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Chemistry LibreTexts

Atomic Orbitals

An atomic orbital is a mathematical function that describes the behavior of electrons. Electrons act as both waves and particles which is known as wave-particle duality. Relating to this, the Heisenberg uncertainty principle states that there is limited precision when trying to locate an electron. The atomic orbitals give us the probability of finding an electron at a specific region of an atom. More specifically, atomic orbitals can describe the quantum states of an electron in the electron cloud around the atom. The position of an individual electron can be described by the four quantum numbers: n, l, ml, and ms. The Pauli exclusion principle states that electrons cannot possess the same set of quantum numbers.

Quantum Numbers

The positions of the electrons in relation to the nucleus are described by quantum numbers. Each electron has four quantum numbers that apply to its shell, subshell, orbital, and spin.

Principle Quantum Number (\(n\))

Shells: n = 1, 2, 3...

The first quantum number is the principle quantum number, n. This describes the energy level and distance from the nucleus of an electron. The greater this number, the further away from the nucleus the electrons will reside. Electrons in shells with higher values are farther away and will have higher enery and less stability than electrons in lower energy shells. The smallest principle quantum number is 1 and increases by integer increments.

Angular Momentum Quantum Number (\(l\))

Subshells: l = 0, 1, 2 ... (n-1)

The angular momentum quantum number describes the subshell and shape of an orbital. Orbital names include s, p, d, f, g, h... where 0=s, p=1, etc.  The standard periodic table contains elements with s, p, d, and f orbitals. The values of the angular momentum quantum numbers range from 0 to n-1.  For example, a hydrogen atom's electron in the ground state would have a n=1 value and therefore must have an l=0 value. This means that the electron is located in the s subshell within the first orbtal.  The shape of an s orbital is shown below. The next orbital, n=2, contains a s (l=0) and a p (l=1) subshell. The shape of a p orbital is shown below, which can be configured in 3 different orientations in space: along the x, y and z axis.  The next orbital, n=3, contains an s, p, and d (l=2) orbital.  The shape of a d orbital is shown below, which has five different orientations in space.  The orientations in space are determined by the next quantum number.

  s orbital.JPG             p orbital.JPG        d orbital (1).JPG

     An s orbital                   One of the three possible p orbitals            One of the five possible d orbitals

Magnetic Quantum Number (\(m_l\))

ml=...(-l)...-1,0,+1...(+l)

The magenetic quantum number, or orbital, describes the orientation of the orbital in space. That means it describes whether the part of the electron lies mostly on the x,y, or z axis of the three dimensional grid.  The values of the magnetic quantum number are the negative and positive values of l.  As described above, the s subshell has one orientation, since it is spherical and l=0. The p subshell has three orientations, represented by ml= -1, 0, 1. The d subshell has five orbitals represented by ml= -2, -1, 0, 1, 2. 

Magnetic Spin Number (\(m_s\))

ms= -1/2, +1/2

For every possible n and s value combination,  two spins values are possible: a negative spin and the other electron with a positive spin.                                                                                                                                                                                                                                       

References

  1. Timberlake, Karen C. Chemistry : An Introduction to General, Organic, and Biological Chemistry. 10th ed. Prentice Hall Higher Education, 2008.
  2. Petrucci, Ralph H. General Chemistry: Principles and Modern Applications.9th ed. New Jersey: Pearson Education Inc. 2007.

Problems

  1. Write an orbital designation corresponding to the quantum numbers n=3, l=1 and ml=1
  2. Write an orbital designation corresponding to the quantum numbers n=4. l=2 and ml=0
  3. Can an orbital have the quantum numbers n=2, l=2, and ml=2?
  4. For an orbital with n=3 and ml=1, what is (are) the possible value(s) of l?