The purpose of this lab is to observe how a real system may be approximated using the simple quantum mechanical model of a one-dimensional particle-in-a-box. Due to the arrangement of the electrons in the conjugated dyes, some electrons are free to move along the molecule's carbon-carbon chain like a particle-in-a-box. Therefore, the main concept of this lab is to relate the UV-vis absorption spectrum of a dye to the particle-in-a-box model, and to find which model fits the experimental values best. Model A is to set the edge of the box as the nitrogen atoms, Model B is to extend the box one bond beyond the nitrogen atoms while keeping the number of electrons the same, and Model C is adjusting the length by an empirical correction factor.
Absorption spectra have particular characteristics wherein the wavelength given is representative of the energy absorbed and is related to the electronic structure of the molecules through a simple quantum mechanical model (for a one dimension schema) known as particle in a box. The color of each of the dyes is due inherently to the chromophore, which is a system of conjugated double bonds.1 The structure of the chromophore can be represented as two equivalent resonant hybrid orbitals (depending on the particular pairs of atoms chosen to be connected with pi-bonds/ and on the dye). The molecular orbitals, constructed from the p-orbitals on the C and N atoms, delocalize the pi-electrons along the entire length of the system. The difference in energy of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is small.1 Therefore, the energy of the photon required to promote the electron between the levels is quite low- and falls within the visible range of the spectrum- giving rise to the colors of the dyes.
[...] These values are not far off from the accepted value of 1.4 E-10m, which is the bond length of a carbon-carbon bond in a benzene ring. The percent difference was for the 4,4 series and for the 2-2 series in both models A and B. The objective of this lab was successfully completed as the conjugated dyes were analyzed using UV-vis spectrum and the quantum mechanical model of one dimensional particle-in-a-box. Model B was seen to approximate the experimental values better than Model as the percent error was lower for all [...]
[...] The percent differences for Model A are significantly greater than the percent differences for Model therefore, Model B is deemed to be a better predictor of wavelengths than Model A. Model B could be a better predictor of wavelengths due to the fact that it expands beyond the conjugated system to include the Nitrogen atoms. Model A's predictions of wavelength were lower than the actual wavelengths; Model B's predictions of wavelength were lower than actual for the 2-2' series, while its predictions of wavelength were greater than actual for the 4-4' series. [...]
[...] Model B is as follows: (Model B)1 Table Comparison of Theoretical and Measured Wavelengths Using Models A and B Dye # λ, Model A λ, Model B λ measured % % Difference, Difference, Model A Model B This compares the closeness of the model values with the measured wavelength The equation for length is as follows: The values for N and the number of bonds are the same as before; thus, bond length can easily be determined using this formula. [...]
[...] Table Theoretical Wavelengths for the Conjugated Dyes using both Models A and B Dye # Bond Bond N λmeasured Box Length Model A Model B The theoretical bond length can be calculated by using the relationship among the length of a bond, the experimental length of the box, and the number of bonds. This relationship is , where γ is an empirical correction factor This relationship is linear; where the dependent variable is L and the independent variable is the number of bonds. [...]
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