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Atomic Orbitals

Orbitals are mathematically derived regions of space with different probability distributions for finding an electron.  There are four different kinds of orbitals, denoted s, p, d, and f each with a different shape. Of the four, the s and p orbitals are the most important to organic and biological chemistry.  The d orbitals are central to organometallic chemistry and are of very importance in biological systems as well.  While f orbitals also exist, they are less commonly encountered in introductory organic and biological chemistry discussions.

Take Action
  • On the applet below, click on the 1s radio button, to show the 1s orbital.
  • Next, click on a 2p to show the 2p orbital.
  • Do you think the electrons are closer to the nucleus in the 2s or the 2p orbital?
Take Note

The electrons are closer to the positively charged nucleus in the 2s orbital than in a 2p orbital. This means that electrons in a 2s orbital are more stable (lower in energy) compared to those in a 2p orbital.

 

s Orbitals

An s orbital is spherical with the nucleus at its center.

Action
  • Successively click on the 1s, 2s, and then 3s radio buttons.  What do you notice about the size of the s orbitals?
  • The probability of finding an electron in an s orbital looks like a sphere having the nucleus as its center.  In two dimensions it looks like a circle.
  • We can say that s-orbitals are "spherically symmetric" having the probability of finding the electron at a given distance equal in all the directions.
  • The size of the s orbital increases with the row or shell number (i.e. principal quantum number (n)).   Thus, 4s > 3s> 2s > 1s.

p Orbitals

A p orbital is dumbbell-shaped.

Action
  • Click on one of the 2p orbitals buttons.  Notice the dumbbell shape?

This implies that the electrons are primarily located in specific regions along the axes for each of the three p orbitals.

d Orbitals

Four of the d orbitals are cloverleaf-shaped. The fifth d orbital is shaped like an elongated dumbbell with a doughnut around its middle.  The d orbitals are important in transition metal and organometallic chemistry.

Take Note
  • Orbitals represent a volume of space within which an electron would have a certain probability of being based on particular energy states and atoms
  • s orbitals are centered around the nucleus and therefore the electrons in them are closer to the nucleus.
  • s orbitals are lower in energy (more stable) than p orbitals in the same row.
  • p orbitals have two lobes that project away from the nucleus.  Electrons in p orbitals spend more time further away from the nucleus than s electrons.

 

Understanding Quantum Numbers: Defining Electron States

To fully describe the properties of an electron within an atomic orbital, a set of four quantum numbers is used. These numbers uniquely define the energy, shape, spatial orientation, and spin of an electron.

Principal Quantum Number (n):

  • This number indicates the electron's main energy level or shell.
  • It can take any positive integer value (n = 1, 2, 3, …).
  • Higher values of n correspond to higher energy levels and larger orbitals, meaning the electron is, on average, further from the nucleus. For example, a 2s orbital is larger and higher in energy than a 1s orbital.

Azimuthal (or Angular Momentum) Quantum Number (l):

  • This number describes the shape of the orbital and defines a subshell within an energy level.
  • Its value depends on n, ranging from 0 to n−1.
  • Specific values of l are designated by letters:
    • l=0 corresponds to an s orbital (spherical shape).
    • l=1 corresponds to a p orbital (dumbbell shape).
    • l=2 corresponds to a d orbital (more complex shapes, often cloverleaf).
    • l=3 corresponds to an f orbital (even more complex shapes).

Magnetic Quantum Number (ml):

  • This number describes the orientation of the orbital in space.
  • Its value depends on l, ranging from −l to +l, including zero.
  • For a given l, there are 2l+1 possible ml values, corresponding to the number of orbitals of that shape in a subshell. For example:
    • If l=0 (s orbital), ml=0 (1 orbital).
    • If l=1 (p orbital), ml=−1, 0, +1 (3 orbitals: px, py, pz).
    • If l=2 (d orbital), ml=−2, −1, 0, +1, +2 (5 orbitals).

Spin Quantum Number (ms):

  • This number describes the intrinsic angular momentum of an electron, often referred to as its "spin."
  • Electrons behave as if they are spinning, generating a magnetic field. There are only two possible spin orientations: ms=+1/2 (spin up) and ms=−1/2 (spin down). This is crucial for understanding how electrons fill orbitals (e.g., the Pauli Exclusion Principle).