Orbitals and the four quantum numbers

From orbits to orbitals

Bohr gave the electron an orbit, which is a path. The quantum model gives it an orbital, which is a region of space. The two words sound alike, and they mean different things.

In $1926$, Erwin Schrödinger wrote an equation for the electron in an atom. The electron is treated as a wave. We do not need to solve the equation here. What matters is what the solutions look like.

• A solution is a wave function, $\psi$ (psi). It depends on the position in space.
• The equation has solutions only for certain energies. So the energy of the electron is quantised, and this comes out of the mathematics, with no extra assumption.
• The wave function itself has no direct meaning. But $\psi^2$ at a point is the probability density: the probability of finding the electron in a small volume there, divided by that volume.

An orbital is a wave function of one electron in an atom. We often picture it as the region around the nucleus where the electron is found with a high probability, about $90\ \%$. This picture is called the electron cloud.

For hydrogen, the equation can be solved exactly. For atoms with more electrons, the orbitals are close to those of hydrogen, and we keep the same names.

The four quantum numbers

Each orbital is described by three whole numbers. A fourth describes the electron itself.

The principal quantum number $n$ names the shell: $n = 1$ is the K shell, $n = 2$ is L, $n = 3$ is M, and $n = 4$ is N. A larger $n$ means a larger orbital with more energy.

The azimuthal quantum number $l$ names the subshell and its shape. We use letters:

So the subshell with $n = 3$ and $l = 1$ is called $3p$.

The magnetic quantum number $m_l$ tells us how the orbital points in space. For a given $l$, there are $2l + 1$ values, so a subshell has $2l + 1$ orbitals. An s subshell has $1$ orbital, p has $3$, d has $5$ and f has $7$.

The spin quantum number $m_s$. An electron behaves as if it spins, and it has two possible orientations, which we call spin up and spin down. This is not a property of the orbital.

Counting the orbitals

The number of orbitals in shell $n$ is $n^2$, and each orbital holds two electrons, so the shell holds $2n^2$ electrons. You saw this rule in Class 9 and 10. Now you see where it comes from.

Nodes

A node is a place where the probability of finding the electron is zero. There are two kinds.

• Radial nodes are spheres around the nucleus. Their number is $n - l - 1$.
• Angular nodes are planes (or cones) through the nucleus. Their number is $l$.

The total number of nodes is $n - 1$. For example, $2s$ has $2 - 0 - 1 = 1$ radial node and no angular nodes. $3p$ has $1$ radial node and $1$ angular node. $3d$ has no radial node and $2$ angular nodes.

You should find that the shell $n$ holds $n^2$ orbitals in all, and that each subshell has two more orbitals than the one before.

Practise with new numbers

Try it in Python

This program lists every orbital of the first four shells with its quantum numbers.

Remember these

• An orbital is a wave function. $\psi^2$ gives the probability density, and the cloud is the region where the electron is likely to be found.
• $n = 1, 2, 3, \ldots$ and $l = 0$ to $n - 1$ and $m_l = -l$ to $+l$ and $m_s = \pm\tfrac{1}{2}$.
• A subshell has $2l + 1$ orbitals. A shell has $n^2$ orbitals and holds $2n^2$ electrons.
• Radial nodes: $n - l - 1$. Angular nodes: $l$. Total nodes: $n - 1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Which values of $l$ are allowed when $n = 3$?

2. What are the values of $m_l$ for an $f$ subshell?

3. How many angular nodes does a $d$ orbital have?

4. Which subshell has $n = 5$ and $l = 1$?