Energy. Position, x 0 L. Spectroscopy and the Particle-in-a-Box. Introduction

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1 Spectroscopy and the Particle-in-a-Box Introduction The majority of colors that we see result from transitions between electronic states that occur as a result of selective photon absorption. For a molecule to absorb a photon, the energy of the incident photon must match the energy difference between the initial state and some excited state of the molecule. We can describe this concept using the equation Ephoton = hυ = ΔEmolecule = Eupper state - Elower state (1) in which E represents the energy of the photon or molecule being studied, h is Planck's constant, and υ is the frequency. To predict the color of a specific molecule from fundamental physical chemistry principles, one must know the array of possible molecular energy levels (quantized rotational, vibrational, and electronic energy levels). Molecules of a colored object absorb visible-light photons when they are excited from their lowest-energy electronic state (called the ground state ) to a higher-energy electronic state (called an excited state ). In principle, the various electronic states of an atom or molecule may be calculated quantummechanically. In fact, quantum mechanics can be used to predict the allowed set of energy levels for an atom or molecule. For larger molecules, determining these energy levels requires making approximations, and it can still be computationally intensive. (For simpler molecules, free software is available that would allow you to carry out such calculations on your home computer.) However, with a carefully chosen set of molecules, we can study some of the principles of quantum mechanics in the general chemistry laboratory. Some ultraviolet (UV) light and visible light-absorbing molecules are members of a special group for which the simple "particle-in-a-box" quantum-mechanical model applies nicely. If we are willing to make some assumptions, this model can be used to predict the energy levels of electrons responsible for UV or visible wavelength transitions. Imagine that a particle of mass m (in this case, that particle is an electron) travels in one dimension (x) between two walls separated by a distance L. Then assume that the potential energy between these walls (i.e., from 0 x L) is constant, while the potential energy jumps to infinity at the walls. This assumption allows us to draw a simple potential energy diagram like that shown at right. Energy 0 0 L Position, x 1

2 Solving the Schrödinger equation for this simple one-dimensional particle-in-a-box system yields the following allowed energies: E n = n2 h 2 8mL 2 n =1, 2, 3,... (2) where h is Planck's constant, m is the particle mass, and L is the one-dimensional length. Note that the different allowed energies are labeled by the quantum number n which can only take on integer values. If the energy of a particle-in-a-box is measured, these are the only results that will be found no other energies are possible results of a measurement of the energy according to quantum mechanics. These energies can be qualitatively understood by considering the wave-particle duality in quantum mechanics, wherein objects we normally think of as particles, e.g., electrons, in some ways behave as waves and vice-versa. This means that an electron in a box must be described as a wave in quantum mechanics with wavelength λ=h/p=h/mv, where p is the electron momentum, m is mass, and v is speed. This wave is related to the probability of finding the electron in fact the electron probability distribution of positions is the square of the wave amplitude or wavefunction and hence the wave must go to zero outside of the box where the potential energy is infinitely high. That is, the wave describing the electron must fit neatly in the box so that it goes to zero at the edges. This is clearly the case when the length of the box, L, is equal to λ/2, λ, 3λ/2, 2λ, 5λ/2,... Combining this with the expression for λ above and using E=mv 2 /2 gives the allowed energies. Note that this somewhat hand-waving explanation also shows how the energy levels, and their spacings, depend on L and m. As L becomes very large the energies get closer and closer together, eventually becoming continuous (no longer quantum ); this is due to L being much, much larger than the wavelength λ for a particle with a typical energy. Similarly, increasing the mass, m, has the same qualitative effect as making the box larger, which is why you (a particle ) do not notice quantum effects when you sit in a room (a box ), even though your motion is fundamentally described by quantum mechanics. Stated another way, discrete energy level spacing is observed for very low-mass particles confined to small quarters (in this case, an electron within an atom or molecule that gives a small value for ml 2 ). For the molecules considered in this experiment the electronic energy level spacing corresponds to the energy of a visible photon. Specifically, there must be high-energy valence electrons capable of traveling "freely" over the length of the molecule, L. These "free" electrons behave approximately like the particles in a one-dimensional box. To understand the electronic structures of the three compounds you will study in this experiment, begin by considering the bonding between the carbon atoms in simple organic molecules. More complex organic molecules containing alternating single and double bonds, such as the ones used in this experiment, are said to be conjugated. Butadiene is a simple example of a conjugated molecule. The chemical structure of butadiene is shown below in both expanded and bond-line formats. The double bonds involve a p-orbital in each participating carbon atom overlapping 2

3 with the p-orbital(s) of its neighbor(s). The electrons in these double bonds can be considered to be delocalized over all the carbon atoms of the conjugated chain, and thus can be thought of as moving somewhat freely along the length of the chain. Take as an example the smallest molecule used in these experiments and drawn below. There are multiple valid ways to place double bonds (shown below on the left) which combine to make the electrons in the double bonds delocalized (indicated on the right by the dashed line). Viewing each electron in a double bond as a particle-in-a-box is then a fairly crude, but physically reasonable, model of those electrons moving along a chain of carbon atoms. For many compounds, modeling the behavior of these delocalized electrons as quantum mechanical particles-in-a-box is remarkably successful. In this experiment, you will do just that. You will measure the light absorption properties of a carefully chosen set of organic molecules - molecules with different conjugated chain lengths and hence box lengths - and relate your absorption spectra to the particle-in-a-box quantum mechanical model for the electrons. 3

4 Pre-lab Review the instructions for the operation of the OceanOptics Spectrometer. Safety: Goggles must be worn at all times. Organic solvents (e.g., cyclohexane) should be collected in a separate container as waste. Do not uncap or empty the contents of the cuvettes at any point. Pre-lab Assignment: Please write out the following in your lab notebook. This assignment must be completed before the beginning of lab. You will not be allowed to start the experiment until this assignment has been completed and accepted by your TA. 1) Briefly describe the objectives of this experiment. 2) Write out the experimental procedure in your lab notebook according to the Guidelines for Keeping a Laboratory Notebook handout. In addition to these pre-lab requirements, a short quiz will be given at the beginning of lab based on the material in this lab write-up. Procedure Part 1 - Measuring the Spectra for Electrons in Boxes In this experiment, you will carry out absorbance measurements on three conjugated dyes for which the particle-in-a-box theory works very well. The compounds are 1,4-diphenyl-1,3- butadiene; 1,6-diphenyl-1,3,5-hexatriene; and 1,8-diphenyl-1,3,5,7-octatetraene. The chemical structures are shown below: 1,4-diphenyl-1,3-butadiene 1,6-diphenyl-1,3,5-hexatriene 1,8-diphenyl-1,3,5,7-octatetraene 4

5 Obtain pre-filled cuvettes of 1,4-diphenyl-1,3-butadiene; 1,6-diphenyl-1,3,5-hexatriene; and 1,8-diphenyl-1,3,5,7-octatetraene. Each compound has been dissolved in cyclohexane, placed in a capped cuvette, and labelled accordingly with "1,4", "1,6", or "1,8." Please do not uncap or discard the solutions in the pre-filled cuvettes. You will use the OceanOptics spectrometer to acquire spectra of these compounds. Be sure to re-calibrate the instrument. What will you use as a calibration blank? After calibration, adjust the wavelength (x-axis) display range to 345 nm nm. This particular spectrometer is not designed for operation below 345 nm, and we are presently not interested in measuring the absorbance above 445 nm. As you acquire and save each spectrum, be sure to record in your notebook, along with a sketch of the spectrum, the lowest-energy (i.e., longest wavelength) local maximum in the absorbance, or λmax, to help you interpret the spectral data. Part 2 - Interpreting the Data For each spectrum, determine the wavelength (λmax) of the lowest energy peak. Considering that the UV and visible light used in this experiment are forms of electromagnetic radiation, what simple equation can you use to convert from wavelength to photon energy, E? You will need to know specific values for E when you apply the energy expression given in Eq. (2). When calculating the length L of your experimental "box" using the expression for allowed energies, En, in Eq. (2), m is taken to be the mass of an electron and ni (nf) is the initial (final) quantum number for the electronic transition. The values for ni and nf are determined by first counting the number of delocalized electrons associated with the carbon double bonds between the phenyl rings, and then constructing an energy diagram in which energy levels are filled with electron pairs. For example, if a molecule has eight electrons in its double bonds, the electronic configuration can be summarized by the following diagram: n=5 n=4 hυ n=5 n=4 Energy Molecule (ground state) n=3 absorption n=2 n=1 Molecule* (excited state) n=3 n=2 n=1 5

6 After filling the lowest quantum levels, it becomes obvious that the lowest energy electronic transition for this particular system involves excitation of an electron from level ni=4 up in energy to the next level, nf=5. Using the expression for the allowed energies in Eq. (2), we find E molecule = E nf E ni where nf and ni are the final and initial quantum numbers associated with the electronic transition. By substituting an experimental value for ΔEmolecule, it is possible to solve this equation for L, the length of your one-dimensional experimental box made up of the conjugated electron pathway. In this experiment, the box length is taken to the distance between the phenyl rings, as the phenyl rings represent the walls of the box. If we assume that the relevant carboncarbon bonds in a conjugated organic molecule are each nm long, we can easily calculate the theoretical box length for each molecule. You will want to compare each of your experimental box lengths to the theoretical values so that you can obtain a rough estimate of experimental error (% error). If the only energy changes accompanying light absorption were strictly electronic in nature, we would expect absorption spectra to show sharp maxima at the predicted wavelengths. In reality, such "line spectra" are generally observed only for isolated (e.g., gaseous) atoms, whereas substances in the liquid phase show broad absorption bands. Since the experiment is done under conditions that do not yield sharp spectral lines, use the lowest-energy (i.e., longest wavelength) local maximum (peak position), or λmax, in your calculations. Question Sheet E molecule = E 5 E 4 = ( )h 2 8mL 2 You must complete and turn in the question sheet provided to you by your teaching assistant before you leave the laboratory. electronic state Glossary excited state a quantum-mechanical state of an atom or molecule associated with the arrangement of the electrons around the nuclei; each electronic state has a corresponding electronic energy; electronic states are to be distinguished from vibrational or rotational states of molecules any state of a quantum mechanical system such as a molecule which has higher energy than the ground state; there are many excited states, the lowest in energy is called the first excited state, the next lowest the second excited state, etc. 6

7 ground state the lowest energy state of a quantum mechanical system, e.g., a molecule organic molecule a molecule containing carbon; typically only used to describe molecules that do not involve elements other than hydrogen and oxygen in addition to carbon; other molecules are called inorganic molecules quantum mechanics the theory of matter in which objects are described probabilistically in a way that combines both particle- and wave-like characteristics; the description exhibits dramatic differences from classical (Newtonian) mechanics for objects which are small in size and/or mass quantum number an index for labeling the different quantum mechanical states, e.g., electronic states of a molecule, which is usually integer (0, 1, 2,...) or half-integer (1/2, 3/2, 5/2,...) spectrometer, spectrophotometer an apparatus used to obtain a spectrum by measuring the intensity of light transmitted through a sample relative to the intensity of the incident light 7

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