Atomic orbitals (s, p, d, f) Knowledge-Based Study

Study notes

1. What is an Atomic Orbital?

   In modern quantum mechanics, electrons do not travel along defined circular paths around the nucleus as proposed in the Bohr model. Instead, their locations are described using probability distributions.

   An atomic orbital is a mathematical function and three-dimensional region of space around the nucleus where there is a high probability (typically around 90–95%) of finding an electron.

   Quantum Numbers Defining Orbitals

   Each atomic orbital is uniquely identified by three quantum numbers:

   • Principal Quantum Number ($n$): Specifies the main energy level or shell ($n = 1, 2, 3, \dots$). As $n$ increases, the orbital size and distance from the nucleus increase.
   • Angular Momentum Quantum Number ($l$): Defines the three-dimensional shape of the orbital, taking integer values from $0$ to $n - 1$. Each value of $l$ corresponds to a specific orbital type:
     • $l = 0$: s orbital
     • $l = 1$: p orbital
     • $l = 2$: d orbital
     • $l = 3$: f orbital
   • Magnetic Quantum Number ($m_l$): Dictates the spatial orientation of the orbital in space, taking integer values from $-l$ through $0$ to $+l$.

   According to the Pauli Exclusion Principle, any single atomic orbital can accommodate a maximum of two electrons, and these electrons must have opposite spins ($m_s = +\frac{1}{2}$ or $-\frac{1}{2}$).

   The s Orbitals ($l = 0$)

   Every principal energy shell contains one s orbital ($m_l = 0$).

   • Shape: An s orbital is spherically symmetrical around the nucleus. This means the probability of locating an electron depends purely on the distance from the nucleus, not the direction.
   • Capacity: Because there is only one s orbital per shell, an s subshell holds a maximum of 2 electrons.
   • Size and Nodes: As the principal quantum number increases ($1s < 2s < 3s$), the orbital increases in size and energy. Higher s orbitals contain spherical radial nodes—regions where the probability of finding an electron drops to zero.

   The p Orbitals ($l = 1$)

   The p subshell appears beginning at the second principal energy level ($n \ge 2$).

   • Shape: p orbitals possess a characteristic dumbbell shape, consisting of two lobes separated by a nodal plane that passes through the nucleus.
   • Orientations: With $l = 1$, the magnetic quantum number can have three values: $m_l = -1, 0, +1$. Consequently, each p subshell consists of three degenerate (equal-energy) orbitals aligned along mutually perpendicular axes: $p_x$, $p_y$, and $p_z$.
   • Capacity: With 3 individual orbitals holding up to 2 electrons each, a p subshell has a maximum capacity of 6 electrons.

   The d Orbitals ($l = 2$)

   The d subshell first appears at the third principal energy level ($n \ge 3$).

   • Orientations and Types: Since $l = 2$, the possible values for $m_l$ are $-2, -1, 0, +1, +2$, giving a total of 5 degenerate orbitals per d subshell:
     • Four orbitals display a four-lobed cloverleaf pattern: $d_{xy}$, $d_{xz}$, $d_{yz}$, and $d_{x^2 - y^2}$.
     • One orbital, $d_{z^2}$, has a unique shape consisting of two lobes aligned along the z-axis surrounded by a central doughnut-shaped torus in the xy-plane.
   • Capacity: With 5 orbitals per subshell, a d subshell can accommodate up to 10 electrons.

   The f Orbitals ($l = 3$)

   The f subshell first appears at the fourth principal energy level ($n \ge 4$).

   • Orientations and Types: For $l = 3$, the allowed values of $m_l$ range from $-3$ to $+3$ ($2l + 1 = 7$). Thus, an f subshell contains 7 degenerate orbitals. These orbitals exhibit complex, multi-lobed geometries with multiple nodal planes.
   • Capacity: Each of the 7 orbitals can hold up to 2 electrons, allowing an f subshell to hold a maximum of 14 electrons.

   Overview of Orbital Subshells

   Understanding the relationships between quantum numbers, orbital shapes, and capacities is essential for writing electron configurations and analyzing chemical bonding.

   • s Subshell: $l = 0$; 1 orbital; maximum 2 electrons; spherical shape.
   • p Subshell: $l = 1$; 3 orbitals; maximum 6 electrons; dumbbell shape.
   • d Subshell: $l = 2$; 5 orbitals; maximum 10 electrons; cloverleaf shape (with $d_{z^2}$ having a torus).
   • f Subshell: $l = 3$; 7 orbitals; maximum 14 electrons; complex multi-lobed shapes.

   General Rules for Any Principal Shell ($n$)

   1. The number of subshells in shell $n$ equals $n$.
   2. The total number of orbitals in shell $n$ equals $n^2$.
   3. The maximum electron capacity of shell $n$ is given by the formula $2n^2$.

   For example, in the third energy level ($n = 3$), there are $3^2 = 9$ orbitals (one 3s, three 3p, and five 3d), with a maximum capacity of $2(3)^2 = 18$ electrons.