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Quantum Numbers and Electron Orbitals

The Four Quantum Numbers of Electrons

By Todd Helmenstine

Diagram of a P Orbital

P Orbital

ACD Labs
Chemistry is mostly the study of electron interactions between atoms and molecules. Understanding the behavior of the electrons in an atom is an important part of understanding chemical reactions. Early atomic theories used the idea that an atom's electron followed the same rules as a mini solar system where the planets were electrons orbiting a center proton sun. Electric attractive forces are much stronger than gravitational forces, but follow the same basic inverse square rules for distance. Early observations showed the electrons were moving more like a cloud surrounding the nucleus rather than an individual planet. The shape of the cloud, or orbital, depended on the amount of energy, angular momentum and magnetic moment of the individual electron. The properties of an atom's electron configuration are described by four quantum numbers: n, ℓ, m, and s.

The first is the energy level quantum number, n. In an orbit, lower energy orbits are close to the source of attraction. The more energy you give a body in orbit, the further 'out' it goes. If you give the body enough energy, it will leave the system entirely. The same is true for an electron orbital. Higher values of n mean more energy for the electron and the corresponding radius of the electron cloud or orbital is further away from the nucleus. Values of n start at 1 and go up by integer amounts. The higher the value of n, the closer the corresponding energy levels are to each other. If enough energy is added to the electron, it will leave the atom and leave a positive ion behind.

The second quantum number is the angular quantum number, ℓ. Each value of n has multiple values of ℓ ranging in values from 0 to (n-1).This quantum number determines the 'shape' of the electron cloud. In chemistry, there are names for each values of ℓ. The first value, ℓ = 0 called an s orbital. s orbitals are spherical, centered on the nucleus. The second, ℓ = 1 is called a p orbital. p orbitals are usually polar and form a teardrop petal shape with the point towards the nucleus. ℓ = 2 orbital is called a d orbital. These orbitals are similar to the p orbital shape, but with more 'petals' like a clover leaf. They can also have ring shapes around the base of the petals. The next orbital, ℓ=3 is called an f orbital. These orbitals tend to look similar to d orbitals, but with even more 'petals'. Higher values of ℓ have names that follow in alphabetical order.

The third quantum number is the magnetic quantum number, m. These numbers were first discovered in spectroscopy when the gaseous elements were exposed to a magnetic field. The spectral line corresponding to a particular orbit would split into multiple lines when a magnetic field would be introduced across the gas. The number of split lines would be related to the angular quantum number. This relationship shows for every value of ℓ, a corresponding set of values of m ranging from -ℓ to ℓ is found. This number determines the orbital's orientation in space. For example, p orbitals correspond to ℓ=1, can have m values of -1,0,1. This would represent three different orientations in space for the twin petals of the p orbital shape. They are usually defined to be px, py, pz to represent the axes they align with.

The fourth quantum number is the spin quantum number, s. There are only two values for s, +½ and -½. These are also referred to as 'spin up' and 'spin down'. This number is used to explain behavior of individual electrons as if they were spinning in a clockwise or counterclockwise. The important part to orbitals is the fact that each value of m has two electrons and needed a way to distinguish them from one another.

These four numbers, n, ℓ, m and s can be used to describe any electron in a stable atom. Each electron's quantum numbers are unique and cannot be shared by another electron in that atom. This property is called the Pauli Exclusion Principle. A stable atom has as many electrons as it does protons. The rules the electrons follow to orient themselves around their atom are simple once the rules governing the quantum numbers are understood.

For review:

  • n can have whole number values: 1, 2, 3, ...
  • For every value of n, ℓ can have integer values from 0 to (n-1)
  • m can have any whole number value, including zero, from -ℓ to +ℓ
  • s can be either +½ or -½
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