Particle in a Box Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, TX 76019, United States ABSTRACT: The molecular length of control samples 1,4-diphenyl-1,3-butadiene, (1,6-diphenyl-1,3,5-hexatriene), and 1,8- diphenyl-1,3,5,7-octatetratriene were examined using spectrophotometric methods. The spectral results were evaluated and compared with theoretical calculations made possible by modification of Schrodinger s equation. Results provided a close approximation to theoretical results with the smallest length of the selected samples, yet greater uncertainty is shown with samples of higher lengths and higher margins of error. Quantum mechanical theory has provided great insights in understanding the physical realm of atomic particles. Extrapolating the atomic nature of biological processes can be fully appreciated in agricultural studies dealing with energy management by photosynthesis, or the toxicological effects in pharmaceutical deployment studies. 1 Current spectroscopic methods can serve as a means to illustrate the feasibility of quantum behavior. To ascertain the molecular dimensions of control samples, a wavelength spectra can be used a general strategy. This method would simplify the interpretation of results and evaluate trends, if any. The results would be used to substantiate theoretical calculations provided by quantum mechanics. The samples selected contain conjugated hydrocarbons capped by two phenyl terminus to provide a particle in a box model for an analytical comparison. Using DPB (1,4-diphenyl- 1,3-butadiene), DPH (1,6-diphenyl-1,3,5-hexatriene), and DPO (1,8-diphenyl-1,3,5,7-octatetratriene) as the samples chosen, the theoretical lengths for each species can be calculated using 0.139 nm as the average length. Moreover, by multiplying the average length with the number of carbon-carbon bonds for each conjugated system between two aromatic rings, the theoretical results can be obtained. 2 These samples were examined using a Shimadzu UV-Vis 2600 spectrophotometer. The benefits of this instrument was well suited for this experimental design. The wavelength range extends to infrared regions above 1300 nm allowing great sensitivity for samples which are UV active. The printout of results contain multiple absorbance measurements to optimize data accuracy and validation. 3 With each additional carbon bond, the energies associated with the unrestricted movement of electrons in a highly conjugated systems should vary as the region of space changes in length. To evaluate the length of each molecule experimentally as a function of energy, preliminary calculations were required prior to obtaining a spectra. Using the Schrödinger equation for discreet energy levels, the calculations reveal the relationship of a wavelength spectrum and length of system. To show this dependency, systematic calculations were carried to provide a useful approximation for procuring the magnitude of the energies involved. (1) HΨ = EΨ (2) ħ2 2m d 2 u(x) dx 2 v(y) ħ2 2m d 2 v(y) dx 2 u(x) + V(x, y)u(x)v(y) = (E x + E y )[u(x) + v(y)] (3) ħ2 2m [ 1 d 2 u(x) u(x) dx 2 + 1 d 2 v(y) v(y) dx 2 + V(x, y)] = (E x + E y ), where u(x) = sin ( nπx ) and v(y) = sin (nπy L L ) (4) E T = h2 2 8m (n f L 2 n 2 i hc L2) = λ (5) L = hλ(n f 2 n i 2 ) 8mc To calculate length, L in equation (5), the simplified Schrödinger equation (1) was expanded to evaluate two separate energy systems with their respective functions u(x)and v(y). To satisfy the boundary conditions in a one dimension box, = 0 is used in terms of n to allow the eigen values of the sin nπx L wave functions to normalize, provided by equation (3). The total energy associated with each discreet orbital, n, (E x + E y ), simplify to form E T. The planks constant, h, and electron mass, m, make up the Planck-Einstein equation, h/2m, and are associated with the energy absorbance of photon energy from the spectrophotometer which can be measure with its respective wavelength, λ, and speed of light, c, in equation (4). 5 It is shown that the length of the molecule is directly dependent on the number of quantum levels present in the molecule. As the UV source excites the species of interest, one electron in the highest filled molecular orbital (HOMO) readily excites to the lowest unfilled molecular orbital (LUMO). This is explicitly demonstrated by equation (5). The increase in conjugated π systems can only be achieved by an increase number of carbon-carbon bonds. Additionally, the energy required to excite the HOMO would exponentially increases as well. 4 The Aufbau principle is used to calculate the quantum number, n, of the highest filled level by taking the number of conjugated π bonds, excluding the phenyl rings then adding 1 to determine the LUMO of that molecule. Concodantly, DPB would have 2 π bonds for n i and 3 π bonds for LUMO, n f.
Absorbance Chart 1. Bond Line Illustration Diphenyl Butadiene Samples Examination of the spectra revealed the wavelengths needed for calculation. An experimental error can be seen upon initial inspection of DPB, compared with the other samples. The concentration of DPB may have been compromised, which would explain the higher intensity of the spectra. However, the intent of the experiments did not depend on the concentration, and repeating the test with a corrected concentration would give collinear results with the previous trial. With each respective compound in Figure 1, the wavelengths of the shortest peak was recorded at 316 nm for DPB, 337 nm for DPH, and 397 nm for DPO. These values, with lowest wavelength peak is known to have the highest energy, and therefore, it s a logical assumption that directly corresponds the HOMO and LUMO, the energy level of interest. 1,4-diphenyl-1,3-butadiene 1,6-diphenyl-1,3,5-hexatriene 200 180 160 140 120 100 80 60 40 20 0 300 350 400 Wavelength (nm) 1,8-diphenyl-1,3,5,7-octatetratriene Figure 1. Complete spectra of the diphenyl hydrocarbon compounds. 1,4diphenyl-1,3-butadiene; 1,6-diphenyl- 1,3,5-hexatriene; 1,8-diphenyl-1,3,5,7-octatetraene. Chart 1. A classification of diphenyl molecules with a varied number of conjugated hydrocarbons. Cyclohexane was the solvent chosen due to its low polarity, making it appropriately miscible with the samples used. Due to the high sensitivity of the spectrophotometer, careful consideration was placed to ensure containers holding the samples were thoroughly washed with cyclohexane. Using the spectrophotometer, initial test runs were conducted prior to sample preparation to confirm the instruments functioned properly. A concentration 1 10-5 M for each sample were used to obtain the final spectra results. Using a Fischer Scientific brand micropipette, a 10.54 µl of 9.489 10-3 M DPB, a 19.30 µl of 1.93 10-2 M DPH and a 19.38 µl of 1.94 10-2 M DPO were quantitatively transferred in separate graduated experimental vessels, then diluted in cyclohexane to 10 ml. Two, 1cm cuvettes filled with cyclohexane were placed in the UV spectrophotometer allowing the spectra to establish a baseline calibration. Leaving the solvent cuvette in the designated blank port, the spectra for each sample were subsequently carried at a preset wavelength range 420-300nm. Final results were submitted for analysis. Table 1. Theoretical Versus Experimental Box Length Results Compound 1,4diphenyl-1,3- butadiene 1,6-diphenyl- 1,3,5-hexatriene 1,8-diphenyl- 1,3,5,7-octatetraene Theoretical (nm) Experimental (nm) Marginal Error (%) 0.695 0.692 0.432 0.973 0.845 13.1 1.25 0.986 21.1 Using equation (5) the result were tabulated into Table 1 to compare with experimental results. The findings conclude a greater uncertainty as the bond length increases. The standard margin of error equation was used: (6) Margin of Error = Theoretical Experimental Theoretical 100 Despite the close approximation to theoretical results, the margin of error does increase, but not exponentially with respect to molecular length as the calculation would suggest. More measurements would need to be conducted using different samples through multiple measurements, with applied statistical techniques to verify the reproducibility of results and validity of theory. Furthermore, it can be concluded with certainty that the equations provided to determine the length sample cannot provide a valid approximation to theoretical results with long molecules, and would therefore breakdown with high errors. This is in agreement with quantum mechanics being better suited for calculations on smaller magnitudes. The particle in a box model expounds this notion, where, as
the length of the box decreases, the intensity of the wave function increases giving a higher probability for a particle to be within a region of space. The converse is true if the box length were to increase. The source of the error can be attributed to the limitations of the calculation, since it is assumed that the solution to the Schrödinger s equation outside the box is trivial and ignored. Where, the potential energy for a particle in a one-dimensional box may reduce the overall error. References 1. Shah, N.; Gao, M.; Tsutsui, K.; Lu, A.; Davis, J.; Scheuerman, R.; Fitch, W. L.; Wilgus, R. L. A novel approach to high-throughput quality control of parallel synthesis libraries. J. Comb. Chem. 2000, 2, 453-460. 2. Aihara, J. Reduced HOMO-LUMO gap as an index of kinetic stability for polycyclic aromatic hydrocarbons. The Journal of Physical Chemistry A 1999, 103, 7487-7495. 3. Allen, F. H.; Kennard, O.; Watson, D. G.; Brammer, L.; Orpen, A. G.; Taylor, R. Tables of bond lengths determined by X-ray and neutron diffraction. Part 1. Bond lengths in organic compounds. Journal of the Chemical Society, Perkin Transactions 2 1987, S1-S19. 4. Nelson, E. Derivation of the Schrödinger equation from Newtonian mechanics. Physical Review 1966, 150, 1079. 5. Rossenaar, B. D.; George, M. W.; Johnson, F. P.; Stufkens, D. J.; Turner, J. J.; Vlcek Jr, A. First Direct Structural Information on a Reactive. sigma.. pi.* Excited State: Time-Resolved UV-Vis and IR Spectroscopic Study of Re (benzyl)(co) 3 (ipr-dab). J. Am. Chem. Soc. 1995, 117, 11582-11583.
Particle in a Box Department of Chemistry and Biochemistry, The University of Texas at Arlington, Arlington, Texas 76019-0065, United States of America No Supporting Information ABSTRACT: Presented here is the report on theoretically determining the length of a conjugated pi system through utilizing the one-dimensional particle in a box equation. We experimentally gathered data through UV-Vis spectroscopy. The obtained spectra allowed us to determine the HOMO to LUMO transition of our molecule through wavelengths. The longest wavelength was used for our experimental calculations. We report our experimental results for the length of our box, the theoretical length of our box, the equations that were derived then utilized for our numbers, the possible reasons for our erroneous results, and possible ways to improve the theoretical results. an equation that is able to mathematically solve, not only, quantum mechanical problems but classical mechanical problems as well. Utilizing this equation, our group theoretically determined the length of three different conjugated π systems, 1,4-diphenyl-1,3- butandiene(a), 1,6-diphenyl-1,3,5-hexatriene(B) and 1,8-diphenyl-1,3,5,7-octatetraene(C) (Figure 1). Figure 1: Structures of Experimental Compounds A B Introduction The particle in a box (PIB) equation is a theoretical way to solve many of the basic problems in quantum mechanics. The beauty of the PIB equation lies within the ability of the equation to work in multiple dimensions. This equation can be used to determine how electrons move over large biological molecules, how a proton moves through space, or how delocalized π electrons move across conjugated double bonds. The PIB equation gives rise to some of the basic ideas of wave mechanics 1. This equation works by effectively locking a particle in a potential well of infinite depth. During the solution to the differential Schrödinger equation (the details of which will not be discussed in depth) something interesting arises, sin(nπ/a)x. This sin function indicates that particles do exhibit wave-like behavior and that n must be equal to an interger. This gives rise to different quantized states. If one were to solve the PIB equation at the infinite quantum state, they would see that particles begin to act as they do when classical mechanical equations are used. This gives rise to the most beautiful part of this equation; we finally have Materials and Methods Jasco UV-Vis spectrometer was used for all spectra. Chemicals were provided by laboratory TA at an initial concentration of 9.489 mm, 5.182 mm, and 5.16 mm for A, B, and C respectively; all three compounds were dissolved in cyclohexane. Next, the three compounds were all diluted to a final concentration of 10E-5 M in cyclohexane. The UV-Vis spectrometer was blanked with cyclohexane and spectra were taken between the wavelengths of 300 to 425 nm. Experiemental calculations were determined through the equation We were then able to our energy equal to hν C
and knowing that ν= cλ -1 we were able to derive our equation for length; n f was taken to be the quatntum number of the LUMO and n i the quantum number of the HOMO. L = [(λ(n f 2 -n i 2 )h)(8mc) -1 ] 1/2 This equation was the basis of all our experimental calculations. To determine our theoretical box length, we simply multiplied the number of carbon-carbon bonds between the phenyl rings in each compound by 0.139nm 2. Wavelengths used in our derived experimental equation are 348nm, 373nm, and 398nm for A, B, and C respectively. Results Theoretical box lengths obtained are 0.695nm, 0.937nm, and 1.25nm for A, B, and C respectively. The three captured spectra, used to determine the wavelength used in our experimental calculations are represented in figure 2. Figure 2: Overlay of Obtained Spectra Experimental calculations produced the results 0.707nm, 0.890nm, and 1.00nm for A, B, and C respectively. Percent errors for each of the compounds are 1.7%, 5.0%, and 20% for A, B and C (Figure 3) Figure 3: Chart Representation of Results Compound Theoretical Experimental % Error A 0.695nm 0.707nm 1.7 B 0.937nm 0.890nm 5.0 C 1.25nm 1.00nm 20 Discussion As you can see from our results, experimental and theoretical values are relatively close to each other. Though as the length of the molecule increases we can see that the equation quickly begins to fall apart, giving greater percent errors. This could be due to the fact that our equation is based off a onedimensional equation. Though the molecules are very rigid, resulting in a lack of rotational energy, the longer the conjugated system becomes the more room the electrons have to move around giving rise to the increasing percent error. It would be interesting if we were able to calculate the experimental bond length of 1,2-diphenyl-1,2-ethene to see if an even shorter bond distance would give greater accuracy for our equation. If one looks at figure 2 the will notice that the longer the conjugated system, the longer the wavelength of light absorbed from the spectrometer. This makes sense; if you consider these conjugated systems as a string on a guitar, the longer the string the lower the tone that is made (larger the wavelength of sound). This is analogous to these conjugated systems. Conclusion This lab utilized UV-VIS spectroscopy of compounds A, B, and C (Figure 1) in order to experimentally determine the bond length between the two phenyl rings. Experimental absorption wavelengths were utilized in our one-dimensional particle in a box Schrödinger equation to give our experimental values. These were compared to theoretical values that were obtained by multiplying the number of carboncarbon bonds by 0.139nm (Figure 3). As the length of the conjugated system increased, our percent error became greater and greater and the equation essentially fell apart. These discrepancies could be resolved through utilizing a two-dimensional equation, which could give rise to more accurate results. Overall the lab was successful at illustrating how PIB can be utilized to experimentally determine bond lengths and also was thorough at demonstrating the limitations of this equation. Knowing how to utilize tools giving to us is extremely important, but what s more important is knowing how, where, and why these tools begin to disintegrate, this allowing for proper use. References Physical Chemistry, Principles and Application in Biological Sciences, 4 th Edition, Tinoco, Sauer, Wang, Puglisi, 2002, Printice-Hall, Inc. New Jersey 07485 Alternative Compounds for the Particle in a Box Experiment. Bruce D. Anderson, J. Chem Ed. 74, 985 (1997)