Nodal planes are regions around atomic nuclei where the likelihood of finding electrons is zero. The coordinates of these planes are found by solving the Schrödinger wave equation for atoms or molecules to find the shape of atomic and molecular orbitals.
The density of the electron cloud that gives the shape of an atomic orbital indicates the likelihood of finding electrons in regions around the atom or molecule. The denser the cloud is, the higher the likelihood of finding an electron. Nodal planes are completely devoid of electron cloud density, indicating that the likelihood of finding electrons in these planes is zero.
A key concept in quantum mechanics is that electrons have both particle and wave properties. The nodal planes around an atomic nucleus are an indication of the wave nature of electrons, as nodes are points of zero amplitude along a standing wave. No amplitude of vibration means there are no electrons present.
The number and position of nodal planes give rise to many of the properties of atoms and molecules. Molecules with few nodal planes are highly stable, whereas those with numerous nodal planes are unstable and reactive. The number of nodal planes is directly proportional with the free energy of a group of bonded atoms.