Editing Particle in a box (section)

=== Conjugated polyenes ===

[[File:Beta-Carotin.svg|thumb|right|β-carotene is a conjugated polyene]]

Conjugated polyene systems can be modeled using particle in a box.<ref>{{Cite journal |last=Autschbach |first=Jochen |date=November 2007 |title=Why the Particle-in-a-Box Model Works Well for Cyanine Dyes but Not for Conjugated Polyenes |url=https://pubs.acs.org/doi/abs/10.1021/ed084p1840 |journal=Journal of Chemical Education |language=en |volume=84 |issue=11 |pages=1840 |doi=10.1021/ed084p1840 |issn=0021-9584}}</ref> The conjugated system of electrons can be modeled as a one dimensional box with length equal to the total bond distance from one terminus of the polyene to the other. In this case each pair of electrons in each π bond corresponds to their energy level. The energy difference between two energy levels, ''n<sub>f</sub>'' and ''n<sub>i</sub>'' is:
<math display="block">\Delta E =  \frac{(n_f^2 - n_i^2) h^2}{8mL^2}</math>

The difference between the ground state energy, n, and the first excited state, n+1, corresponds to the energy required to excite the system. This energy has a specific wavelength, and therefore color of light, related by:
<math display="block">\lambda = \frac{hc}{\Delta E}</math>

A common example of this phenomenon is in [[β-carotene]].{{citation needed|date=October 2019}} β-carotene (C<sub>40</sub>H<sub>56</sub>)<ref name=":0">{{Cite web|url=https://pubchem.ncbi.nlm.nih.gov/compound/beta-carotene|title=beta-carotene {{!}} C40H56 – PubChem|last=Pubchem|website=pubchem.ncbi.nlm.nih.gov|access-date=2016-11-10}}</ref> is a conjugated polyene with an orange color and a molecular length of approximately 3.8&nbsp;nm (though its chain length is only approximately 2.4&nbsp;nm).<ref name=":1">{{Cite journal|last1=Sathish|first1=R. K.|last2=Sidharthan|first2=P. V.|last3=Udayanandan|first3=K. M.|title=Particle in a Box- A Treasure Island for Undergraduates}}</ref> Due to β-carotene's high level of [[Conjugated system|conjugation]], electrons are dispersed throughout the length of the molecule, allowing one to model it as a one-dimensional particle in a box. β-carotene has 11 [[carbon]]-carbon [[double bond]]s in conjugation;<ref name=":0" /> each of those double bonds contains two π-electrons, therefore β-carotene has 22 π-electrons. With two electrons per energy level, β-carotene can be treated as a particle in a box at energy level ''n''=11.<ref name=":1" /> Therefore, the minimum energy needed to excite an [[electron]] to the next energy level can be calculated, ''n''=12, as follows<ref name=":1" /> (recalling that the mass of an electron is 9.109 × 10<sup>−31</sup> kg<ref>P.J. Mohr, B.N. Taylor, and D.B. Newell, "The 2014 CODATA Recommended Values of the Fundamental Physical Constants". This database was developed by J. Baker, M. Douma, and [[Svetlana Kotochigova|S. Kotochigova]]. Available: [http://physics.nist.gov/constants]. National Institute of Standards and Technology, Gaithersburg, MD 20899.</ref>):
<math display="block">\Delta E = \frac{(n_f^2 - n_i^2) h^2}{8 m L^2}= \frac{(12^2 - 11^2) h^2}{8 m L^2}= 2.3658\times10^{-19} \text{ J}</math>

Using the previous relation of wavelength to energy, recalling both the [[Planck constant]] ''h'' and the [[speed of light]] ''c'':
<math display="block">\lambda = \frac{ hc }{ \Delta E }= 0.00000084 \text{ m} = 840 \text{ nm}</math>

This indicates that β-carotene primarily absorbs light in the infrared spectrum, therefore it would appear white to a human eye. However the observed wavelength is 450&nbsp;nm,<ref>β-Carotene http://www.sigmaaldrich.com/catalog/product/aldrich/855553?lang=en®ion=us (accessed Nov 8, 2016).</ref> indicating that the particle in a box is not a perfect model for this system.