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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 1,8diphenyl-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.
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 𝒅𝟐 𝒖(𝒙)
ℏ𝟐 𝒅𝟐 𝒗(𝒚)
𝒗(𝒚) −
𝒖(𝒙)
𝟐
𝟐𝒎 𝒅𝒙
𝟐𝒎 𝒅𝒙𝟐
+ 𝑽(𝒙, 𝒚)𝒖(𝒙)𝒗(𝒚)
= (𝑬𝒙 + 𝑬𝒚 )[𝒖(𝒙) + 𝒗(𝒚)]
𝟏 𝒅𝟐 𝒖(𝒙)
𝟏 𝒅𝟐 𝒗(𝒚)
ℏ𝟐
[
+
+ 𝑽(𝒙, 𝒚)]
𝟐
𝒗(𝒚) 𝒅𝒙𝟐
𝟐𝒎 𝒖(𝒙) 𝒅𝒙
= (𝑬𝒙 + 𝑬𝒚 ),
𝒏𝝅𝒙
𝒏𝝅𝒚
𝒘𝐡𝒆𝒓𝒆 𝒖(𝒙) = 𝐬𝐢𝐧 (
) 𝒂𝒏𝒅 𝒗(𝒚) = 𝐬𝐢𝐧 (
)
𝑳
𝑳
𝐡𝟐 𝒏𝟐𝒇 𝒏𝟐𝒊
𝐡𝒄
(𝟒) 𝑬𝑻 =
( − )=
𝟖𝒎 𝑳𝟐 𝑳𝟐
𝝀
(𝟑) −
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-diphenyl1,3-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
(𝟓) 𝑳 = √
𝐡𝝀(𝒏𝟐𝒇 − 𝒏𝟐𝒊 )
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,
𝑛𝜋𝑥
= 0 is used in terms of n to allow the eigen values of the
sin
𝐿
wave functions to normalize, provided by equation (3). The
total energy associated with each discreet orbital, 𝑛, (𝐸𝑥 +
𝐸𝑦 ), simplify to form 𝐸𝑇 . 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 𝑛𝑖 and 3 π bonds for LUMO, 𝑛𝑓 .
Chart 1. Bond Line Illustration Diphenyl Butadiene
Samples
length 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.
200
180
160
1,4-diphenyl-1,3-butadiene
Absorbance
140
1,6-diphenyl-1,3,5-hexatriene
120
100
80
60
40
20
0
300
350
Wavelength (nm)
400
1,8-diphenyl-1,3,5,7-octatetratriene
Figure 1. Complete spectra of the diphenyl hydrocarbon compounds.
1,4diphenyl-1,3-butadiene;
1,6-diphenyl1,3,5-hexatriene;
1,8-diphenyl-1,3,5,7-octatetraene.
Chart 1. A classification of diphenyl molecules with a varied
number of conjugated hydrocarbons.
Table 1. Theoretical Versus Experimental Box Length
Results
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.
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 wave-
Compound
Theoretical (nm)
Experimental
(nm)
Marginal
Error (%)
1,4diphenyl-1,3butadiene
0.695
0.692
0.432
1,6-diphenyl1,3,5-hexatriene
0.973
0.845
13.1
1,8-diphenyl1,3,5,7-octatetraene
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, 453460.
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 theo-
retically 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-1,3butandiene(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 mechanics1. 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
C
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
and knowing that ν= cλ-1 we were able to derive our
equation for length; nf was taken to be the quatntum
number of the LUMO and ni the quantum number of
the HOMO.
L = [(λ(nf2-ni2)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
% Error
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.
Compound
Theoretical
Experimental
A
0.695nm
0.707nm
1.7
B
0.937nm
0.890nm
5.0
References
C
1.25nm
1.00nm
20
Physical Chemistry, Principles and Application in Biological Sciences, 4th Edition, Tinoco, Sauer, Wang,
Puglisi, 2002, Printice-Hall, Inc. New Jersey 07485
Discussion
As you can see from our results, experimental and
theoretical values are relatively close to each other.
Alternative Compounds for the Particle in a Box Experiment. Bruce D. Anderson, J. Chem Ed. 74, 985
(1997)