Quantum Numbers and Orbitals: MDCAT Chemistry notes
Quantum Numbers and Orbitals for MDCAT: principal, azimuthal, magnetic and spin quantum numbers, allowed values and electron capacities.
From orbits to orbitals
Bohr's model gave the electron a fixed circular orbit with definite position and speed. Heisenberg's uncertainty principle contradicts this, since position and momentum cannot both be known exactly. Schrödinger's wave equation instead gives the probability of finding an electron in a region of space. A region of high probability is an orbital. Solving the equation gives three quantum numbers ($n$, $l$, $m$); the fourth (spin, $s$) was added separately to explain the fine structure of spectral lines.
The four quantum numbers
| Quantum number | Symbol | Allowed values | Tells us |
|---|---|---|---|
| Principal | $n$ | 1, 2, 3, 4 ... (K, L, M, N shells) | Shell; size and energy of orbital, distance from nucleus |
| Azimuthal (angular momentum) | $l$ | 0 to $n-1$ | Shape of orbital and the subshell |
| Magnetic | $m$ | $-l$ to $+l$ including 0 ($2l+1$ values) | Orientation of orbital in space |
| Spin | $s$ | $+\frac{1}{2}$ or $-\frac{1}{2}$ | Direction of electron spin |
Subshells
| $l$ | Subshell | Number of orbitals ($2l+1$) | Max electrons $2(2l+1)$ |
|---|---|---|---|
| 0 | s | 1 | 2 |
| 1 | p | 3 | 6 |
| 2 | d | 5 | 10 |
| 3 | f | 7 | 14 |
So the value of $l$ depends on $n$: for $n=1$ only 1s exists (no 1p); the M shell ($n=3$) has s, p and d; $n=4$ has s, p, d, f. A 4p orbital has $l=1$, whatever its shell.
Key formulas
$$l_{\max}=n-1\qquad \text{orbitals in subshell}=2l+1\qquad \text{electrons in subshell}=2(2l+1)$$
$$\text{orbitals in shell}=n^2\qquad \text{electrons in shell}=2n^2$$
Pauli exclusion principle
No two electrons in an atom can have all four quantum numbers the same. Hence an orbital holds a maximum of two electrons, with opposite spins. The orbital $n=1, l=0, m=0$ is 1s and holds 2 electrons.
Checking a set of quantum numbers
- $n$ must be a positive integer (no negatives).
- $l$ must be less than $n$.
- $|m|$ must not exceed $l$.
- $s$ must be $\pm\frac{1}{2}$.
Worked examples. $n=3, l=2, m=0, s=+\frac{1}{2}$: allowed (a 3d electron). $n=2, l=2$: not allowed, $l$ cannot equal $n$. $n=4, l=3, m=4$: not allowed, $m$ cannot exceed 3. For $l=2$, $m$ has five values: −2, −1, 0, +1, +2.
Counting electrons with $l=2$. Vanadium ($Z=23$) is $[Ar]3d^34s^2$, so 3 electrons have $l=2$.
Shielding and distance
Lower $n$ means closer to the nucleus: 2p is closer than 3s, 3p or 3d. Inner electrons reduce the nuclear pull felt by outer ones; this is the shielding or screening effect.
Common MDCAT traps
- Shape is decided by $l$, not $n$; orientation by $m$.
- Spin is not obtained from Schrödinger's equation.
- $2(2l+1)$ gives electrons in a subshell; $2n^2$ gives electrons in a shell.
- 1p and 2d do not exist.
- Uncertainty principle, not Hund's or Pauli's rule, contradicts Bohr's orbits.
Quick revision
- $l$ ranges from 0 to $n-1$.
- $m$ ranges from $-l$ to $+l$.
- One orbital holds 2 electrons.
- f subshell: 7 orbitals, 14 electrons.
- 2p orbital: $n=2$, $l=1$.