### Introduction

The proposed particle in a box experiment uses theories from quantum mechanisms in order to determine and prove the behavior of a molecule. In the experiment performed, various dye molecules were observed using a Spectronic 21 spectrophotometer to determine their wavelengths of maximum absorption. The spectrophotometer emits light through the sample and uses a detector to measure the absorbance through a range of wavelengths. The “box” is considered to be the space between the nitrogens of the dye molecules, where the behavior of the molecules is observed. Upon measuring different concentrations of the dye molecules with the Spectronic 21, standard linear plots of absorbance versus concentration could then be graphed. These plots could then be used to establish that the dyes follow Beer’s law, A = εbc. The measured wavelengths of maximum absorption could then be compared to the theoretical values, which can be found using the following equations:

λ_{max }= 8ml^{2}c (p + 3)^{2} / [h(N+1)]

λ_{max }= 8ml^{2}c (p + 3 + α)^{2} / [h(p+4)]

where h = Planck’s constant, m = mass of an electron, c = speed of light, l = distance between the nitrogens, N = number of electrons in the entire molecular orbital π system, p = number of carbon atoms, and α = extra distance the conjugate electrons go beyond the terminal nitrogens. These equations are derived from the basics of quantum mechanics. By comparing the measured values to the calculated values, one can validate the theories of quantum mechanics and also determine α.

### Procedure

To begin, 1 L of 1 x 10^{-6} M solutions were prepared using the following compounds: 1,1’-diethyl-4,4’-cyanine iodide, 1,1’-diethyl-4,4’-carbocyanine iodide, and 1,1’-diethyl-4,4’-dicarbocyanine iodide. Then, the following dilutions of each of those solutions were prepared: 100%, 50%, 25%, 10%, and 5%. Each of the dilutions was then analyzed using a Spectronic 21 spectrophotometer to determine their measured absorbance and wavelength of maximum absorption. These measured values were saved on the computer and later analyzed.

### Results

Cyanine Iodide | Carbocyanine Iodide | Dicarbocyanine Iodide | |

λ max _{(measured) }(nm) |
595 |
707 |
814 |

λ max _{(calculated)} (nm) |
604 |
736 |
868 |

α |
-0.0732 |
-0.236 |
-0.441 |

% error |
1.49 |
3.94 |
6.22 |

c (m/s) |
2.998 x 10 |
2.998 x 10 |
2.998 x 10 |

m (kg) |
9.1 x 10 |
9.1 x 10 |
9.1 x 10 |

p |
7 |
9 |
11 |

h (m^{2}kg/s) |
6.626 x 10 |
6.626 x 10 |
6.626 x 10 |

N |
10 |
12 |
14 |

ε |
80281 |
226691 |
127172 |

l (m) |
1.42 x 10 |
1.42 x 10 |
1.42 x 10 |

### Calculations

λ max _{(calculated)} =

8 (9.1 x 10^{-31 }kg) (1.42 x 10^{-10} m)^{2} (2.998 x 10^{8 }m/s) (7 + 3)^{ 2} / [ (6.626 x 10^{-34} m^{2}kg/s) (10 + 1) ]

λ max _{(calculated)} = 6.04 x 10^{-7} m (1 nm / 10^{-9} m) = 604 nm

λ max _{(measured) }=

8 (9.1 x 10^{-31 }kg) (1.42 x 10^{-10} m)^{2} (2.998 x 10^{8 }m/s) (7 + 3 + α)^{ 2 }/ [ (6.626 x 10^{-34} m^{2}kg/s) (10 + 1) ]

α = 0.001628

Percent Error = | λ max _{(calculated) }– λ max _{(measured) }| / λ max _{(calculated) }x 100%

| 604 -595 | / 604 x 100% = 1.49%

### Conclusions

The calculated λ max values were extremely close to the measured λ max values. The percent error was 1.49% for cyanine iodide, 3.49% for carbocyanine iodide, and 6.22% for dicarbocyanine iodide. These results seem very good considering the laboratory conditions; they are nearly the best results we could have hoped for. This seems to show the equations derived from quantum mechanics are indeed valid. However, the measured λ max values are all less than the calculated λ max values, which should not have been the case. The measured λ max values should have been greater than the calculated λ max values in order to account for α, the extra length past the terminal nitrogens. This in turn made the values for α all negative. The dyes all seems to follow Beer’s law, as shown by the high R^{2} values calculated by plotting the absorbance versus concentration for each dye. It was odd that the absorbance reading for dicarbocyanine iodide exceeded a value of 1. Normally this should never happen; 1 should be the maximum absorbance value. This may be accounted for by the spectrophotometer not being correctly calibrated.