Active Learning Classroom – Activity: Orbital Shapes

Background

According to Heisenberg’s uncertainty principle, we cannot measure an electron’s position and velocity (directly related to energy) precisely at the same time.

Δx × mΔv ≥ h/4π (Heisenberg’s uncertainty principle)
As a result, we cannot locate the precise position of an electron of a specified energy, instead we can describe the electron’s position as a probability distribution map showing where the electron is likely to be found – an orbital. The wave function ψ is a mathematical function that describes the wavelike nature of the electron. The square of a wavefunction |ψ|² represents the probability density distribution of the electron. The solutions to Schrӧdinger equation are a set of possible wave functions – corresponding to a set of orbitals.

Hψ = Eψ (Schrӧdinger equation)

Probability density (|ψ|2) distribution of hydrogen 1s orbital.	The 1s orbital surface: a contour surface that encloses 90% of the electron probability.

Each orbital is specified by a set of three interrelated integers – quantum numbers. These quantum numbers, and their constraints, arise from the conditions under which the Schrödinger equation is solved and, thus, the solutions (wavefunctions) themselves.

The principal quantum number (n) determines the overall size and energy of an orbital for the hydrogen atom. The allowed values are: n = 1, 2, 3, …… As hinted in the Bohr model, electrons have discrete energy levels. For the hydrogen atom, the energy of an electron is En = -2.18 × 10^-18 J (1/n²), where n is the principal quantum number of the orbital.

The angular momentum quantum number (l) determines the shape of the orbital. The allowed values are: l = 0, 1, 2, … (n-1). For example, if n = 1, the possible value of l is 0; if n = 2, l = 0, 1. By convention, the values of l are represented by letters s, p, d, f, g, …… Orbitals with l = 0 are called s orbitals, orbitals with l = 1 are called p orbitals, and so on.

The magnetic quantum number (ml) determines the orientation of the orbital. The allowed values are: ml = -l, … 0, … l. For example, if l = 0, m_l = 0; if l = 1, m_l has three possible values -1, 0, 1. In other words, there are 2l +1 possible orbitals with the same value of l.

There is a forth quantum number, the spin quantum number (ms), that specifies the spin of the electron. ms has only two possible values, ½ or –½. An electron either spins up (m_s = ½) or spins down (m_s = - ½).

There are spatial locations in an orbital where the probability density of finding an electron is zero (i.e. |ψ|² = 0), and such a location is called a node. Nodes are classified as either radial nodes or angular nodes.

Related Reading

1. LibreTexts: Chapter 2, section 2.3, Subsection “Understanding Quantum Theory of Electrons in Atoms”.