Shapes of Orbitals: MDCAT Chemistry notes
Shapes of Orbitals for MDCAT: spherical s orbitals, dumb-bell p orbitals and their orientation, d and f orbitals, nodes and Bohr radius.
What an orbital shape means
According to Schrödinger's wave mechanical model, the exact path of an electron cannot be known; we can only find the probability of finding it in a region of space. An orbital's "shape" is the boundary surface enclosing the region where the probability of finding the electron is high (about 90 to 95%). The shape is fixed by the azimuthal quantum number $l$, the size by $n$, and the orientation by $m$.
s orbitals
- $l=0$, $m=0$: one orbital per shell.
- Spherical and symmetrical around the nucleus; the probability depends only on distance, not direction.
- Size increases with $n$: 1s < 2s < 3s. A 2s orbital has one spherical node (a region of zero probability); a 3s orbital has two. In general the ns orbital has $n-1$ radial nodes.
- For the hydrogen 1s electron the maximum probability of finding the electron lies at 0.529 Å (0.053 nm) from the nucleus, the same as Bohr's radius.
p orbitals
- $l=1$, $m=-1, 0, +1$: three p orbitals in each shell from $n=2$ onward.
- Dumb-bell shaped, with two lobes on opposite sides of the nucleus and a nodal plane through the nucleus where the probability is zero.
- The three orbitals, $p_x$, $p_y$ and $p_z$, are identical in size, shape and energy (degenerate) and differ only in orientation: they lie along the x, y and z axes, at right angles to each other.
- The first element with an electron in a p orbital is boron ($1s^22s^22p^1$); H, He and Li have only s electrons.
d and f orbitals
| Subshell | $l$ | Number of orbitals (orientations) | Shape |
|---|---|---|---|
| s | 0 | 1 | Spherical |
| p | 1 | 3 | Dumb-bell, two lobes |
| d | 2 | 5 | Mostly double dumb-bell (four lobes); $d_{z^2}$ is a dumb-bell with a ring |
| f | 3 | 7 | Complex, multi-lobed |
The number of orientations equals the number of values of $m$, which is $2l+1$: so there are 7 f orbitals, 5 d orbitals and 3 p orbitals.
Worked examples
- Which element among H, He, Li and B has a dumb-bell orbital occupied? Boron, the only one with a 2p electron.
- How do $2p_x$ and $2p_y$ differ? Only in orientation (x-axis vs y-axis).
- How many lobes does a p orbital have? Two.
Common MDCAT traps
- s is spherical; p is dumb-bell; "double dumb-bell" describes most d orbitals.
- $p_x$, $p_y$, $p_z$ differ in orientation, not size, shape or capacity.
- The most probable distance for a hydrogen 1s electron is 0.053 nm, not 0.53 nm.
- The probability idea comes from Schrödinger, not Bohr or de Broglie.
Quick revision
- Shape is set by $l$.
- s orbital: spherical, one per shell.
- p orbital: two lobes, three orientations.
- d: five orbitals; f: seven orbitals.
- Higher $n$ gives larger orbitals.