# Orbitals, Quantum Numbers and Electron Configuration

Chemistry I · Atoms, Bonds and Reaction Rates · https://tryals.app/learn/chemistry-i/orbitals-quantum-numbers-and-electron-configuration

## From Orbits to Orbitals

De Broglie proposed that matter is also wave-like, with $\lambda = h/mv$. Heisenberg then showed that position and momentum cannot both be sharp: $\Delta x \, \Delta p \geq h/4\pi$. Together these destroy the idea of an electron on a definite path. **Schrödinger's equation** replaces the orbit with a wavefunction $\psi$, whose square $|\psi|^2$ gives the probability of finding the electron at a point. An **orbital** is that probability cloud, not a track.

Solving the equation for hydrogen produces exactly three quantum numbers, plus spin:

| Number | Symbol | Allowed values | Meaning |
|---|---|---|---|
| Principal | $n$ | $1, 2, 3, \dots$ | Shell, size and energy |
| Angular momentum | $l$ | $0$ to $n-1$ | Subshell shape (s, p, d, f) |
| Magnetic | $m_l$ | $-l$ to $+l$ | Orientation in space |
| Spin | $m_s$ | $\pm 1/2$ | Intrinsic spin direction |

So a shell $n$ contains $n$ subshells, a subshell $l$ contains $2l+1$ orbitals, and each orbital holds 2 electrons, giving $2n^2$ electrons per shell.

Filling obeys three rules. The **Aufbau principle** fills the lowest-energy orbital first. The **Pauli exclusion principle** forbids two electrons in one atom from sharing all four quantum numbers, capping any orbital at two electrons of opposite spin. **Hund's rule** says degenerate orbitals each take one electron, all with parallel spin, before any of them pairs, because paired electrons repel.

In a hydrogen atom, energy depends on $n$ alone. In every other atom, electron-electron repulsion splits the subshells, so $2s$ lies below $2p$ and, famously, $4s$ fills before $3d$.

> **Common pitfall:** treating an orbital as a path. Nothing orbits. An orbital is a region where the electron is likely to be found, and a $p$ orbital's two lobes are one orbital, not two, the electron is not travelling between them.

## Practice questions

8 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. How many electrons can the shell $n = 3$ hold in total?

**Answer:** 18

**Why:** The $n = 3$ shell contains $3s$, $3p$ and $3d$, 1 + 3 + 5 = 9 orbitals at 2 electrons each, which is $2n^2 = 2 \times 9 = 18$.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/how-many-electrons-can-the-shell-n-3-hold-in-total

### 2. Why does a $p$ subshell contain exactly three orbitals?

A. Because each orbital holds two electrons and six fit in total
B. Because $m_l$ runs from $-l$ to $+l$, giving three values when $l = 1$
C. Because Hund’s rule requires three orbitals before pairing
D. Because there are three axes in space and one orbital per axis by definition

**Answer:** B. Because $m_l$ runs from $-l$ to $+l$, giving three values when $l = 1$

**Why:** A $p$ subshell has $l = 1$, so $m_l$ takes $-1, 0, +1$, three orbitals. The alignment with three axes is a consequence of that count, not the reason for it, and the six-electron capacity follows from it rather than causing it.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/why-does-a-p-subshell-contain-exactly-three-orbitals

### 3. Two electrons in the same atom may share all four quantum numbers if they occupy different shells.

**Answer:** False

**Why:** False, the Pauli principle forbids any two electrons in one atom from having an identical set of all four quantum numbers, with no exception. Electrons in different shells already differ in $n$, so the premise is self-contradicting.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/two-electrons-in-the-same-atom-may-share-all-four-quantum-numbers-if

### 4. Which statements about orbitals are correct?

A. An orbital describes a probability distribution, not a trajectory
B. The two lobes of a p orbital are two separate orbitals
C. Degenerate orbitals fill singly before any pairs form
D. Each orbital can hold at most two electrons

**Answer:** A. An orbital describes a probability distribution, not a trajectory; C. Degenerate orbitals fill singly before any pairs form; D. Each orbital can hold at most two electrons

**Why:** Orbitals are probability clouds, cap at two electrons by Pauli, and fill singly first by Hund. The two lobes of a $p$ orbital belong to one and the same orbital.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/which-statements-about-orbitals-are-correct

### 5. Complete the statement of the three filling rules.

**Answer:** The **Aufbau** principle fills the lowest-energy orbital first, the **Pauli exclusion** principle limits each orbital to two electrons, and **Hund's** rule spreads electrons singly across degenerate orbitals before pairing.

**Why:** Aufbau sets the order, Pauli sets the capacity, and Hund governs how degenerate orbitals share electrons. Heisenberg’s uncertainty principle is unrelated to filling.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/complete-the-statement-of-the-three-filling-rules

### 6. An orbital is defined as a probability distribution $|\psi|^2$ rather than a classical trajectory. What follows from this distinction when interpreting the two lobes of a single $p$ orbital?

A. The electron oscillates rapidly between the lobes along a fixed nodal path
B. They represent one continuous probability region rather than two destinations
C. Each lobe constitutes an independent orbital sharing identical quantum numbers
D. Finding an electron in one lobe excludes its probability density in the other

**Answer:** B. They represent one continuous probability region rather than two destinations

**Why:** Treating lobes as separate destinations resurrects classical trajectories. A wavefunction describes an indivisible, static probability cloud; finding an electron at one coordinate never implies physical transit from another.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/an-orbital-is-defined-as-a-probability-distribution-rather-than

### 7. In a hydrogen atom the $2s$ and $2p$ orbitals have equal energy, but in lithium the $2s$ lies lower. Why?

A. Hund's rule dictates that orbitals of lower angular momentum fill before higher ones
B. The greater nuclear charge in lithium selectively pulls spherical s orbitals inward
C. Electron-electron repulsion splits subshells once more than one electron is present
D. The 2p orbital contains an additional radial node that raises its baseline energy level

**Answer:** C. Electron-electron repulsion splits subshells once more than one electron is present

**Why:** Hydrogen has a single electron, so nothing breaks the degeneracy and energy depends on $n$ alone. Add a second electron and repulsion plus shielding make the more penetrating $2s$ lower than $2p$.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/in-a-hydrogen-atom-the-2s-and-2p-orbitals-have-equal-energy-but-in

### 8. The 4s subshell is filled before the 3d subshell in the neutral atoms of the fourth period.

**Answer:** True

**Why:** True, although $n$ is larger, the $4s$ orbital penetrates more effectively toward the nucleus and lies slightly below $3d$ in the neutral atom, so it fills first. It also ionises first, which surprises many students.

Page: https://tryals.app/practice/chemistry-i/orbitals-quantum-numbers-and-electron-configuration/the-4s-subshell-is-filled-before-the-3d-subshell-in-the-neutral-atoms