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This experiment aims to investigate the quantized nature of molecular electronic states through spectroscopy. The focus will be on studying a homologous series of molecules, examining the variation in electronic energy levels during the experiment. The investigation will involve both experimental and theoretical approaches, with theoretical models used to explore how the electronic absorption energy of molecules changes with size. Additionally, the experiment aims to develop skills in comparing theoretical values obtained from simple models with more complex and robust ones.

The understanding of spectroscopic transitions necessitates the application of quantum mechanics. In this experiment, quantum mechanics will be employed to model the electronic transitional energy of a molecule as it transitions from its ground state to its first excited state. Colored compounds like cyanine and polymethine exhibit absorption in the visible region when excited. The experiment will involve obtaining absorption spectra of various dyes, determining the wavelength of maximum absorption, and using it to calculate the energy difference between the excited and ground states.

The experimental results will be compared with theoretical predictions.

The absorption band in the visible spectrum, corresponding to the transition from a molecular state to an excited electronic state, typically ranges from 170 kJ/mol to 300 kJ/mol above the ground state. Dyes absorbing in the visible spectrum often have weakly bound or delocalized electrons, such as free radicals or π electrons, in conjugated systems. Electronic transitions in polymethine dyes involve electrons along a conjugated chain with alternating double and single carbon bonds, denoted by the nomenclature P(#carbon-carbon bonds).

The wavelength of absorption bands depends on the spacing of electronic energy levels, necessitating knowledge of the associated transitions.

The free-electron model (Kuhn) is a precise model for explaining absorption maxima energy (λmax), assuming π electrons move freely along a conjugated carbon system. The relationship between the length of the conjugated system and λmax will be explored in this laboratory exercise. The absorbance wavelength is determined by the Boltzmann distribution equation, considering the population average of absorbance for both structures. The conjugated chain's length (L) is defined as the shortest chain from nitrogen to nitrogen, and it can be determined for each structure using a known value for a C=C bond. The quantum mechanical solution for the energy level of this model is represented by equation.

The main objective of this experiment is to probe the quantized nature of molecular electronic states using spectroscopy. The focus is on studying a homologous series of molecules and examining the variation in electronic energy levels during the experiment. The investigation includes both experimental and theoretical approaches, with a specific emphasis on the Particle in a Box model for conjugated systems like hexatriene. The experiment aims to compare theoretical values with experimental results obtained through spectrophotometry.

**Theoretical Background:**

The electronic transition from the Highest Occupied Molecular Orbital (HOMO) to the Lowest Unoccupied Molecular Orbital (LUMO) is crucial in understanding molecular spectroscopy. Equation (2) establishes the relationship between the energy of transition and the electron levels involved, where n1 is the HOMO and n2 is the LUMO.

The Particle in a Box model is applied to conjugated systems like hexatriene. For hexatriene, with six carbon atoms and six pi electrons, the S1 to S0 transition corresponds to a change from n=4 to n=3 in the Particle in a Box model (Equation 3). The energy of discrete level En is given by Equation (4), and the energy difference (ΔE) is related to the frequency (ν) by Equation (5).

The number of pi electrons (N) in a conjugated system with p carbon atoms is given by N = p+3, and the length of the conjugated chain (L) is L = (p+3)l, where l is the bond length. The polarizable groups at the end of the chain can lengthen the chain by a factor of α, with α ranging between 0 and 1.

**Experimental Setup:**

- The computer and spectrometer were turned on, and the spectrometer program was opened when the amber light turned green.
- The spectrum range was set from 360 nm to 900 nm.
- Using a 2.5 mL graduated pipette, 1.00 mL of the dye stock solution was dispensed into a 10 mL volumetric flask and diluted to the mark with methanol.
- Visible spectrum readings were taken using plastic disposable cuvettes.
- Aliquots of the solution were diluted until an absorbance reading of one was obtained to form the working concentration.
- The sample was further diluted, and spectra for all dyes were recorded as overlays.

**Calculations:**

- Calculate the number of pi electrons (N) in each dye using N = p + 3.
- Determine the length of the conjugated chain (L) for each dye using L = (p + 3)l.
- If polarizable groups are present, adjust the chain length by multiplying L by the polarizability factor (α).
- Calculate the energy of transition using Equations (2), (3), and (4).

**Discussion:**

- Discuss the relationship between the number of carbon atoms, pi electrons, and the length of the conjugated chain.
- Analyze the impact of polarizable groups on the chain length and transition energy.
- Compare the experimental transition energies obtained from spectrophotometry with the theoretical values calculated using the Particle in a Box model.

This laboratory experiment successfully combined theoretical concepts with experimental techniques to investigate molecular electronic states. The comparison between theoretical predictions and experimental results provides valuable insights into the quantized nature of electronic transitions in conjugated systems.

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