Quantum Numbers, Sublevels, and Electron Orbital Capacities

Quantum Numbers and the Mathematical Model of Atomic Structure

• Uncertainty Principle and Electron Measurement:

  • The simultaneous determination of an electron's precise location and velocity is physically constrained.

  • A measurement can determine where an electron is located or how fast it is moving, but both properties cannot be known simultaneously.

  • Full arrow representations can take values of $2$, $1$, or $0$.

• Magnetic Quantum Number ($m_l$):

  • The magnetic quantum number ($m_l$) describes the spatial orientation of an orbital.

  • For an angular momentum quantum number value of $l = 3$, the allowed magnetic quantum number values are:     $m_l \text{ values} = \text{-3, -2, -1, 0, 1, 2, 3}$

  • Each of these states corresponds to electron spin values of $+\frac{1}{2}$ or $-\frac{1}{2}$.

• Physical Significance of the Quantum System:

  • This mathematical system fully accounts for and explains the spatial behavior, orbitals, and location of electrons within an atom.

Sublevel Designations and Orbital Counts

• Relabeling Azimuthal Quantum Numbers ($l$):

  • Numeric $l$ values ($0$, $1$, $2$, $3$) are relabeled to correspond directly to atomic sublevels.

  • Sublevels are designated using four primary letters:

  • $s$ sublevel ($l = 0$)

  • $p$ sublevel ($l = 1$)

  • $d$ sublevel ($l = 2$)

  • $f$ sublevel ($l = 3$) — forming the basis of the $f$ block.

• Relationship Between $m_l$ and Orbital Count:

  • The orbital count of a given sublevel corresponds directly to the total number of permissible $m_l$ values available for that sublevel's $l$ value.

Sublevel Orbital and Electron Capacities

• $s$ Sublevel:

  • Azimuthal Quantum Number: $l = 0$

  • $m_l$ Value Options: $1$ option ($m_l = 0$

  • Total Orbital Count: $1$ orbital

  • Maximum Electron Capacity: $2$ electrons

• $p$ Sublevel:

  • Azimuthal Quantum Number: $l = 1$

  • $m_l$ Value Options: $3$ options ($m_l = \text{-1, 0, 1}$

  • Total Orbital Count: $3$ orbitals

  • Maximum Electron Capacity: $6$ electrons

• $d$ Sublevel:

  • Azimuthal Quantum Number: $l = 2$

  • $m_l$ Value Options: $5$ options ($m_l = \text{-2, -1, 0, 1, 2}$

  • Total Orbital Count: $5$ orbitals

• $f$ Sublevel:

  • Azimuthal Quantum Number: $l = 3$

  • $m_l$ Value Options: $7$ options ($m_l = \text{-3, -2, -1, 0, 1, 2, 3}$

  • Total Orbital Count: $7$ orbitals

Electron Spin and Orbital Capacity Principles

• Electron Spin ($s$ or $m_s$):

  • Within any given spatial orbital, electrons can possess one of two spin orientations: positive ($+\frac{1}{2}$) or negative ($-\frac{1}{2}$).

  • Because an orbital can accommodate exactly one positive spin and one negative spin, the total capacity of any single orbital is $2$ electrons.

• Mathematical Calculation of Sublevel Electron Totals:

  • If $1$ orbital holds $2$ electrons, the total electron capacity for a sublevel with $N_{\text{orbitals}}$ orbitals is calculated as:     $\text{Electron Capacity} = 2 \times N_{\text{orbitals}}$

  • For the $s$ sublevel:     $1 \times 2 = 2\,\text{electrons}$

  • For the $p$ sublevel:     $3 \times 2 = 6\,\text{electrons}$

Questions & Discussion

• Question: What are the permissible $m_l$ values when $l = 3$?

  • Answer: The permissible values are $-3$, $-2$, $-1$, $0$, $1$, $2$, and $3$, with spin states of $+\frac{1}{2}$ or $-\frac{1}{2}$.

• Question: How many orbital options exist for an $s$ sublevel ($l = 0$)?

  • Answer: There is $1$ option ($m_l = 0$), meaning the orbital count for $s$ is $1$

• Question: How many orbital options exist for a $p$ sublevel ($l = 1$)?

  • Answer: There are $3$ options inside the sublevel, giving an orbital count of $3$

• Question: How many orbital options exist for a $d$ sublevel and an $f$ sublevel?

  • Answer: The $d$ sublevel has $5$ options, and the $f$ sublevel has $7$ options.

• Question: If $1$ orbital can hold $2$ electrons, how many electrons can $3$ orbitals hold?

  • Answer: $3$ orbitals hold $6$ electrons.