Demonstrating Beer’s Law with BeerPHYS 3330 (Modern Optics) Term Project

George Barron and Aditya Mewar(Dated: December 10, 2016)

In this work we exhibit an implementation of a method of spectroscopic analysis of several typesof beer using Beer’s law. This is achieved with 3D printing technology, smartphone cameras, andstatistical analysis. Statistical analysis is performed on the data found to determine estimates of thelinear attenuation coefficient within some error bounds. Overall, the results are satisfactory withrelatively low error. This enables us to make some general conclusions about types of beer basedoff of these data.

CONTENTS

I. Introduction 1

II. Setup and Procedure 1A. System Calibration 1B. Design 2C. Materials 2D. Procedure 3

III. Results and Analysis 4A. Recordings and Data 4B. Statistical Analysis 4C. Discussion 5D. Interpretation 5

IV. Final Words 5A. Recommendations 5B. Conclusion 5

V. Figures 6

References 6

I. INTRODUCTION

Assuming a uniform attenuation in a medium, the ra-diant flux φ(h) in a medium with a constant attenuationcoefficient follows the law

dφ(h)

dh= −µ · φ(h) (1)

which is easily solved (via separation of variables) withthe solution

φ(h) = φ(0)e−µh (2)

We can define φ(h)φ(0) = T (h) = T to be the transmit-

tance at depth h in the medium. Hence, we have theBeer-Lambert law, given by

T = e−µh (3)

This law is used in spectroscopy of solutions. Thisderivation is adapted from [3] for the case of a constantlinear attentuation coefficient.

The goal of this project is to demonstrate this law us-ing a smartphone and a 3D printer. Specifically, we testand analyze the transmittance of different types of beersat different wavelengths. In this way, we 3D print theapparatus used to demonstrate this and collect data ona smartphone camera.

II. SETUP AND PROCEDURE

A. System Calibration

There are several components of the apparatus thatneed calibration.

1. We first need to set the location of the laser on thecamera lens. This is accomplished by taking thecavity portion of the apparatus off and shining thelaser straight down onto the smartphone throughthe base. Be sure to place the button on the laserpointer under one of the clamps so that the laserstays on during the experiment. Do not tightenthe clamp too much to the point where it damagesthe button. After this, the experimenter can ad-just the laser in the vice so that it points directlydownwards onto the camera.

Once this is done, the experimenter can remove thephone and mark a place with sharpie so that thedirection of the laser may be adjusted without thephone in place. After this is complete, the laser iscalibrated.

2. The exposure settings of the camera must be cali-brated as well, because by default the camera willadjust in between shots and ruin any ability to com-pare frames.

To do this, set the camera phone face-down ona flat surface so that no light enters the camera.Launch Open Camera (freely available on GooglePlay Store, not owned by experimenters) and lockexposure, lock focus, set no flash, set white balanceto incandescent. This is a consistent process that

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will allow us to obtain a consistent configuration ofthe camera exposure from trial to trial.

3. The polarizer needs to be adjusted as well. To dothis, when the smartphone is not in the housing,simply rotate the polarizer such that the transmit-ted intensity of light is at a minimum. We are ableto achieve the minimum angle within 2◦ error usingtechniques implemented in previous labs. We notethat this does not have an impact on our measure-ments for the linear attenuation coefficients, as thisis fixed for each trial. In this way, we are only con-cerned with the intensity of each data point relativeto the others in its set.

4. The side of the liquid cavity has a scale that mea-sures the depth of the liquid. This was constructedby making markings on a piece of paper using aruler known to be accurate within 0.1cm. Hence,this dictates the error on each measurement madefor height. Note that this is the scale by which wemeasure the height in the chamber, not the volumeleft in the syringe.

5. The beer needs to be prepared for use in this setup.Straight out of the container, most beers are toocarbonated to not produce bubbles on the top thataffect the transmittance. The most effective waywe found was to let the beer sit overnight so a lotof carbonation dissipates.

6. The imaging program needs to be calibrated to de-termine the error in the intensity as the videos arebeing recorded. To do this, we take a several secondvideo of each wavelength with no beer in the cham-ber. Then we split this video into separate framesand calculate teh intensity of each and normalizeas described in the analysis section. Then, we maycalculate the mean and the variance of set of datafor each wavelength. In the case of indigo light, wehave a mean of 1.00 and a variance of 3.47 · 10−6.In the case of red light, we have a mean of 1.00and a variance of 5.70 · 10−6. In this way, the ver-tical error bars on our plots are present, but barelyvisible.

B. Design

The purpose of the apparatus is to allow the experi-menter to vary the optical path length of the light in theliquid. Hence, we 3D print a column that holds liquid,under which there is a base that holds it and houses asmartphone. Above this is a laser mounted in place witha polarizer in between. A picture of the design with allcomponents except the smartphone is shown in Figure 1.

The 3D model itself was designed in Autodesk 123DDesign [1], and 3D printed on a Lulzbot Mini. Theseaspects are discussed more in the materials section.

Here are some notes on the design.

FIG. 1. Complete setup with some samples of beer.

1. Base allows the smartphone to slide in just enoughso that it does not move from experiment to ex-periment. In this way, the laser points at the sameplace on the camera in each trial. The laser is cal-ibrated in a specific, consant orientation as such.

2. The polarizer in between the output of the laserand the liquid cavity is present to allow the ex-perimenter to decrease the intensity of the beam,so that the camera phone can distinguish betweenheights. This, too, is calibrated according to ourspecifications.

3. The base includes an opening for the light to passthrough from the liquid cavity. There must be somebarrier preventing the liquid from pouring onto thephone, so on the bottom of the cavity we have at-tached a cut out portion of a piece of an overheadtransparency.

Since the original design was printed, we have madesome improvements to the original design to improveportability. This includes printing a case that houses im-proved versions of the original apparatus. Improvementsinclude pegs to secure the base and the cavity, a housingon the top to hold the laser in place. A picture of thepieces in the box is shown in Figure 2, and a picture ofthe apparatus assembled is shown in 3.

C. Materials

1. The most expensive component of the setup wouldbe the 3D printed materials, were we required topay for these. The quality of the componentsprinted are fairly good. The degree to which theapparatus was accurate in terms of lengths. Morespecifically, the error in the x and y directions wasat max 0.2mm [2]. The only problems experienced

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FIG. 2. Improved design in container.

FIG. 3. Improved design in assembled form.

involved liquid leaking through the seams in be-tween each layer of printed material. This was re-solved using super glue that was not water or alco-hol soluble. The 3D printed materials were createdin the optics lab, which we did not have to pay for.The total cost was $0.

2. The most expensive material used was the actualbeer. The choice of beer was a variety pack by NewBelgium Brewing Company that included severalbeers. The beers purchased were Citradelic (IndiaPale Ale), Ranger (IPA) , Rampant (IPA) and FatTire (amber ale). We also purchased Old Rasputin(Russian imperial stout) as it is very dark, and verylittle light can be transmitted. This was to get alarger range of transmittance values. The total costof the beer was $25 at a local liquor store.

3. We needed a plastic syringe to put the beer into theliquid cavity. The readings on the volume in thesyringe are not accurate enough for measurements.Hence we have another method of measuring theheight. This is discussed in the calibration portionof the report. The syringe chosen has maximumreadable volume of 60mL. The total cost of this

was $5 on Amazon.

4. The lasers used were provided by the instructor.The original plan was to perform all trials at threedifferent wavelengths. The red laser has wave-length 650nm±10, the indigo laser has wavelength405nm ± 10, and the green laser has wavelength532nm±10. Unfortunately, the button on the greenlaser provided was very difficult to control, andpressing the button in different ways would yielddifferent intensities of the outgoing beam. Even-tually, the button gave out altogether before thetrials involving water were concluded. Hence, onlythe red and the indigo wavelengths are used. Thetotal cost of these lasers is $0, thanks to the in-structor.

5. The material used to cover the bottom of the liq-uid cavity is a small, cutout portion of overheadtransparency paper. This was chosen as it shouldnot cause much distortion, as it is used in opticalapplications. This was provided by Dr. Wiegert,free of charge. The total cost was $0.

6. Most importantly, we used a Nexus 5X smartphoneto take the pictures and Open Camera (available onthe Google Play Store) for the camera software.

7. All other materials used were basic componentsused in the optics lab. This includes braces forholding the phone, clamps for setting up and hold-ing the polarizer and laser, and the polarizer itself.

D. Procedure

Here we describe the procedure for gathering onedataset for a single beer and a single wavelength. Someof the steps included are calibration steps, and are refer-enced as such.

1. This step is preferably done the night before. Openbeer and let it sit until the carbonation dissipates.This can be done in several ways. This is describedin the Calibration section.

2. Calibrate the laser so that the laser points directlyon the smartphone camera. This is described indetail in the Calibration section.

3. Calibrate the polarizer so that minimal light passesthrough. This is described in detail in the Calibra-tion section.

4. Fill the syringe with the largest possible reading of60mL, and place upright (with needle facing up) ontable.

5. Calibrate the phone to the specifications in the Cal-ibration section.

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FIG. 4. Sample of low transmission indigo light through wa-ter.

6. Start the video recording and place the phone inthe apparatus so that it is fixed in place accordingto how the orientation of the laser is calibrated.

7. Start to fill the apparatus.

(a) Announce the current reading on the scale onthe side of the apparatus. This is given inmillimeters.

(b) Insert the beer into the liquid cavity fromabove until the you reach the next desiredheight.

(c) Wait until the bubbles have died out, and theliquid has stopped moving. Alternatively, takea paper towel or something else and place itjust above the surface of the beer as to pop andabsorb bubbles. If this step is taken, however,make sure that this does not affect the readingon the height on the side of the apparatus, orat least take this into consideration.

(d) Repeat this process until a desired height isreached.

8. Set down the syringe somewhere liquid safe.

9. Remove the liquid cavity from the apparatus anddispose of beer.

10. Remove the phone from the apparatus and stop therecording.

11. Wash the syringe and liquid cavity with water forthe next experiment.

A sample of what should be seen in the video isprovided in Figure 4.

III. RESULTS AND ANALYSIS

A. Recordings and Data

The primary data collected are images collected fromthe videos recorded. This data is provided in the projectDropbox. The link to all the data sets and analysis cod-ing files are in http://bit.ly/beerslawoptics.

Because of this, the lab notebook kept during experi-ments does not contain as many measurements as a typ-ical notebook. However there are still procedures andsmall notes about how to collect and process data. Thisis provided in http://bit.ly/beerslawnotebook.

B. Statistical Analysis

To obtain information about the linear attenuation co-efficient in each case, we perform curve fittings in thefollowing way.

1. Specify a beer and a wavelength that is being used.

2. For this video, take the set of frames for whichheights are called out, as in the procedure.

3. Create a pair that includes the height hi and theimage intensity Ti. The image intensity is obtainedby summing over the entire RGB array of pixels ineach image.

4. Normalize the dataset by dividing each intensity bythe intensity for which hi = 0. This defines a pointfor which the transmission is unity.

5. Fit the dataset containing pairs (hi, Ti) to the func-tion Ti = T0e

−µhi+d. The value of T0 is a scalingfactor close to 1, and the value of d is an offset pa-rameter that accounts for the optical path lengthof light through the air. This parameter is small aswell. Finally, µ is the linear attenuation coefficientthat we are most interested in.

After this process, for each beer and wavelength a valueof µ is obtained and the associated error. Moreover, weare able to provide plots of each fitting overlaid with theoriginal data points, as well as the residuals. These areshown in Figures 5 - 12. As we can see in most cases, thefits are acceptable. Details of each fitting are discussed inthe the captions of respective fits. Note that in each case,the vertical error bars are present, yet trivially small, asdiscussed in the calibration section. The horizontal errorbars are a constant 0.5cm, as this is the most precisescale we used for the rule on the side of the apparatus.The results of the fittings are discussed in detail in thediscussion section.

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C. Discussion

The results of the fittings for each beer and wavelengthare presented in the following table along with each error.Each value presented is the appropriate linear attenua-tion coefficient, µ, given in cm−1.

Beer Indigo RedRasputin 5.15 ± 0.30 0.98 ± 0.06Citradelic 0.99 ± 0.07 0.19 ± 0.02

As is quantiatively clear in the values of µ obtained,the errors are fairly small compared to the values, andare what we would expect.

D. Interpretation

The interpretation of the values of µ obtained are asfollows. The larger a value of µ is, the less light passesthrough the medium in total. This is what we wouldmathematically expect by the law presented, but alsomakes intuitive sense based on what the beers used looklike. The Rasputin is a Russian Imperial stout, and isvery thick and dark to the eye. Hence, the values of µare in general larger than those of the Citradelic beer.Conversely, the Citradelic beer is fairly lighter to lookat than the Rasputin, as it is an IPA and brewed withcitrus.

Furthermore, we see that the error scales somewhatproportionally with the value of µ itself. This makessense, as for larger µ, it is more difficult to distinguishbetween different intensities of light, because very littleamounts of light are passing through the medium.

IV. FINAL WORDS

A. Recommendations

To minimize error in this procedure and implementa-tion, we suggest the following changes and some problemsexperienced.

1. The amount of bubbles caused by carbonation inthe beer was very inconvenient and difficult to getreliable results from. This was ameliorated by flat-tening the beer overnight and absorbing bubbleson the top with a paper towel. Ideally this shouldnot have been such a large problem, but this was amajor factor in obtaining reliable results for us. Ifpossible, less ”heady” beers should be avoided.

2. The way the 3D printer chooses a pattern to printmade it so that our design came out to have gapsin the interior walls of the apparatus. This made itso that when modifying the apparatus, there werelarge holes in the sides, causing it to leak. The holes

were patched up with superglue in the end. If wewere to re-do this experiment, we would choose abetter 3D printer so that this would be avoided.

3. Calibrating the orientation of the laser was fairlyeasy, but in reality the laser beam refracts offthe surface of the liquid because it is not flat inall places, and causes different refraction patternswhen entering the smartphone camera. This seemsto cause erratic behavior in the data in some places,as the refracted beam could land above, below, tothe left of, or to the right of the smartphone cam-era. A possible improvement would be to use adiffuser so that at no point the beam is pointingdirectly onto the camera lens, but instead comingdiffusely from all directions.

4. It would be interesting to obtain more possiblewavelengths of light using a monochromator so thefull visible spectrum of a beer could be measured.In particular, one could explore the linear attenu-ation coefficient by introducing wavelength depen-dence as T = e−µ(λ)h, where λ is the wavelength.

5. To improve the methods described, it would be use-ful to perform this process on a sample of water forwhich we know the value of µ at each wavelengthtested. In this way, we could have a more concisemeasure of error with which we could propagate ourother errors. This was not analyzed, as these valuesof µ for water were not available at the wavelengthswe are testing. Water was used to run test trialson the device before purchasing the beer, however.An example of this is shown in Figure 4.

6. We note that the recorded values are not publicallyavailable, and so we have nothing to compare thesemeasurements to. We highlight the importance ofthis note, as it would be possible to use water todetermine the accuracy of our instrument, but thevalues mesaured for water are not available. Hadwe done this, we could propogate the error as de-scribed above.

7. Finally, it would be interesting to see a large num-ber of beers compared with this method. This wasnot carried out because of time restrictions, as theflattening process for the beer takes a long time.

B. Conclusion

Finally, we conclude that this procedure of finding thelinear attenuation coefficient is acceptable and yields er-rors within a tolerable range for estimation purposes.The sources of error are understood, and methods toimprove on are provided in the Recommendations sec-tion of the report. Overall, the researchers are pleasedwith the results and did not expect errors as small as the

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1 2 3 4 5 6Height (cm)

0.2

0.4

0.6

0.8

1.0

Intensity

FIG. 5. Fit versus actual data points from Citradelic IPAusing red laser.

1 2 3 4 5 6Height (cm)

-0.04

-0.02

0.02

0.04

0.06

0.08

Intensity Residuals

FIG. 6. Residuals of fit from Citradelic IPA using red laser.The first point h = 0 is omitted from the plot, as it is at unityand throws off the scale of the rest of the graph.

ones obtained. In this way, we conclude that 3D print-ing apparati for optical experiments (at least similar tothis method) and analyzing data with smartphones is aviable method of experimentation.

V. FIGURES

[1] Autodesk. Autodesk 123d design, 2016.http://www.123dapp.com/design.

[2] D. P. for Beginners. Lulzbot mini review: Compact, pow-erful yet easy to use, 2016.

[3] Wikipedia. Beer - lambert law - wikipedia, the freeencyclopedia, 2016. https://en.wikipedia.org/wiki/Beer-Lambert law.

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1 2 3 4 5 6Height (cm)

0.1

0.2

0.3

0.4

Intensity

FIG. 7. Fit versus actual data points from Citradelic IPAusing indigo laser. The first point h = 0 is omitted from theplot, as it is at unity and throws off the scale of the rest ofthe graph.

2 4 6 8Height (cm)

-0.04

-0.02

0.02

0.04

Intensity Residuals

FIG. 8. Residuals of fit from Citradelic IPA using indigo laser.The first point h = 0 is omitted from the plot, as it is at unityand throws off the scale of the rest of the graph.

1 2 3 4 5 6Height (cm)

0.2

0.4

0.6

0.8

1.0

Intensity

FIG. 9. Fit versus actual data points from rasputin IPA usingred laser.

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1 2 3 4 5 6 7Height (cm)

-0.02

-0.01

0.01

0.02

Intensity Residuals

FIG. 10. Residuals of fit from rasputin IPA using red laser.The first point h = 0 is omitted from the plot, as it is at unityand throws off the scale of the rest of the graph.

1 2 3 4 5 6Height (cm)

0.002

0.004

0.006

0.008

0.010

0.012

Intensity

FIG. 11. Fit versus actual data points from rasputin IPAusing indigo laser. The first point h = 0 is omitted from theplot, as it is at unity and throws off the scale of the rest ofthe graph. Moreover, we see that the fit is shifted verticallyslightly for most points. We attribute this to the fact thatthe transmission is so small that the images are essentiallypitch black for these hi. In this way, the minimum sum ofeach intensity is smaller than what the fit predicts.

2 4 6 8Height (cm)

0.0005

0.0010

0.0015

0.0020

0.0025

Intensity Residuals

FIG. 12. Residuals of fit from rasputin IPA using indigo laser.The first point h = 0 is omitted from the plot, as it is at unityand throws off the scale of the rest of the graph.