Quantum Mechanical Model of the Atom: Atomic Orbitals and Quantum Numbers

Ch 2.6 The Quantum Mechanical Model of the Atom: Orbitals and the Quantum Numbers

Introduction to Atomic Structure

The quantum mechanical model of the atom describes the arrangement and behavior of electrons within atoms. Understanding where electrons are most likely to be found is essential, as electrons play a key role in chemical reactions. These regions of high probability are called atomic orbitals.

• Nucleus: Contains protons and neutrons; extremely small compared to the overall atom.

• Electrons: Occupy regions around the nucleus, described by orbitals.

• Orbitals: Most probable locations for electrons; crucial for understanding chemical bonding and reactivity.

• Analogy: If the nucleus is the size of a pea at the center of a stadium, the electrons are found somewhere in the stadium, but not in fixed orbits.

How Do We Know What Orbitals Look Like?

The shapes and properties of atomic orbitals are not directly observed but are determined through quantum mechanical calculations. These calculations are based on fundamental principles and equations of quantum mechanics.

• Experimental Evidence: Some properties of orbitals are inferred from experiments (e.g., atomic spectra).

• Calculation: The detailed shapes of orbitals are calculated using the Schrödinger equation.

• Schrödinger Equation: The fundamental equation of quantum mechanics for atoms:

$\hat{H}\Psi = E\Psi$

• Wavefunctions (Ψ): Solutions to the Schrödinger equation; describe the probability distribution of electrons (i.e., orbitals).

• Wave-Particle Duality: Electrons exhibit both wave-like and particle-like properties.

Quantum Numbers

Atomic orbitals are described by a set of three quantum numbers, each providing specific information about the electron's state within an atom.

Principal Quantum Number (n)

• Symbol: n

• Possible Values: Positive integers (1, 2, 3, ...)

• Meaning: Indicates the main energy level (shell) and relative distance from the nucleus.

• Energy: As n increases, the energy and average distance of the electron from the nucleus increase.

• Grouping: Orbitals with the same n are in the same shell (e.g., n = 2 is the second shell).

Angular-Momentum Quantum Number (l)

• Symbol: l

• Possible Values: Integers from 0 up to (n-1) for each value of n.

• Meaning: Determines the shape of the orbital.

• Shapes:

  • l = 0: s orbital (spherical)

  • l = 1: p orbital (dumbbell-shaped)

  • l = 2: d orbital (cloverleaf-shaped)

  • l = 3: f orbital (complex shapes)

• Grouping: Orbitals with the same n and l are in the same subshell.

Magnetic Quantum Number (ml)

• Symbol: ml

• Possible Values: Integers from -l to +l, including zero.

• Meaning: Specifies the orientation of the orbital in space (relative to an external magnetic field).

• Number of Orbitals: For each value of l, there are (2l + 1) possible values of ml, corresponding to the number of orbitals in a subshell.

Summary Table: Quantum Numbers and Subshells

The following table summarizes the allowed combinations of quantum numbers for the first few shells and subshells:

n	l	Subshell
1	0	1s
2	0	2s
2	1	2p
3	0	3s
3	1	3p
3	2	3d
4	0	4s
4	1	4p
4	2	4d
4	3	4f

Subshell notation: s (sharp), p (principal), d (diffuse), f (fundamental)

Allowed Combinations of Quantum Numbers

Not all combinations of quantum numbers are allowed. The following rules must be satisfied:

• n: Positive integer (n = 1, 2, 3, ...)

• l: Integer from 0 to (n-1)

• ml: Integer from -l to +l

Example: For n = 3:

• l = 0 (3s), ml = 0

• l = 1 (3p), ml = -1, 0, +1

• l = 2 (3d), ml = -2, -1, 0, +1, +2

Shapes of Atomic Orbitals

The shape of an orbital is determined by the angular-momentum quantum number (l):

• s orbitals (l = 0): Spherical shape

• p orbitals (l = 1): Dumbbell shape, oriented along x, y, or z axes (px, py, pz)

• d orbitals (l = 2): Cloverleaf shapes, with more complex orientations

• f orbitals (l = 3): Even more complex shapes

Visualization: The images in the notes show the typical shapes of s, p, and d orbitals.

Practice: Valid Quantum Number Sets

To determine if a set of quantum numbers is valid, check the following:

• n must be a positive integer.

• l must be an integer from 0 to (n-1).

• ml must be an integer from -l to +l.

Example: Is n = 2, l = 1, ml = -1 valid?

• n = 2 (valid)

• l = 1 (0 ≤ l ≤ n-1 = 1; valid)

• ml = -1 (−1 ≤ ml ≤ +1; valid)

Therefore, this is a valid set of quantum numbers.

Summary Table: Subshells and Number of Orbitals

Subshell (n, l)	Subshell Notation	Number of Orbitals
n=1, l=0	1s	1
n=2, l=0	2s	1
n=2, l=1	2p	3
n=3, l=0	3s	1
n=3, l=1	3p	3
n=3, l=2	3d	5
n=4, l=0	4s	1
n=4, l=1	4p	3
n=4, l=2	4d	5
n=4, l=3	4f	7

Note: The number of orbitals in a subshell is given by (2l + 1).

Key Takeaways

• Atomic orbitals are regions of high probability for finding electrons, described by quantum numbers.

• The quantum mechanical model uses the Schrödinger equation to calculate orbital shapes and energies.

• Quantum numbers (n, l, ml) define the energy, shape, and orientation of each orbital.

• Understanding allowed quantum number combinations is essential for describing electron configurations.