Spectroscopy and The “Particle in a Box” DEPARTMENT OF CHEMISTRY UNIVERSITY OF KANSAS 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 impinging 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 quantum- mechanically. 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 be computationally intensive (For simpler molecules, free software is available that would allow you to carry out such calculations on your home computer). With a carefully chosen set of molecules, however, 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. This model can be used to predict the energy levels of electrons responsible for UV or visible wavelength transitions — if we are willing to make some assumptions. Imagine that a particle of mass m (in this case, the particle is an electron) travels in one dimension (x) between two walls separated by a distance L. Then assume that the potential energy between 1 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 on page one. Solving the Schrödinger equation for this simple one-dimensional particle in a box system yields the following allowed energies: 2 2 n h En = 2 n = 1,2,3,... 8mL where h is Planck's constant, m is the mass of the particle, and L is the length of the one- dimensional box. 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 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 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. This “wave” is related to the probability of finding the electron — in fact the electron probability distribution of positions is the square of the wavefunction that describes the particle — 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=mv2/2 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 larger than wavelength λ for a particle with a typical energy. Similarly, increasing the mass 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 mL2). 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. 2 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 a very simple organic molecule, ethene (C2H4). A line-bond structure for ethene is shown above, together with a more detailed orbital cartoon. Each carbon atom utilizes sp2 hybrid orbitals to overlap with the 1s orbitals from two hydrogen atoms, while the remaining sp2 hybrid orbital overlaps with an sp2 hybrid orbital from the adjacent carbon atom. This "end-on" overlap is called σ-bonding (“sigma bonding”). This experiment is particularly concerned with a second type of bonding in which the p-orbitals on adjacent carbon atoms overlap. For purposes of electron bookkeeping, each p-orbital can be assumed to contain one electron. The overlap between p-orbitals on the adjacent carbons in ethene is "side-on" and termed a π-bond (“pi bond”). Hence, the double bond between the ethene carbon atoms actually consists of two distinct components, a σ-bond and a π-bond. More complex organic molecules containing alternating single and double bonds are said to be conjugated. The compounds used in this experiment are indeed conjugated, but the chemical structure of butadiene (shown below) is a useful, simpler example of a conjugated molecule. An examination of the π-bonding in butadiene shows that a p-orbital in each carbon atom overlaps with the p-orbital(s) of its neighbor(s). Again, each p-orbital is formally said to contain 1 electron. More extensively conjugated molecules called polyenes have formal structures with alternating single and double bonds along the carbon atom chain. The electrons in the π-bonds of polyenes can be considered to be delocalized over all the atoms of the conjugated chain, and thus can be thought of as moving somewhat freely along the length of the chain. Viewing each electron as a particle in a box is then a fairly crude, but physically reasonable, model of electrons moving along a chain of carbon atoms. For many compounds, it is remarkably successful in modeling the behavior of the quantum mechanical particle in a box. In this experiment, you will measure the light absorption properties of a carefully chosen set of organic molecules and relate the absorption spectra to the particle in a box quantum mechanical model for the electrons. 3 Pre-lab Review the instructions for the operation of the OceanOptics Spectrophotometer. You should have used this instrument in a previous lab and it will be used during this and future laboratory exercises. Thus, the operation of this instrument should be familiar to you. This pre-lab assignment is due at the beginning of lab. You will not be allowed to start the experiment until this assignment has been completed and submitted to your TA. List all of the chemicals you will use for this week's experiment. For each chemical, list specific safety precaution(s) that must be followed. To find specific safety information, please obtain a Materials Safety Data Sheet (MSDS) on the chemical of interest. MSDSs can be found through an internet search (e.g., google) or from the following website: www.hazard.com Read the MSDS and find specific safety concerns for each chemical. Be sure to include the route(s) of entry and the possible acute and chronic effects of exposure, if given. Using your own words, complete the OBJECTIVE and PROCEDURE sections in your lab notebook. (See the Maintaining Your Laboratory Notebook link on the lab website to learn what these lab notebook sections entail.) Also, please write out answers to the following questions: 1) The particle in the box energy, En, involves Planck’s constant, h, and the particle mass, m. What particle is involved in the transitions you will measure in this experiment? What are the values of h and m? 2) According to the assumptions of the particle-in-a-box experiment, how many freely moving π- electrons are there between the phenyl rings in each of the three organic compounds you will be studying? 3) Using the particle in the box equation and a wavelength of 400 nm, what is the length of the box of 1,4-diphenyl-1,3-butadiene? 4) The particle in a box theory assumes that potential energy is constant along the entire conjugated carbon-carbon chain. Are there any flaws in this assumption? Procedure Safety: Goggles must be worn at all times. Organic solvents (e.g., cyclohexane) should be collected in a separate container as waste.