A Laser Refraction Method for Measuring Liquid --
Mackenzle E. Kina. Robert W. Pitha, and Stephen F. Sonturn Middlebury College, Middlebury, VT 05753 Laws governing diffusion are fundamental to our understandine of t r a n s ~ o rin t living - cells, chemical reactions, and analytical separation techniques. In this paper we describe a simole . .ohvsical - chemistw experiment for measuring diffu~ion constants by the defiectibn of a laser beam. Experiments on free diffusion have previously been described ( 1 3 ) .This experiment differs in a pedagogical sense by allowing the student to view concentration gradients directly. The purpose of the experiment is to give students an introduction to the theory and measurement of diffusion and to introduce statistical distribution functions. This experiment has been successfully used in our physical chemistry laboratory course where collection of the experimental data can easily he done in two hours. In the following sections we briefly discuss: the theories of diffusion, distribution functions, and refraction; the experimental procedure for this experiment; and finally the results for the diffusion of KC1 into water. Theory Diffusion T h e w When pure water is layered upon an aqueous salt solution, an uoward mieration of salt and a downward migration of water occur. he process by which the concentration difference between the solutions decreases is called diffusion. Diffusion is empirically described by Fick's first and second laws (4)
~.
J = -D(Jc/Jx), (JelJt), = D ( d e l J x 2 ) ,
We may also obtain the concentration gradient (aclax), by differentiating eq 5 (the derivative of a step function is a delta function) to give (JcIJx), = c&,
0,t )
(6)
Graphs of c(x, t ) and (aclax), are given in Figure 1. Statistical Theory Diffusion occurs because of the constant thermal motion of the molecules in the fluid. In their continual microscopic wanderings molecules are more likely in a statistical fashion to leave the more concentrated regions of the solutions. In addition to the underlying statistical process of diffusion there is a formal connection between the solutions of the diffusion eauation and the normal or Gaussian distrihution function of statistics. The normal distribution function is the familiar bellshaped curve (see Fig. 2) used todescrihe such diverse things as the distribution of student grades to the random errors of measurement. The equation for the Gaussian distribution is where p is the average value of x and u is the standard deviation of x. Graphically the average and standard devi-
(1) (2)
where the J is the flux, r is the concentration of salt, x is the
nosition. t is the time. and D is the diffusion coefficient of the .~ salt. ~ v e after k 130years Fick's laws still play a major role in ~
~
the study of diffusion (5). Fick's second law, eq 2, which determines bow fast the concentration is changing a t a given position or depth in the salt solution, is the basis for most measurements of the diffusion constant. Fick's second law is a consequence of the first law eq 1under the assumption that the diffusion constant D is independent of concentration (6). Equation 2 can be solved using Fourier transform methods (7l to give the time and position dependence of solution concentration c(x, t ) in terms of a convolution integral
where f(u) is a function that describes the initial concentration profile of solute andg(x, u,t ) is a Gaussian function
For o w experiment the initial concentration profile is a step function at x = 0 where the concentration goes from zero in the water layer to a value of co in the salt solution giving
Depth x Flgve 1. The concent'atim end its gradiem are shown as a function of Um depth xhom an innial boundsry between water and a salt soluHon.
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ation are reflected, respectively, in the position of the maximum and spread of the curve. Comparison of G ( x ,a, p) to the diffusion equation g(x, u, t ) implies that s = (wt)'"
( f i x ) ) of a function of x is given by a weighted average where the probability is used to weight each value of x ( f ( x ) )= \ r n x ) ~ ( xo, , r)&
(8)
In a physical sense this comparison leads to interpreting the average displacement c of the diffusing molecules as being directly proportional to the square root of the time t. This como&i~onalsofurnishes a wav to determine the diffusion cons'tant D if the standard deviation of our diffusion curves can he determined. In order to determine a from our experimental data we will summarize some pertinent properties of distributions. For a more complete description see ref 8. Both equations associated with the continuous variahle x and the approximate equations associated with the tabulated experimental points xi will be expressed. The definition of the Gaussian distribution as a linear probability density has two consequences. First, the area under the distrihution curve represents the prohahility. That is to say the linear probability density must he multiplied by an increment in x in order to find the probability Second, the total area under the curve which represents the total probability must he one. If the total area is not one, it may be normalized to one by dividing by its area. The factor ( 2 ~ 9 ) - ' " is the normalization factor for the exponential function exp(-(x - p)2/2c2)and also represents the maximum height that the normal distrihution will reach. For the discrete experimental data g(xi) the normalization may he done by numerical integration.
(f(x)) 2f(xi)G(xi)k
(11)
In particular the average x value ( x ) and the average are related to the mean p squared displacement ( ( x and standard deviation a. ( x ) = p a 2xiG(xJAx
((I- d 2 ) = 2 a. 2(xi - P ) ~ G ( x J A ~
(12) (13)
Students were provided with a short Minitab (9) procedure file to imnlement eas 12 and 13. a listine of which mav he obtained from the aithors. Optical Theory
Quantitative measurement of diffusion constants requires the ability to follow the concentration or its gradient as a function of time and position. The refractive index n provides an indirect way of measuring concentration gradients. When light passes from one medium to another, it is hent toward the medium with the bigger refractive index according toSnell's law [ n , sin(/?,)= n?sin(&)1.I n a medium with a refractive index gradient the light is hent into an arc toward the more concentrated solution. 1)erivations of the relevant eauations mav be found in a numher of articles (10-121, and results in a vertical deflection AX' on a screen, it a distance B from the center of a cuvette with solution width w, of approximately AX' = wB(an/ax),
(14)
The time dependence of this deflection is determined directly by the diffusion eauation because the index of refraction is nearly linearly dependent on concentration (13).
then Distribution functions contain information about average properties of the variable x. For example, the average value
An=n-noa.K(e-eo)
(15)
Substitution of this equation into the diffusion eq 6 yields the refractive index gradient due to free diffusion (anlax), = And%,0, t )
(16)
Here An denotea the total difference in refractive index across the initial boundary. Becauseg(x, 0,t ) is normalized, An also represents the total area under the (anlax), curve. Two basic-measurements of the refractive index gradient can he used to measure the diffusion constant. First the area-to-height ratio 4rDt = [~n/(an/az),J
(17)
where An represents the area and (anlax),, denotes the maximum height of the anlax curve. And second the standard deviation a of the normalized (an/ax) curve. Wt=$
(18)
In both cases, because of an undetermined starting time to,a linear regression of [An/(anlax)J2 or S versus time must be done to determine D. Experimental Procedure
Experimental Setup
The experimental components consist of a low-power helium neon laser.. a cvlindrid lens made from a elass rod. a cuvette on a " Laborau,ry jack, and a pereen. The glass rod, the cuvette, and the screen are mounted perpendicular to the laser beam as shown in Flgure 3. A 0 5.mW Arerotech model PS05 laser producing a 4-mmdiameter beam was used fur a light source. In order to scan several depths in the cuvetce simultaneously, the beam of light from the laser is fanned through a Pyrex glass rod (6 mm diameter)mounted
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Flgue 2. Gaussian w mrmal distribution fvnctlon superimpxed onanappmximate histogram ofthe disbibution;s is the standsrd devlatlon.p is the average xvalw, and &xisthe finilehiswarn Interval in X. 788
Joumal of Chemical Education
Fig.e 3. Line hawlng of me experimentalapparatus
at approximately a 45' angle to the vertical. The rod is placed so that its edge isadistance A from the center of a cuvette. Theeuvette was wnstruded from two 3- X 3-in. glasa plates epoxied to 0.5- X 0.5-in. Plexiglas spacers sealing three sides of the cuvette. The cuvette was mounted on a laboratoryjack to facilitate vertical pwitioning. The screen consists of a piece of 10-divisians-per-centimeter graph paper taped to avertical support. Distance A and B can he adjusted to regulate the amplification of the respnse. Typical values used in our lab were A = 40.0 cm and B = 200 an. A 2 M KCl(aq) salt solution is added to the cuvette until it is half full. The cuvette is raised or lowered using the lab jack so that the unfanned laser heam strikes the meniscus. The vertical refraded Line of light from the menism defines the vertical direction on the graph paper while the unrefracted laser spot ia used to set the origin an the graph paper. After marking the origin, the glass rod is pwitioned to fan the laser beam through the cuvette. The initial undefleeted line is d r a m on the screen. A small cork is then floated on the solution, and water is slowly (1-2 min) layered on the salt solution by allowing the water to flow in over the cork from a 60-mZ. syringe or pipet. Some practice is usually needed to generate a sharp interface, which results in a maximum deflection of 1&20 cm. It is sometimes possible to sharpen up the interface by withdrawing solution from the interface with a syringe. The required experimental data include: the distances A, B, and w, the solution temperature, indexes of refractions of both the water and the salt solution using an Abbe refmetometer, the salt solution wncentration calculated from its density, and the recorded defleetion curves. The laser deflection curves should he traced at5-min intervals for 40 min alternately using a pencil and then a pen. Students were provided with and required to follow manufacturer's safety guidelines for the laser and chemicals used including chemical disposal procedures. To reinforce these safety sheets, the students were explicitly warned about direct eye contact with the h e r beam. AU positioning of the laser beam was done from the indirect scattered light off the acreen.
Data Analysis The experimental deflection w e s Z' and the undeflected laser line X' (see Fie. 4) are tabulated at wnstant Y increments of 1cm. NO& all ekdinates were - ~that ~-~relative to themdeflected laser beam mark and included the maximum deflectionpoint at Y' = 0. These data were then translated and scaled using the geometry of the experiment (see Fig. 3) to reflectthe actual depth x and gradient Jnl Jx in the cuvette. ~
-
~
~
~
~
p ~~~~~~ ~ ~
~~ ~ ~
~~
~~~ -
-
~
-
~
T h e above two equations transform the screen coordinates into values related to diffusion in the cuvette. The JnlJx-vs.-x data may he analyzed in a variety of ways as discussed above. The simplest method is to use the area-to-maximum-height ratios. There are
Flgye 4. Screen det!.3clion curve for KC1 dllluslm. Solm curve rssults horn fmingw 1610Um experimental data with An = 0.0189ard Dt = 0.0285 cm2.
Area-maxlmurn helghl ratlo Area frwn An 1.88' IndlvMual curve area Average curve area
1.88
1.688 (19)b 1.957 (17) 1.72 (5) 2.27 (3) 1.740 (19) 2.053 (18)
Stati~tlcalanalysis of c Distributionturnion linear regrssslon nonlinear regression
1.909 (12) 1.925 (24) 1.68 (4) 1.841 (3) 1.87 (6) 1.86 (9) 1.88 (6) 1.876 (14) 1.789 (14)
2.16 (4) 2.32 (9) 2.12 (4)
literatwe
1.8ar
1.9994 HI'
1.8W
1.66 1316
S e ref 15.vol. 7, pp 13 am 75, mi m&ed for wavelength. aSeemf16. VOI. 2. pea, werags of o = 2.7 Msnd c = 1.3 M. Seeref I7.c = 2.1 MdMuslnglmoq= 1.9M.
three ways to estimate the area of these curves: (1) from the experimentally determined An, (2) from the area of each m e , and finally (3) from the average area of all the curves. Or the diffusion wnstant may he determined by analyzing for the standard deviationa of each of the m e s . Again there are several ways of implementing this analysis: (1) a statistical analysis of the curves as outlined above, (2) a direct Linear least-sauares remession of the Lo(Jn/Jz) M. x2 far earh curve followed by a regression of one over the resultant dopes w. t to determine I), and (3) a nonlinear least-squarer fit of all the curves to the diffusion eq 16 whore An, t o the starting rime, and I ) are the regression fit parameters. Bevington's (8)programs were used to do this analysis. Students were only asked to use method (1) of the area-height and standard deviation analysis as implemented with a ~initabprocedurefile.
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~
~
~
~
-
Results and Dlscusslon Using these methods, students were able to measure successfully diffusion constants for NaC1, KOH, and KC1 solutions to about 3% accuracy as compared to values in the International Critical Tables (14).T h e table contains two students' results for the index of refraction and diffusion coefficient of KC1. This table indicates that the accuracy is
Volume 66
Number 9
September 1989
789
limited by systematic errors due to analysis and experimental design. For example, the numerical integration method, eq 11,systematically overestimates the area near the peak of the Gaussian curve (see Fig. 2). Because of this overestimate, using An determined from measurements with an Abbe refractometer gave the hest comparison to the literature ues. The large values for the diffusion constant using nonlinear regression indicates that the experimental data is not strictly Gaussian. This could he due to an initial concentration profile that was not a step function, to temperature changes, to a nonlinear relationship between and c, and finallv to the concentration de~endenceof D. Methods that correct for these systematic errors are present in the literature (5,lO)hut are beyond the scope of this paper. In conclusion, this method offers a rapid, visually instmctive, and accurate method for studying diffusion. This experiment also functions as a pedagogicil t w l for reinforcing the concepts of distribution functions, which are important to the statistical descriptions of nature.
790
Journal of Chemical Education
Literature Clted
I. ~ i n d e P. s w.;~ a a s i m b n iL. , R; poison. A,; R ~A h ~J. c k m ~ . E ~. U .~ s n53, , 3~~332. 2. CWf~rd.B.:0chiai.E. I.J. C k m . E d u . L9SO.57,87~. 3. Irina.J. J.Chem.Edu. 1980,57,676671. 4. ~ k kA., A ~ O . P ~ Y chew. S ; ~ 1855.84.ss-88. 5. ~ o m a kJ. , P.:~oeeieki,J. K.;schoeider.D. J.; reed. J. K J , them. phys. 1986.84, 3381.3395. 6. Nborty.R. A . Phyaieol Ckolirfry. 7tbd.;W h y : Near Yark. 1981:~823-824. T. Butkov. E.MatkmotkolPhysics; weder N~~ YO&, 1968:p 523. 8. B"'n@an. P. R. Data Reduction and Ermr Amlysll for the Physical Sciences; Mffiraa: New York. 1969. 9. ~ i ~ i l ratatistical rb =meadah& ~ioilrrb~n~.. 3081 ~ ~ t e r n~r.. ~ i ststecouepe. a~ PA -
~
~
~
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~
10. costing,L.J.Ad".Rofein.Ch.m. 1951,11,4m51. 11. Bsmard.A. J.;Ahlbom, B. Am. J.Phys. lWS,43,5'773-54. 12. Krueger, D.A . Am. J Phy.. 1980,48,1IWlss. 13. Gostinga, L.J.. Fujita, H. 5. J. Am. Chem. Sm. 1981,79.135%1366. 14. Washburn, E. W. Ed. I n t e r n t i o m 1 Critical Tables of Nurnoriroi Chemhfryand Technology;McCraa: New York, 1930. 15. Bemeola, V. J. An. Sac. Quim. A r g ~ n t i m1920.8.11-31.13-99. 16. Clack, 6. W . Phyr. Soe. Proe.London L916,29,4%58. 17. Gostinga, L.J. J . Am. Chem. Soc. 1950.7% 441E4422.
