Rhodopsin is the pigment in the retina rod cells responsible for vision, and consists of a protein and the co-factor retinal. Retinal is a π-conjugated molecule which absorbs light in the blue-green region of the visible spectrum, where photon absorption represents the first step in the visual process. Absorption of a photon results in retinal undergoing a transition from the ground- or lowest-energy state of the molecule to the first electronic excited state. Therefore, the wavelength of light absorbed by rhodopsin provides a measure of the ground and excited-state energy gap.

a. The absorption spectrum of rhodopsin is centered at roughly 500. nm. What is the difference in energy between the ground and excited state?

b. At a physiological temperature of 37º C, what is the probability of rhodopsin populating the first excited state? How susceptible do you think rhodopsin is to thermal population of the excited state? The partition function will contain only two terms corresponding to the ground state (ε = 0 J) and the excited state (ε = 3.97 × 10⁻¹⁹ J):