Which set of quantum numbers is invalid?

An example of an invalid set of quantum numbers is (n = 2, l = 2, m = 0, s = +1/2). This set is invalid because the angular momentum quantum number (l) cannot be equal to or greater than the principal quantum number (n).

Understanding Quantum Numbers

Quantum numbers are a set of values that describe the unique quantum state of an electron in an atom. They help define the electron's energy, orbital shape, spatial orientation, and spin. There are four primary quantum numbers:

1. Principal Quantum Number (n): Describes the electron's energy level and distance from the nucleus.
2. Angular Momentum (Azimuthal) Quantum Number (l): Describes the shape of the electron's orbital.
3. Magnetic Quantum Number (m_l): Describes the orientation of the orbital in space.
4. Spin Quantum Number (m_s): Describes the intrinsic angular momentum (spin) of the electron.

Rules for Valid Quantum Numbers

For a set of quantum numbers to be valid, they must adhere to specific rules based on their interdependencies:

• Principal Quantum Number (n)

  • Must be a positive integer (1, 2, 3, ...). Higher values indicate higher energy levels and orbitals further from the nucleus.

• Angular Momentum (Azimuthal) Quantum Number (l)

  • Must be an integer ranging from 0 up to (n - 1).
  • This is a critical rule: l cannot be equal to or exceed n.
  • The value of l corresponds to subshells:
    • l = 0 corresponds to an s subshell (spherical shape).
    • l = 1 corresponds to a p subshell (dumbbell shape).
    • l = 2 corresponds to a d subshell (more complex shapes).
    • l = 3 corresponds to an f subshell (even more complex shapes).
      For further reading on orbital shapes, refer to resources like LibreTexts Chemistry on Atomic Orbitals.

• Magnetic Quantum Number (m_l)

  • Must be an integer ranging from -l to +l, including zero.
  • This number indicates the specific orbital within a subshell. For example, if l = 1 (p subshell), m_l can be -1, 0, or +1, representing three different p orbitals (p_x, p_y, p_z).

• Spin Quantum Number (m_s)

  • Can only be +1/2 or -1/2.
  • This describes the two possible intrinsic spin states of an electron.

Identifying an Invalid Set: The Case of (n = 2, l = 2, m = 0, s = +1/2)

Let's examine the given set: (n = 2, l = 2, m = 0, s = +1/2).

1. n = 2: This is a valid principal quantum number (a positive integer).
2. l = 2: This is where the invalidity occurs. According to the rules, l must be less than n (i.e., l < n). In this set, n = 2 and l = 2, meaning l is equal to n. This violates the rule that l must be a value between 0 and (n-1). For n = 2, l can only be 0 or 1.
3. m_l = 0: If l were valid, m_l values from -l to +l would be allowed. For l = 2, m_l = 0 would be permissible. However, since l itself is invalid, the entire set becomes invalid.
4. s = +1/2: This is a valid spin quantum number.

Because the value of l is not allowed to equal or exceed the value of n, the set (n = 2, l = 2, m = 0, s = +1/2) represents an impossible state for an electron and is thus an invalid set of quantum numbers.

Examples of Valid and Invalid Quantum Number Sets

Understanding these rules is crucial for correctly describing electron configurations and orbital characteristics in chemistry and physics.

n	l	m_l	m_s	Validity	Reason for Invalidity
1	0	0	+1/2	Valid
2	1	0	-1/2	Valid
3	2	-2	+1/2	Valid
2	2	0	+1/2	Invalid	l cannot be equal to n (must be < n).
1	1	0	+1/2	Invalid	l cannot be equal to n (must be < n).
3	1	2	-1/2	Invalid	m_l cannot exceed l (must be between -l and +l). For l = 1, m_l can only be -1, 0, or +1.
0	0	0	+1/2	Invalid	n must be a positive integer (cannot be 0).

Practical Insights

These quantum number rules are fundamental to understanding how electrons occupy orbitals in atoms. They underpin principles like the Pauli Exclusion Principle, which states that no two electrons in an atom can have the exact same set of four quantum numbers. Adherence to these rules ensures that atomic models accurately reflect observed chemical and physical properties.