Learning Lab

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Quantum numbers are used to describe orbitals.

• Principal quantum number (\(n)\): It describes the energy of the electron. This can only be positive integers, such as \(n=1,2,3\) and so on.
• Orbital quantum number (Azimuthal quantum number) (\(\ell)\): This gives 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
  \(\ell=1=p\) orbital: dumb-bell shape.
  \(\ell=2=d\) orbital: four-leaf clovers
• Magnetic quantum number (\(m_{\ell}\)) : \(m_{\ell}=0,\pm1,\pm2,......\pm\ell\) (integers between \(-\ell\) and \(+\ell\)). This indicates the number of orbitals in the subshell and their orientation in space.
  \(m_{\ell}=0=\) one \(s\) orbital,
  \(m_{\ell}=0,\pm1=\) three \(p\) orbitals,
  \(m_{\ell}=0,\pm1,\pm2=\) five \(d\) orbitals.
• Spin quantum number (\(m_{s}\)) : \(m_{s}=+\frac{1}{2}\) or \(-\frac{1}{2}\), often represented as \(\uparrow\) or \(\downarrow\), to indicate spin up or down. Electrons are often designated as arrows in orbital “boxes”. 

For an electron \(n=2\). What are the possible values for \(\ell\) and \(m_{\ell}\)?
Answer:
For \(\ell\) permitted values are \(0,1,2,3\ldots\left(n-1\right)\). If \(n=2\), \(\ell\) can be \(0,1\).

\(\ell=0\) indicates \(s\) orbital. \(\ell=1\) indicates \(p\) orbital.

For \(\ell=0,\:m_{\ell}=0\) only: this defines one orbital. In this case, the \(2s\)-orbital.
For \(\ell=1,\:m_{\ell}=0,+1\) or \(-1\) only: this defines three separate orbitals. In this case, the three \(2p\) orbitals.