Quantum Numbers, Electron Configuration, and Periodic Trends: Study Notes for Introductory Chemistry

Quantum Numbers and Electron Configuration

Quantum Numbers

Quantum numbers are used to describe the unique quantum state of an electron in an atom. There are four quantum numbers:

• Principal quantum number (n): Indicates the main energy level or shell. Values: n = 1, 2, 3, ...

• Angular momentum quantum number (l): Indicates the subshell or shape of the orbital. Values: l = 0 to (n-1).

• Magnetic quantum number (m): Indicates the orientation of the orbital. Values: m = -l to +l (including zero).

• Spin quantum number (s): Indicates the spin of the electron. Values: s = +1/2 or -1/2.

Allowed Combinations: For a given n, l must be less than n, and m must be between -l and +l.

Subshell Designations

Each subshell is designated by a combination of n and l values. The common subshells are:

• s: l = 0

• p: l = 1

• d: l = 2

• f: l = 3

Some subshell designations do not exist because the value of l cannot be equal to or greater than n.

Periodic Trends and Atomic Structure

Ionization Energy

Ionization energy is the energy required to remove an electron from a gaseous atom or ion. It generally increases from left to right across a period due to increasing nuclear charge and decreases down a group due to increasing atomic radius.

Electron Affinity

Electron affinity is the energy change that occurs when an electron is added to a neutral atom in the gaseous state to form a negative ion. A higher (more negative) electron affinity means the atom more readily accepts an electron.

Core Electrons

Core electrons are the electrons in an atom that are not in the outermost shell (valence shell). They are closer to the nucleus and do not participate in chemical bonding.

Atomic Orbitals and Nodes

5f Orbitals

The 5f orbital is a type of atomic orbital found in the fifth energy level (n = 5) with l = 3. The number of nodes in an orbital is given by:

$\text{Number of nodes} = n - l - 1$

For a 5f orbital: $n = 5$, $l = 3$

$\text{Number of nodes} = 5 - 3 - 1 = 1$

Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in the same atom can have the same set of four quantum numbers. This means that an orbital can hold a maximum of two electrons, and they must have opposite spins.

Electron Configuration

Writing Electron Configurations

Electron configuration describes the arrangement of electrons in an atom's orbitals. The order of filling is based on the Aufbau principle, which fills lower energy orbitals first:

• 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p

There are two common notations:

• Long notation: Lists all occupied orbitals (e.g., 1s2 2s2 2p6 ...).

• Short (noble gas) notation: Uses the previous noble gas in brackets, then continues (e.g., [Ar] 4s2 3d5).

Examples of Electron Configurations

• Manganese (Mn, Z=25): [Ar] 4s2 3d5

• Lead (Pb, Z=82): [Xe] 6s2 4f14 5d10 6p2

• Barium ion (Ba2+, Z=56): [Xe] (since Ba loses its two 6s electrons to form Ba2+)

Subshell Existence Table

The following table summarizes which subshells exist for given principal quantum numbers (n):

n	Allowed l values	Possible Subshells
1	0	1s
2	0, 1	2s, 2p
3	0, 1, 2	3s, 3p, 3d
4	0, 1, 2, 3	4s, 4p, 4d, 4f
5	0, 1, 2, 3, 4	5s, 5p, 5d, 5f, 5g

Additional info: Subshells such as 1p, 2d, 3f, etc., do not exist because l cannot be equal to or greater than n.