Shapes and energies of orbitals

The shapes of the orbitals

The shape of an orbital is decided by $l$. We cannot draw a cloud exactly, so we draw a surface that holds most of the probability.

• s orbitals ($l = 0$) are spheres. There is one orientation. The probability depends only on the distance from the nucleus, not on the direction.
• p orbitals ($l = 1$) have two lobes, one on each side of the nucleus, like a dumbbell. There are three orientations, along the $x$, $y$ and $z$ axes: $p_x$, $p_y$ and $p_z$. The probability is zero at the nucleus, in a nodal plane between the two lobes.
• d orbitals ($l = 2$) have five orientations. Four of them have four lobes. $d_{xy}$, $d_{xz}$ and $d_{yz}$ have lobes between the axes. $d_{x^2-y^2}$ has lobes on the axes. The fifth, $d_{z^2}$, has two lobes along the $z$ axis and a ring around the middle. A d orbital has two nodal surfaces.
• f orbitals ($l = 3$) have seven orientations and still more complex shapes.

The two signs of the wave function (blue and orange in the picture) matter when orbitals overlap to form bonds. They do not change the probability.

Where is the electron?

The graph shows the probability of finding the electron at a distance $r$ from the nucleus in hydrogen. It counts all the directions together.

• The $1s$ orbital has its peak at one Bohr radius, $52.9\ \text{pm}$. Bohr's first orbit is the place where the electron is most likely to be found. It is not a fixed path.
• A higher $n$ spreads the cloud farther out. $2s$ has its main peak at about $5$ Bohr radii, and $3s$ at about $13$.
• The number of dips to zero is the number of radial nodes, $n - l - 1$. $2s$ has $1$, $3s$ has $2$, $3p$ has $1$, and $3d$ has none.
• An $s$ electron has a small peak close to the nucleus, and a $p$ electron does not. We use this in a moment.

Energy in hydrogen

In an atom with one electron, the energy depends on $n$ only: $E_n = -\dfrac{13.6}{n^2}\ \text{eV}$. The $2s$ and $2p$ orbitals have the same energy. Orbitals with the same energy are called degenerate.

Energy in an atom with more electrons

With more than one electron, the electrons repel each other, and the picture changes. An electron in the outer part of the atom does not feel the full charge $+Ze$ of the nucleus, because the inner electrons shield it. It feels a smaller effective nuclear charge,

where $S$ is the shielding by the other electrons.

Orbitals of the same shell have different shapes, so they are shielded by different amounts. An $s$ electron has a small peak close to the nucleus, so it penetrates the inner shells and feels more of the nuclear charge. A $p$ electron penetrates less, a $d$ electron still less. So in the same shell,

Now $2s$ and $2p$ are no longer degenerate. The energy depends on both $n$ and $l$.

The $(n + l)$ rule

The energy of an orbital rises with the value of $n + l$. If two orbitals have the same $n + l$, the one with the lower $n$ has the lower energy.

Using this rule, the order of increasing energy is

The $4s$ orbital has a lower energy than $3d$ ($n + l = 4$ against $5$), and so it fills first. This is why potassium has its last electron in $4s$, and not in $3d$. The orbitals $2p$ and $3s$ have the same $n + l$, and $2p$ comes first because its $n$ is lower.

You should find that $4s$, with $n + l = 4$, comes before $3d$, with $5$, and that ties go to the smaller $n$.

Practise with new numbers

Try it in Python

This program draws the radial probability of four orbitals of hydrogen and prints the distances of their peaks.

Remember these

• s is a sphere. p has two lobes, with three orientations. d has five orientations, usually four lobes.
• In hydrogen, the energy depends only on $n$. With more electrons, shielding and penetration make $ns < np < nd < nf$.
• The orbitals fill in the order of increasing $n + l$, and for the same $n + l$, the lower $n$ first: $1s\ 2s\ 2p\ 3s\ 3p\ 4s\ 3d\ 4p\ 5s\ \ldots$
• The $1s$ orbital has its peak at the Bohr radius, $52.9\ \text{pm}$.

Check yourself

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

1. In an atom with many electrons, which is lower in energy, $3p$ or $3d$?

2. Put $5p$, $4d$ and $6s$ in the order they fill.

3. What is the shape of an $s$ orbital?

4. How many nodal planes does a $p$ orbital have?