PARTICLE IN A BOX: VISIBLE SPECTRA OF DYES

Schrödinger developed quantum mechanics based on the use of classical wave equations to describe subatomic particles. One form of the Schrödinger equation for a particle moving in one dimension is

where

y = wave function (No physical meaning, but y² proportional to probability of finding particle.)

m = mass of particle

E = total energy

V = potential energy

The "particle-in-a-box" is a description of a small particle moving in a box in which the potential energy, V, is zero in the box, but is infinite outside the box. The length of the box is "a".

In order to keep the particle in the box y must be zero outside the box. Because y must be continuous, y must also be zero at x = 0 and x = a.

One solution to this problem is

At x = a, y = 0; thus

But also

Where n = 0, 1, 2, ...

So,

or

or

(n is a quantum number 1, 2, 3, ...)

According to our results, the energy levels are quantized!

The Pauli exclusion principle requires that there be no more than two electrons in any energy level. For molecules with N p electrons there will be N/2 levels. If an electron jumps from the level n = N/2 to the lowest empty level, N/2 + 1, the change in energy, DE, is

Since

Where

n = frequency

c = speed of light

l = wavelength

or

For the series of compounds we will study

where j is the number of double bonds in the polyene chain between two rings and the C---C bond length of order 1.5.

Also N = 2j + 4. In addition the highly polarizable benzene rings require that we increase the length of the box by L.

So that

or

Then

By increasing the box length by L

The above equation may be used to estimate the wavelength of light absorbed by an electron of mass, m, in a polyene chain.

PURPOSE

The purpose of the experiment will be to test the particle-in-a-box model. For an electron in a box the wavelength of maximum absorption is given by

EQUIPMENT AND CHEMICALS

Spectrophotometer (Turner 350, Coleman 124, P. E. Lambda 3)

Methanol

Cyanine dyes (1.00 x 10^-3 M stock solutions in methanol, referred to as

#I, #II, #III.)

Compound (I)

Compound (II)

Compound (III)

#II is also known as pinacyanol chloride.

PROCEDURE

From 1.00 x 10^-3 M stock solution, prepare the following solutions (Use 100 microliter micropipet.)

#I Dilute 0.10 ml to 10 ml in methanol

Dilute 0.10 ml to 25 ml in methanol


#II Dilute 0.10 ml to 25 ml in methanol

Dilute 0.10 ml to 50 ml in methanol


#III Dilute 0.10 ml to 25 ml in methanol

Dilute 0.10 ml to 50 ml in methanol

For each dye you will have two solutions of different concentrations. Scan each solutions as follows:

#I 470 - 550 nm


#II 575 - 635 nm


#III 675 - 735 nm

Use the graph showing absorbance (A) vs. wave length (l) for each solution.

Use Beer's law to determine the molar absorptivity for each solution of each dye.

A = Absorbance


a = molar absorptivity


b = path length in cm


c = concentration, mol/L

For each solution calculate a. For a given dye the two values of a should agree within five per cent.

For each dye calculate maximum l for the electron-in-a-box and compare it with your experimental values.

Literature values for maximum l are the following:

Dye #I 525 nm

Dye #II 610 nm

Dye #III 705 nm

(From Introduction to Quantum Concepts in Spectroscopy, W. G. Laidlow, 1970, p. 35.)