Question

What’s the difference between a p-orbital of the second shell and a p-orbital of the third shell? Thank you!

Answer 1

The number of radial nodes, which are spherical nodal shells.

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Recall that

• the principal quantum number `n` gives the energy level of the orbital.
• the angular momentum quantum number `l` labels the shape of the orbital (`s,p,d,f, . . .`).

Well, the total number of nodes (regions where electrons cannot be found) is given by `n - 1`, and the number of angular nodes (nodal planes or conical nodes) is given by `l`.

By subtraction, the number of radial nodes (spherical nodal shells) is given by:

`overbrace(n - 1)^"Total nodes" = overbrace((n - l - 1))^"Radial nodes" + overbrace(l)^"Angular nodes"`

So having an orbital of one higher `n` increases the number of radial nodes.

`bb2p -> n - l - 1`

`= bb2 - 1 - 1 = ul(bb0 " radial nodes")`

`bb3p -> n - l - 1`

`= bb3 - 1 - 1 = ul(bb1 " radial node")`

And this can be visually seen:

Circled in green is the `3p`'s radial node. They still both have one angular node (nodal plane).

This is also seen in radial density distributions:

The radial node shows up at the point where the graph dips down to `bb(a_0r^2R_(nl)^2(r) = 0)`, with `r > 0` (but not at `r -> oo`).

That shows where the orbital wave function goes to zero (here, close to `6a_0`), which indicates the distance away from the nucleus where electrons cannot be found.