Absorption of Light: a differential equation and some physical realizations

 

As a beam of light passes through a medium, it commonly gets weaker, because some light gets absorbed by the atoms or molecules it passes. The goal of this lab is to compare the measured behavior of absorption with distance and the predicted behavior. (The theory section appears first, but it you prefer, you can take data first and tackle theory later.)

 

I. Theory: light absorption as a population process

 

If one thinks of light as composed of photons, one can treat light absorption as a population process and model it by a differential equation.

 

Consider the following statements:

§ Light intensity is proportional to number of photons per second passing through a unit area;

§ Each atom or molecule absorbs a certain (small) proportion of the photons that pass by it;

§ If you think of a material as composed of many thin layers, you can see that each layer subtracts an amount of intensity proportional to the intensity that reaches it from the preceding layer, and proportional to the number of absorbing atoms or molecules in the layer.

Using these statements,

 

II. Visible Light Absorption, Part I (distance)

 

The first experimental task is to measure the fall-off of light intensity as a function of distance, using the Ocean Optics spectrometer. Since you are familiar with the instrument, we can go straight to the questions to answer: (1) how does light intensity fall off with distance into an absorbing medium? (2) does it fall off the same way at each wavelength? and (3) does the fall-off at a given wavelength follow the exponential decay predicted by the differential equation?

 

The light-absorbing medium is weighing paper, and you can easily vary the distance light travels through it by adding thicknesses to the light path. You can use the cuvet holder to hold the papers in place, taking trouble to ensure that the paper is flat and perpendicular to the light path. Make sure the spectrometer is in ‘Intensity’ mode (blue I on toolbar).  It may be necessary to calibrate the instrument first, as we did last week, using a modified version of the dark and light reference current procedure (see last week’s handout), where the light reference current is taken when the lamp is on but the sample holder is empty, to reference the maximum measured intensity.

 

Record intensity at several different wavelengths as a function of number of thicknesses of weighing paper, then convert to intensity as a function of distance, by measuring the thickness of the paper. Then answer the questions above.

 

 

III. Absorption of Nuclear Radiation

As nuclear radiation passes through matter, it experiences absorption, too. Again, intensity is proportional to number of particles, and absorption is a certain small proportion of the particles/photons which pass by; so the differential equation is the same one, except that the parameters have different numerical values.

 

You can measure intensity of nuclear radiation from the source we provide, by using the radiation counter in the Logger Pro system. You can try using different absorbing materials placed between the source and the counter, such as aluminum foil, weigh paper, Kimwipes, etc, making sure you keep the distance between source and counter constant, varying only the amount of material (as measured by its thickness).  In this case, we have no way of differentiating wavelengths, so we deal with the whole spectrum at once. There are just two questions: (1) how does radiation intensity fall off with distance into an absorbing medium? and (2) does it follow the predicted exponential decay?

 

IV. (optional) The Beer-Lambert Law (a.k.a. Beer’s Law)

The theory of part I also predicts that the intensity of light falls off exponentially as the concentration (number per unit volume) of absorbing atoms or molecules increases. This result forms the basis for measuring the concentration. One passes light through a fixed distance of absorbing material, for example a standard sized container of liquid, and compares the received intensity of a standard, known concentration with the received intensity for an unknown sample.

 

Use your solution functions for the differential equation to show that the logarithm of the ratio of the two intensities is linearly related to the ratio of the two concentrations. This is a version of what is known as the Beer-Lambert Law, or Beer’s Law.

 

Prepare a few samples of differing concentrations of the organic dyes we used last week and measure the received intensities.  Make a graph of the logarithm of intensity as a function of concentration.