Quantum numbers are used to describe orbitals. They serve as an "address" for locating an electron within an atom, specifying its energy level, shape, orientation, and spin. Therefore, they provide a detailed understanding of the regions in an atom where electrons are likely to be found, which is crucial for predicting and explaining chemical bonding and molecular geometry.
There are four quantum numbers:
The principal quantum number (\(n)\) describes the energy of the electron. This can only be positive integers, such as \(n=1,\,2,\,3\) and so on.
The orbital quantum number (also known as the Azimuthal quantum number) (\(\ell)\) represents the shape of the subshell electron cloud (orbital shape). \(\ell=0,\,1,\,2,\,.....n-1\) (positive integers less than \(n\)).
\(\ell=0=s\) orbital: spherical shape
\(\ell=0=s\) orbital: spherical shape
\(\ell=2=d\) orbital: four-leaf clovers.
The magnetic quantum number (\(m_{\ell}\)) indicates the number of orbitals in the subshell and their orientation in space. \(m_{\ell}=0,\,\pm1,\,\pm2,\,......\pm\ell\) (integers between \(-\ell\) and \(+\ell\)).
\(m_{\ell}=0=\) one \(s\) orbital
\(m_{\ell}=0,\,\pm1=\) three \(p\) orbitals
\(m_{\ell}=0,\,\pm1,\,\pm2=\) five \(d\) orbitals.
The spin quantum number (\(m_{s}\)) represents the direction of electron spin. These are often shown as arrows in orbital "boxes". \(m_{s}=+\dfrac{1}{2}\) or \(-\dfrac{1}{2}\), often represented as \(\uparrow\) or \(\downarrow\), indicate spin up or down.
Example – determining quantum numbers
For an electron with a quantum number of 2, \(n=2\), what are the possible values for \(\ell\) and \(m_{\ell}\)?
Step 1: Recall what \(\ell\) is and how it is determined.
\(\ell\) is the orbital quantum number. The possible values for \(\ell\) are any positive integers less than \(n\). Therefore, if \(n=2\), then \(\ell=0,\,1\).
Step 2: Recall what \(m_{\ell}\) is and what orbitals they represent.
The \(m_{\ell}\) is \(\ell\). Since \(\ell=0,\,1\):
\(m_{\ell}=0\) for \(\ell=0\). This indicates one orbital. Since \(n=2\), it is one \(2s\) orbital.
\(m_{\ell}=0,\pm1\) for \(\ell=1\). This indicates three separate orbitals (\(m_{\ell}=0,-1,+1\)). Since \(n=2\), it is three \(2p\) orbitals.
