Physics Lab 4 - Diffusion

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Physics

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Apr 3, 2024

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I Lab 4: More Random Processes: Diffusion In this lab, you will continue to examine the random process of diffusion described in the previous lab. This time you will study diffusion by examining Brownian motion. BEFORE COMING TO RECITATIO N Review OpenStax College Physics Section 12.7 (Diffusion)+ READ IT!: https://openstax.org/books/college- physics/pages/12-7-molecular-transport-phenomena-diffusion-osmosis-and-related-processes LAB WRITE UP Complete all of the activities described in on the following pages and answer all of the questions. Review your work with your lab instructor before leaving the lab. LEARNING OBJECTIVES To successfully complete this lab you should be able to • Observe Brownian motion of small particles and estimate the diffusion distance and the diffusion coefficient for the suspended particles • Calculate the diffusion coefficient from histograms of diffusion distances measured at various times Lab 4 p.1

V the screen (if not trying re t' h US ) · b . h ? connec m? t e . B • Cltck once on the MU300 text and the main window should egm to s ow a hve camera feed. Adjust the image resolution to the highest level. Setting the Calibration To _get some familiarity with using the microscope, your first task is to calibrate the camera's field of view. All of the bead videos will be taken at 40X ~a _gnific~tion and so we will need to know the true length scale for the vts1ble width of our video window at that magnification in order to set the pixel-to-length conversion in Tracker. 0 10iv. -0 .01mm C~efully pla~e the calibration slide onto the stage and, using the 40X magnification objective lens, bring the pnnted scale mto focus and center in the viewing window. Now, take a "Snapshot" of this focused image for your record-keeping. Each tiniest division equals 10 microns. What then is tlie size of the field of view of the camera ( either horizontal or vertical) image at 40X magnification? Enter this distance here: Length of Image = __ --'-\ S=..,, Q ,.._ ____ microns Carefully remove the calibration slide and return it to its plastic case. SA VE the image you took to a perm~ent file location with a meaningful file name. Lights, Camera, Action! An Excel workbook has been developed specifically for this lab to help streamline the data collection. You can download this workbook from Blueline. This workbook was designed to accept just 7 frames of tracked video (no less!) and any deviation in the formatting will foul up the entire collation process. On the bright side, the Excel workbooks will automatically carry out most all of the calculations! Now work to gather your video. Instructions: WARNING: READ THIS CAREFULLY! l. As you prepare each slide, remember to gently shake the vial of solution before you extract a sample with the pipette. Large diameter spheres will gradually settle and so your video should be collected fairly quickly after the sample is deposited on the slide. If it takes too much time ( e.g., much more than 15 minutes), you may want to prepare another sample. 2. Place a single drop of solution into the depression on the slide and overlay a cover slip. Gently blot up any excess solution and be sure to avoid bubbles as these will create convection current that adversely alter the bead motion. Do NOT touch the table of your lab station while recording data. lfyou fail to follow In struction 1 and Instruction 2, you may get "drift" effects instead of "diffusion" and will need to repeat the whole experiment! Lab 4 p.3

3. Use the 40X objective for these bead videos. 4. Use the phase-conjugated filter (slider just below lamp housing) and try adjusting the camera contrast (under Color Adjustment tab in the software) to obtain the sharpest image of the beads. 5. In AmScope, under "Camera and Resolution" on the left toolbar. Set the resolution to the highest possible value. Press "Record," Name your file and save to the desktop, and set the time limit to 1 minute. 6. When completed, run the Tracker program (see explanation on Tracker on the next page). Drag the movie of the beads on the desktop to the Tracker window. Wait for the video to load. 7. In Tracker, go to the "Clip Settings" button on the top toolbar (looks like a film strip). 8. Take the number of the end frame, divide by six and enter that number into the "Step Size" entry. Click . - 170 , r;../ - . OK. This changes the step size to 1 O seconds. .._, b - :z.q~ • \ \) lo lob 9. Place the calibration stick along the length of the side of the video ( either horizontal or vertical - whatever you chose for the calibration slide), and change the number to the value you recorded above. This value is in microns. 10. Run your video and zoom in on a bead that appears to be moving freely (Some beads will stick to the glass and not move at all). You will want to zoom in on the video to 100% at least! 11 . Return to the beginning of the video. Create a point mass for this bead and shift+click to mark the position at O seconds, 10 seconds, 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds. 12. Repeat steps 10-11 for at least 5 different beads. Copy and paste the data tables for each point mass for into the Excel workbook provided on Blueline. When you have finished, be sure to clean up the slide and cover slip for the next group. Show your instructor your work before continuing. Lab 4 p.4

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Tracker If you want to install Tracker in your own laptop, go to : https://physlets.org/tracker/ • If you are installing Tracker in a MacBook, you may need to set permissions to install Tracker: Go to Settings> System Preferences> General> Security and Privacy, and click "Open anyway". You may be asked for the password you have as user of the Mac Book, so type in your password. Remember to calibrate Tracker based on your movie . Click the icon shown below: Data Analysis Two of the columns in the excel spreadsheet compute the x and y displacement from the first recorded positions. Make an x-y plot showing the trajectory of at least 5 of the beads . , ......... - I. a) What do you notice about the bead trajectories? \,N vv6 V' C<...,(l do VVL Lab 4 p.S

<'I.. 2.1 = :2..D t., L_v 1-.)::: .LO t b) How does the path (i.e. distance) followed by each bead compare for all beads? Th£., d...\ s.-ta._n Le_ o tv\A..S S D VV CA..S \e,SS C\k\.d., Mμ.,s.s tl~~ -ty\O fuv~st- c) How does the displacement followed by each bead compare for all beads? lla...~s £ e) V°'I~ =f:r,.Q_ -hi\ Y~ t? r {:) l (,,l U6\.. d) What is range of velocities of each bead? How do the ranges compare for all beads? ~s D l ~t) -= '°. 1 '<l 2..,'e, fa\ 1CKU~4 s M C-\S ::> E. (,rt(.,?t) -=- o 4t- cs, 4i~J 5 "R_p..n~e 0 . 1. to 2...-<o - O. 4S7 M i (.J('<JVL~/-s 2. The distance traveled can be calculated as r = (Llx) 2 + (Lly) 2 , including only the squares of Lb; and L1y - which are always positive and so add up to something bigger than zero. This is also calculated in the excel spreadsheet. Make a plot of the average distance traveled vs. time (r vs. t) for the beads that you measured. We expect a non-straight-line profile. Check with your lab TA how to interpret the profile observed. Write down a short summary of the interpretation (it is OK if you have to exclude a couple of data points from your analysis. Why?). L,v'\..QCAJ 'Y ... s pee_d.. '~ L-OV\s-O:Al(\,-\.., - owr-- fov Y v~. t ,~ a ~~,~f\t _ - ~t $0l vJ :5 It ~S. n -t \\~ {)fUll . ~ V\ (/ ex-pt,&Wl Lab 4 p.6

3. What happens to the averag d 'th . speed? Explain. e spee WI time? Do the beads slow down, speed up, or maintain constant t ons-trmt C)V\_ w b I c \ o , s co n~ -t l.lY\ * L, , ~( A;' () cli 'i. p I 61. Vs · t.,i Y't'lt, 1 v aR"7 The "diffusion constant," D, is defined to be the proportionality constant between the average displacement ~quared, r2, of the diffusing object and the measurement time interval, t, over which the diffusion occurs. There ts a_ lso a factor of 4 for geometric reasons (in general, the factor is 2n where n is the number of dimensions in which the particle diffuses, so 2n=4 in 2-dimension, 2n=6 in 3-dimensions, and a 2n=2 in I-dimension): r 2 = 4Dt 4. Make a plot of the average value of r2 vs t for the beads that you measured . If the bead is moving by diffusion, then this formula suggests that r2 vs t should be linear with a slope equal to 4D. Add a linear trendline and determine the diffusion coefficient for the beads. Save your plot and show it to your instructor. 0(\ e)(a!,\ (_,0~1cJWt -=-+ \Ojc., -t1re.. ; VI A )..0 , '(MtY1t (,)( , d) 5. Is your data consistent with the predict i ons of diffusion theory? Is the diffusion coefficient constant, or does it change in time? Explain . ~\), O\J )f ct.AAct. i n l:: ()) yl si ~-l. ,rt,rt ~'x pon'{)Vt t ioJ tytno{ \ n .stea.,l .- of . c,c_ {:YllVLof - -me, 0 CA YJ fJCh1Jf 5 --\;, \ 1 b w e, ff1 v t-evl t- o.A w,is i Vt 2-0 Show your instructor your work before continuing. I 'S an ' 11 r \l,tlV Lab 4 p.7

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\ . . . d. uss with your lab Diffusion Histograms [Do as time permits - 1sc instructor] . d. • for I micrometer diameter The following figures show histograms of the distance traveled (r) m two tmenSions y beads at three different times. Find the average distance traveled for each of the three tim~. h 0 :, C:, 0 m assume the number of entries in each histogram bin are at a displacement equal to the center O t e ts gra bin. The histogram below shows: On the horizontal axis: the distance traveled in 30 seconds by the beads. On the vertical axis: The number of beads that traveled each distance 20 18 16 14 12 10 8 6 4 2 0 ii II I I I I I I Ill I I I ~ ~ ~ ~~~ ~~ ~ -NNNNNNNNNNMM M MM M MMMMV Distance Traveled (displacement in micrometers) Calculate the weighted average of the squared distance traveled (Excel might be helpful). The calculation will look like: (0.5 2 X 1) + (1.5 2 X 9) + (2.5 2 X 12) + (3 .5 2 X 14) + (4.5 2 X 18) + (5.5 2 X 17) + (6.5 2 X 4) . .. 1 + 9 + 12 + 14 + 18 + 17 + 4 ... Calculate the diffusion constant based on this histogram . ('/ .... '-') 29 . S1o I t>~ 2 .,_ - ) --=- o . '°I-Gt2.1 2. l :, o.) Show your instructor your work before continuing. Lab 4 p.8

I The histogram below shows the dist,nce tra,eJed in l00 '"'""' by the J,..,k. On the horizontal axis: the distance traveled in 30 seconds by the beads. On the vertical axis: The number of beads that traveled each distance 20 18 16 14 12 10 8 6 4 2 0 I II I I I I I I I II II Distance Traveled (in micrometers) Calculate the average squared distance traveled. ) Lil) -=- Ll) . "}~)(?)4 ll ?l-)('l)r(_z,?)('7 1".,., D. 5-tl . 'J-+2 .'J •. _ Lv')::. 4l . i55j7 Calculate the diffusion constant. ()(l/ --==O 20 '() -=--2 - ~Gf2.x 10, Show your instructor your work before continuing. Lab 4 p.9

The histogram below shows the distance traveled in 300 seconds by the beads. On the horizontal axis: the distance traveled in 30 seconds by the beads. On the vertical axis: The number of beads that traveled each distance 20 18 16 14 12 10 8 6 4 2 0 I I I I I I I II II I I I I II I I ii II I II I I I I I I I I I I 11 Distance Traveled (in micrometers) Calculate the average squared distance traveled. Calculate the diffusion constant. 1.'&3bxtoz. -=- 4 llb 2loo) · II , I I I Lab 4p.10

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F~d the mean value of the ditrus· histograms. Show your c 1 1 . ton constant and the error on the mean based on the data in the three a cu attons and express these values with the appropriate units . . Q.4CJ27-i- 2 092 xlOJ "¥4 .12.b 3 -::. 7-'f t/ X 102. Mean diffusion constant = '7 q q XI O 1.. Error on the mean for the diffusion constant= 1. · 5 f'] X \ 0 "2.. I am confident at the 68% level that if I repeated the Brownian motion measurements for 1 micron beads in water that I would obtain a mean two dimensional diffusion constant that is 7- .S Sx 1(Yμm 2 /s. Last question Write down at least two (2) take-home messages for you from this lab activity . Use declarative sentences in your answer . μ_,~'A s i m ·, , 9.r, -l.i e.s io I w 01..J.X Vr1 o\AlW\.l41 ts Ct 11 VU WVl I CU1 VVU? V'tW\Vt, t . 0\.A.. ! e atJG We_ YVl rvl DV 11'18 I VI 0\ VGtl'lGLovvl cmtA VI IJ V 6l ti O \Nltu'l V'eY Show your instructor your work before continuing. Lab 4p.11

Physics Lab 4 - Diffusion

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