Many quantum mechanical models are discussed as part of the undergraduate physical chemistry course to help students understand the connection between eigenvalue expressions and spectroscopy. Typical examples covered include the particle in a box, the harmonic oscillator, the rigid rotor, and the hydrogen atom. This article demonstrates that cyclic polyynes, C[subscript 10], C[subscript 14], and C[subscript 18], effectively illustrate the quantum mechanical model for rotation in two dimensions: the particle on a ring. The exercise presented can be used to facilitate a classroom discussion on the particle-on-a-ring model or it can be used as a homework assignment. In either case, students compare a wavelength calculated from the eigenvalue expression for the particle on a ring to a UV-vis spectrum calculated via molecular modeling software. Spartan is used to perform a density functional theory calculation on the cyclic polyynes. Discussion focuses on how the particle-on-a-ring model works, its limitations, and how the model could be improved. (Contains 1 table and 4 figures.)