The effective length of the $\pi$ system in the conjugated linear hydrocarbon hexa-1,3,5-triene, C₆H₈, is 7.4 Å. By treating the $\pi$ electrons as particles in a one-dimensional box, calculate the wavelength of the lowest energy absorption in the electronic spectrum of hexa-1,3,5-triene. A) 260 nm B) 1820 nm C) 140 nm D) 202 nm

Question

Quizplus · Accepted Answer

The wavelength of the lowest energy absorption can be calculated using the particle in a box model, where the energy levels are given by $E_n = \frac{n^2h^2}{8mL^2}$, with $n$ being the quantum number, $h$ the Planck's constant, $m$ the mass of the electron, and $L$ the length of the box. For the lowest energy transition ($n = 1$ to $n = 2$), the energy difference ($\Delta E$) corresponds to the energy of the absorbed photon, which can be related to the wavelength ($\lambda$) by the equation $E = \frac{hc}{\lambda}$, where $c$ is the speed of light. Given the effective length of the box (7.4 Å or 7.4 x $10^{-10}$ m), and using the constants for $h$, $c$, and $m$, one can solve for $\lambda$. The calculation yields a wavelength in the UV range, which for hexa-1,3,5-triene corresponds to approximately 260 nm, making option A the correct choice.

The Effective Length of The π\piπ System in the Conjugated Linear Hydrocarbon Hexa-1,3,5-Triene, C6H8, Is 7

The Effective Length of The $\pi$ System in the Conjugated Linear Hydrocarbon Hexa-1,3,5-Triene, C₆H₈, Is 7