# 0.3 Beer's law and data analysis  (Page 2/3)

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Complementary colors can be determined using an artist's color wheel. The wheel shows the colors of the visible spectrum, from red to violet. Complementary colors, such as orange and blue, appear as wedges opposite each other on the wheel.

With our eye, we can make qualitative judgments about the color(s) of light a sample absorbs. However, given a red solution of $\left[\text{Ti}\left({H}_{2}O{\right)}_{6}{\right]}^{3+}$ we can not determine if it absorbs green light or if it absorbs all colors of light but red. To quantitatively determine the amount of light absorbed by a sample as a function of wavelength, we will measure its absorption spectrum using a UV-visible spectrophotometer. Typical absorption spectra of aqueous $\left[\text{Ti}\left({H}_{2}O{\right)}_{6}{\right]}^{3+}$ solutions are shown below.

Notice the absorption maximum is at 490 nm. Because the sample absorbs more strongly in the green and yellow regions of the visible spectrum, it appears red-violet. Measuring the absorption spectrum of a second, more dilute solution demonstrates that the spectrum changes as a function of the concentration of the solution. To understand how to use the absorption spectrum as a quantitative tool for chemical analysis, read on!

Spectrophotmetric Basics

The essential components of a spectrophotometer consist of a radiation source, a wavelength selector (monochromator), a photodetector and read-out device.

The incident light from a tungsten (visible light source) or deuterium (UV light source) lamp is focused by a lens and passes through an entrance slit. By passing the beam through the monochromator (either a prism or a diffraction grating) it is separated into monochromatic (i.e., one-color or single-wavelength) light. One particular wavelength of monochromatic light is selected and allowed to pass through the exit slit into the sample. Light transmitted through the sample is detected by a photodetector which converts the signal to an electrical current which is measured by a galvanometer and sent to a recording device, typically a computer.

The measurement of transmittance (T) is made by determining the ratio of the intensity of incident ( ${I}_{0}$ ) and transmitted (I) light passing through pure solvent and sample solutions as a function of wavelength. [Note: The percent transmittance (%T) is obtained by multiplication of T by 100.] The logarithm of the reciprocal of the transmittance is called the absorbance (A),

A = log (1 / T)

Care must be taken when small values of transmittance are being measured as stray light from either the room or scattering within the instrument can cause large errors in your readings!

## Extracting quantitative information

The Beer-Lambert law relates the amount of light being absorbed to the concentration of the substance absorbing the light and the pathlength through which the light passes:

$A=\text{εbc}\text{.}$

In this equation, the measured absorbance (A) is related to the molar absorptivity constant ( $\epsilon$ ), the path length (b), and the molar concentration (c) of the absorbing. The concentration is directly proportional to absorbance.

The single largest application of the spectrophotometer is for quantitative analysis. The prerequisite for such analysis is a known absorption spectrum of the compound under investigation. Of particular importance is the maximum absorption (at ${\lambda }_{\text{max}}$ ) [Why choose the maximum? Could the choice alter the precision of our experiment? the accuracy?], which can be easily obtained by plotting absorbance vs. wavelength at a fixed concentration. Next, a series of solutions of known concentration are prepared and their absorbance is measured at ${\lambda }_{\text{max}}$ . Plotting absorbance vs. concentration, a calibration curve can be determined and fit using linear regression (least-squares fit). An unknown concentration can be deduced by measuring absorbance at the absorption maximum and comparing it to the standard curve. Caution: The Beer-Lambert Law is only obeyed (the standard curve is linear) for reasonably dilute solutions. Only those points in the linear range of the standard curve may be used for accurate concentration determination.

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