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Quantum Numbers in Chemistry: A Comprehensive Guide, Exams of Organic Chemistry

The azimuthal, or angular momentum, quantum number, l, defines the type or shape of the orbital where the electron most probably resides. It is the second ...

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Quantum Numbers
Chem12A, Organic Chemistry I
Quantum numbers work like addresses for electrons in an atom. To narrow down your intended
recipient, you would write the state city street street number to tell the post office
where to go. Quantum numbers work the same way: energy level type of orbital
orientation of orbital orientation of electron. Every electron in an atom will have a unique
set of quantum numbers in the form [n, l, ml, ms].
Principal Quantum Number (n = 1, 2, 3, 4, …∞)
The principal quantum number, n, defines the energy level or shell
of the electron. It is the first quantum number that must be
determined, as it defines the available values for all other quantum
numbers. Think of Bohr’s model for the atom; n defines the ring on
which the electron resides.
The value of n tells you how many levels away from the nucleus the electron is. The valence shell
of an atom is the highest occupied shell and is mathematically equivalent to the period of the
element (i.e., H is in period 1, so the valence shell for H is n = 1).
Azimuthal (Angular Momentum) Quantum Number (l = 0, 1, 2, …(n-1) )
The azimuthal, or angular momentum, quantum number, l, defines the type or shape of the
orbital where the electron most probably resides. It is the second quantum number to be
determined as its value relies only on n (i.e., if n = 2, then l = 0 or 1 since n 1 = 1). Each type of
orbital has a unique number of nodes (e.g., where the wave changes phases) which is
mathematically equivalent to l.
If l = 0, the electron is in an s-orbital that has 0 nodes.
If l = 1, the electron is in a p-orbital that has 1 node.
If l = 2, the electron is in a d-orbital that has 2 nodes.
If l = 3, the electron is in an f-orbital that has 3 nodes.
s-orbital
p-orbital
d-orbital
f-orbital
pf2

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Quantum Numbers

Chem12A, Organic Chemistry I

Quantum numbers work like addresses for electrons in an atom. To narrow down your intended recipient, you would write the state  city  street  street number to tell the post office where to go. Quantum numbers work the same way: energy level  type of orbital  orientation of orbital  orientation of electron. Every electron in an atom will have a unique set of quantum numbers in the form [n, l, m l, ms].

Principal Quantum Number ( n = 1, 2, 3, 4, …∞)

The principal quantum number, n , defines the energy level or shell of the electron. It is the first quantum number that must be determined, as it defines the available values for all other quantum numbers. Think of Bohr’s model for the atom; n defines the ring on which the electron resides.

The value of n tells you how many levels away from the nucleus the electron is. The valence shell of an atom is the highest occupied shell and is mathematically equivalent to the period of the element (i.e., H is in period 1, so the valence shell for H is n = 1).

Azimuthal (Angular Momentum) Quantum Number ( l = 0, 1, 2, …( n-1 ) )

The azimuthal, or angular momentum, quantum number, l , defines the type or shape of the orbital where the electron most probably resides. It is the second quantum number to be determined as its value relies only on n (i.e., if n = 2, then l = 0 or 1 since n – 1 = 1). Each type of orbital has a unique number of nodes (e.g., where the wave changes phases) which is mathematically equivalent to l. If l = 0, the electron is in an s-orbital that has 0 nodes. If l = 1, the electron is in a p-orbital that has 1 node. If l = 2, the electron is in a d-orbital that has 2 nodes. If l = 3, the electron is in an f-orbital that has 3 nodes.

s-orbital p-orbital d-orbital f-orbital

Magnetic Quantum Number (ml = - l …0…+ l)

The magnetic quantum number, ml, defines the orientation of the orbital in which the electron

inhabits. The number of ml values is called the degeneracy, or the number of orbitals in the

atom that are at the same energy level. Note that to have the same energy, the orbitals must be on the same energy level as well as be the same type of orbital ( n and l values are equal). A

2s orbital is not degenerate (e.g., the same energy) with a 2p or a 1s orbital.

The ml values are entirely dependent on the l values; each type of orbital has a set degeneracy.

For an s-orbital, ml = 0, and degeneracy = 1. For a p-orbital, ml = -1, 0, +1, and degeneracy = 3. For a d-orbital, ml = -2, -1, 0, +1, +2, and degeneracy = 5. For an f-orbital, ml = -3, -2, -1, 0, +1, +2, +3, and degeneracy = 7.

NOTE: A set of 2p orbitals and 3p orbitals may have the same degeneracy value , but they are NOT degenerate with each other. They both include three total orbitals, but they are not at the same energy: they have different n values.

Spin Quantum Number (ms = - ½, + ½)

The spin quantum number, ms, is completely independent of the values of n, l, and ml. It is always

  • ½ for an electron who has its spin oriented up, and – ½ for an electron who has its spin oriented down. This means that there are only two electrons allowed per orbital, no matter the energy level, shape, or orientation (i.e., a 1s orbital holds 2 electrons, as does 4d orbital).

As of yet, we cannot determine whether an electron is pointed up or down, so it can be assumed always that ms = ± ½

TO SUMMARIZE:

If n is: then l can be: so ml is:

with a

degeneracy

of:

and ms as:

2 01 - 1, 0, 1^013 - -^ ½ , + ½½, + ½