Text page 271 5.4 Applying Quantum Mechanics to Electrons in Atoms Quantum Numbers In an atom, these three-dimensional electron waves or orbitals are more

Text page 271

5.4 Applying Quantum Mechanics to Electrons in Atoms

Quantum NumbersIn an atom, these three-dimensional electron waves or orbitals are more complicatedthan a standing wave on a guitar string. To describe them, quantum mechanics employsthe use of special numbers called quantum numbers Each quantum number specifiessomething different about the orbitals and electrons.

The Principal Quantum Number (n)

When hydrogen’s electron is in the lowest allowed energy state or ground state,then n = 1. Schrödinger’s equation shows us where an electron possessing that amountof energy will most likely be found. When we represent this pictorially, we see an electron“probability density diagram” resembling a spherical cloud

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First Quantum Number (s)

1s orbitalWhen we enclose the cloud in a volume representing about a 90% probability offinding the electron, we call this the 1s orbital. The number “1” represents the principalquantum number, telling us the size of the orbital, and the letter “s” refers to the type or“shape” of the orbital. Figure 5.4.1(a) shows an artistic representation of what a spherical1s orbital might look like if viewed from the outside. Figure 5.4.1(b) is a cross-sectionalview showing radial probability (indicated by greater dot density a slight distance fromnucleus).



Second Quantum Number

l angular momentum

Each new energy level has one new orbital shape in addition to those existing in theprevious level. So if hydrogen’s electron is “excited” and absorbs enough energy to reachthe second allowed energy state, then n = 2 and two orbital shapes or sublevels exist.There is an s orbital with a shape identical to the 1s, except larger, called the 2s orbital.This means that the electron with this greater amount of energy will spend more of itstime farther from the nucleus. There is also a new shape: a p orbital. As n = 2, we call it a2p orbital and it resembles a dumbbell or long balloon pinched in the middle (where thenucleus is located).



Third Quantum Number

ml – Magnetic Quantum Number

The Three NumbersThe three quantum numbers, taken together, will always specify a particular atomicorbital because they tell us all we need to know about that orbital: its size, shape, andorientation in space.

s sublevel

p sublevel

d sublevel



Orbital Shapes for Energy Levels



Orbital Shapes for Energy Levels II

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Fourth Quantum Number

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Pauli Exclusion Principle

Order of sublevel energies:


5.4 Applying Quantum Mechanics to Electrons in AtomsAufbau Principle

The second rule for describing electrons in multi-electron atoms

Order for filling orbitals is given in Figure 5.4.6. Start at lowest energy orbitals and move up.

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5.4 Applying Quantum Mechanics to Electrons in AtomsHund’s Rule

The third rule for describing electrons in multi-electron atoms

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Text page 271 5.4 Applying Quantum Mechanics to Electrons in Atoms Quantum Numbers In an atom, these three-dimensional electron waves or orbitals are more