individual electrons in an atom. To place electrons in orbitals, we use several general principles: Aufbau Principle “Aufbau” literally means “build up” in German. This means we first place electrons in the orbitals with the lowest energy continue working our way up to the higher energy levels. Pauli Exclusion Principle This principle states: No two electrons in an atom can have the same set of quantum numbers. The solution to the Schrödinger equation gives orbitals described by three quantum numbers: n, , m . There is a fourth quantum number needed to describe electron configurations. This fourth quantum number is called the spin quantum number, m s , which can have values of either + 2 1 or - 2 1 . This is related to the magnetism inherent in electrons: the electron can have a magnetic moment with two possible values. It can be interpreted as the electron spinning in two opposite directions, although this is not a literal description. Applying the Aufbau Principle and the Pauli Exclusion Principle to the first two elements, hydrogen and helium:
n m m s Configuration H 1 0 0 + 2 1 1s He 1 0 0 - 2 1 1s 2 Helium has two electrons. The first electron goes into the 1s orbital with spin + 2 1 , the second electron goes into the same 1s orbital with spin - 2 1 , which, due to the opposite spin, is not the same set of quantum numbers as the first electron. The Pauli Exclusion Principle thus implies that each orbital can have a maximum capacity of two electrons, filling that orbital with opposite spin. The notation 1s

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Lecture 15sf - The energy levels of the hydrogen atom...

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