More on Beer's Law:
All parties involved in the research and discussion regarding the determination of beer color are to be commended on the useful information presented to date. However, the application of Beer's Law to the determinations of beer color is yet to be clarified. Beer does conform to Beer's Law and Mr. Ed Hitchcock is correct in his assertion that the non-linearity observed in absorbance curves of dark beer dilutions is due to measurement error caused by limitations in the technique of spectrophotometry.
Spectrophotometers measure the intensity of a beam of light that exists in a sample. This value is mechanically or electronically converted to a percent transmittance (%T) by comparison to the intensity of the light entering the sample. This is transformed to an absorbance using Beer's Law (Mr. Daniels actually provided a portion of the Beer-Lambert Law which corrects for pathlength):
Where: c = concentration of sample A = concentration of sample in absorbance units T = transmittance as a decimal fraction %T = transmittance as percentage k = a constant, the value of which depends on how the solution interacts with the lightThis equation predicts an infinitely linear relationship between absorbance and concentration. In practice this is not observed due to limitations in the spectrophotometer itself. These devices split intense white light into the spectrum of wavelengths using a prism or diffraction grating. Mechanical imperfections cause individual wavelengths to become contaminated with light of other wavelengths. This contaminating light is called stray light and may include wavelengths that do not interact with the sample. As the percent transmittance values decrease (that is, absorbance increases) the stray light will make up a more significant portion of the measured percent transmittance. The error is magnified by the log transformation causing depression in the observed absorbance values when compared to the expected absorbance values. This stray light value is important in determining the linear range of a spectrophotometer. It also explains why labs dilute dark samples before taking an absorbance measurement.
The analysis of Dr. Fix's data is flawed. The error arises from applying the absorbance of an undiluted sample to Beer's Law to calculate the expected absorbances of beer dilutions. This undiluted absorbance is incorrect due to the effects of stray light discussed above. Figure 1a shows a reproduction of Dr. Fix's data from reference 1. The x-axis scale has been converted to absorbance readings for clarity. An undoubtedly linear relationship is evident up to an absorbance of 1.0 (100 degrees L). In this region absorbance does vary linearly with the concentration suggesting that Beer's Law does hold true for this analysis. Above an absorbance of 1.0 units linearity begins to break down due to a %T stray light value that can be quantified. Assuming that Beer's Law holds true an absorbance for an undiluted sample can be calculated by multiplying a value from the linear portion of the curve by the dilution factor. A dilution consisting of 20% Michelob Dark produced an absorbance of 0.6 which falls on the linear portion of the curve. Multiplying this by the dilution factor produces a predicted absorbance of 3.0 units in the undiluted sample (not far off the 2.8 predicted by Ed Hitchcock's experiment). An absorbance of 3.0 units corresponds to a transmittance of 0.1%. The observed transmittance, calculated from an absorbance of 1.7 units, can be calculated to be 2.0%. This indicates a value of 1.9% transmittance due to stray light. Consider the following form of Beer's Law that is more pertinent to the use of a spectrophotometer:
Where: Aobserved = observed/measured concentration in absorbance units %TPure = %T of pure wavelength TStrayLight = %T resulting from stray light (=1.9% in this case)Beer's Law predicts a straight line that extends throughout the linear region of Dr. Fix's data through and absorbance of 3.0 in an undiluted sample. By calculating %T values from this predicted relationship and correcting for stray light with the equation above, Dr. Fix's data can be mathematically reproduced. This is shown in Figure 1B. A comparison of Figures 1A and 1A shows that the absorbance curve for Michelob Dark can be predicted by Beer's Law when considering the effects of stray light. This analysis does suggest a large stray light value for Dr. Fix's data which may be due to the use of an inaccurate spectrophotometer or incorrect use of the spectrophotometer.
The analysis of Dr. Fix's data implies two things: (1) that if a spectrophotometer with a lower stray light specification were used, the linearity of the data would hold up to higher absorbance readings and (2) that a calculated stray light value from this data will be close to the stray light specification of the machine. Figures 2a and 2b show the dilutions of two dark beers (unfortunately Michelob Dark is not readily available where I live). The spectrophotometer used is a relatively high precision Hitachi U2000 with a stray light specification of 0.05% and the path length used was one centimeter. Samples were done in duplicate to estimate error in the dilution (error bars shown on the figures represent the standard deviation of the duplicates). Both figures show a clearly linear relationship up to an absorbance reading of 2.5 units. The breakdown in linearity is due to the stray light limitation of the spectrophotometer being exceeded. This can be mathematically demonstrated. Extrapolation of the linear region of figure 2a predicts an absorbance of 3.79 units at %100 beer (no dilution). Using Beer's Law, this corresponds to a %T of 0.0162. The measured %T was 0.058 (calculated from the absorbance reading) in the undiluted beer sample. This indicates a %T due to stray light of 0.042. This value is close to the 0.05% stray light specification of the machine. The small error can be explained if one assumes that a portion of the stray light will be absorbed by the sample.
Dr. Fix's data is easily explained using Beer's Law and suggests nothing more than the use of an inaccurate spectrophotometer. In any case, nonlinearity of absorbances is routinely observed and to be expected in spectrophotometer assays of every kind. This nonlinearity results from a well-known and quantitative limitation in the spectrophotometer hardware which varies from machine to machine. Beer does obey Beer's Law and, once again, this demonstrates the need to always question the accuracy of the measuring tools we use and to work within their limitations.
University of British Colombia
(1) G. Fix and L. Fix, Vienna, Marzen, Oktoberfest, Classic Beer Style Series (Brewers Publications, Colorado, 1992), p. 121.