How does the shape of an orbital get determined?

The shape of an orbital is determined by the quantum numbers associated with an electron in an atom.

In more detail, the shape of an orbital is fundamentally determined by the solutions to the Schrödinger equation for the hydrogen atom. This equation describes the behaviour of quantum systems and its solutions give the possible energy levels and spatial distributions of an electron in an atom, which we interpret as the shape of the orbital.

The solutions to the Schrödinger equation are characterised by three quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m). Each of these quantum numbers provides a different piece of information about the electron's state.

The principal quantum number (n) determines the energy level of the electron and the size of the orbital. It can take any positive integer value. The larger the value of n, the higher the energy level and the larger the orbital.

The azimuthal quantum number (l) determines the shape of the orbital. It can take any integer value from 0 to n-1. For a given value of n, l determines the number of nodes in the orbital - regions where there is zero probability of finding the electron. The different shapes associated with different values of l are usually referred to as s, p, d, and f orbitals.

The magnetic quantum number (m) determines the orientation of the orbital in space. It can take any integer value from -l to +l. For a given value of l, different values of m correspond to different orientations of the same shape in space.

In summary, the shape of an orbital is determined by the quantum numbers associated with an electron in an atom, which are solutions to the Schrödinger equation. The principal quantum number determines the size of the orbital, the azimuthal quantum number determines the shape, and the magnetic quantum number determines the orientation.