A Simple Educational Method for the Measurement of Liquid Binary

Jun 24, 2014 - A simple low-cost experiment has been developed for the measurement of the binary diffusion coefficients of liquid substances. The expe...
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A Simple Educational Method for the Measurement of Liquid Binary Diffusivities Nicholas P. Rice,†,§ Martin P. de Beer,† and Mark E. Williamson*,‡ †

Centre for Catalysis Research, Department of Chemical Engineering, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa ‡ Department of Chemical Engineering & Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA, United Kingdom S Supporting Information *

ABSTRACT: A simple low-cost experiment has been developed for the measurement of the binary diffusion coefficients of liquid substances. The experiment is suitable for demonstrating molecular diffusion to small or large undergraduate classes in chemistry or chemical engineering. Students use a cell phone camera in conjunction with open-source image processing software to measure concentrations of a colored species in a clear liquid, as a function of position and time. Three fundamental principles in mass transfer and spectrophotometry are demonstrated: Fick’s first and second laws of diffusion and the Beer-Lambert law for absorption of light. The measured value for the binary diffusion coefficient for potassium permanganate in water using this method was found to be within 10% of literature values at the 95% confidence interval. KEYWORDS: Second-Year Undergraduate, Analytical Chemistry, Physical Chemistry, Laboratory Instruction, Hands-On Learning/Manipulatives, UV−Vis Spectroscopy, Dyes/Pigments, Liquids, Laboratory Equipment/Apparatus, Transport Properties

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Meir et al.2 demonstrated improved understanding after students performed simulated experiments on a computer. These studies highlight the effectiveness of practical activities in the learning of diffusion concepts. Many experimental methods for the study of diffusion have been reported, all of which use costly apparatus, and/or require knowledge of various physical properties of the species which are not readily available. Kraus and Tye4 used a backlit capillary tube and cathetometer telescope, having first measured the refractive characteristics of the apparatus and the solvent. Nishijima and Oster5 developed an optical method using a specialized interferometer. Watts6 reported a method using a capillary tube filled with an agar gel to minimize the effects of convection, together with specialist microanalytical techniques. Similarly, Crooks7 and Fate and Lynn8 used gels with spectrophotometers or conductance meters. Linder et al.9 developed a free-boundary technique using a Polson diffusion cell and a spectrophotometer. The method presented in this article eliminates the need for expensive analytical equipment, such as spectrophotometers, by making use of

olecular diffusion is a core component of the syllabus for the teaching of undergraduate chemistry, physics, and chemical engineering. It has been reported that students find this particular concept difficult to grasp,1−3 and therefore, practical demonstrations of the phenomenon are valuable teaching aids. This article describes a simple, low-cost experimental technique which enables students to measure binary diffusion coefficients (DAB) of a colored species dissolved in a clear liquid. Many studies have shown that students find molecular diffusion difficult to conceptualise, even at third- and fourth-year undergraduate level. Westbrook and Marek1 showed that, despite advanced instruction, undergraduate-level science students had no better understanding of diffusion than seventh-grade life science students. In an investigation into the effect of simulated experiments, Meir et al.2 identified the following specific misconceptions and lack of knowledge: • the concept of concentration • the relationship between an amount of substance and its concentration • kinetic energy and random molecular motion of a species • species supposedly having a directional motion toward regions of lower concentration of that species • equilibrium and the effect on diffusion rates as equilibrium is approached Marek et al.3 showed convincingly that students who underwent a laboratory-based interactive learning cycle showed a 94% improvement in understanding. Students who attended expository teaching activities (such as lectures and discussions) showed only a 58% improvement in understanding. Similarly, © XXXX American Chemical Society and Division of Chemical Education, Inc.

• a cell phone camera • inexpensive, readily available, nontoxic reagents • open source (free) image analysis software Both steady-state diffusion (Fick’s first law) and transient diffusion (Fick’s second law) can be demonstrated, as well as the absorbance of light (Beer−Lambert law using the RGB digital color model).

A

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Kohl et al.10 showed that data from digital images, when analyzed using software such as ImageJ, conform to the Beer− Lambert law. A recent study by Kehoe and Penn11 reported the use of cell phone cameras, with backlighting provided by a blank laptop screen, for the colorimetric analysis of solutions of household chemicals. Their method allowed the concentrations of unknown samples of various colored liquids to be determined. The study presented in this article extends this method in ways that are useful for the investigation of diffusion at the undergraduate level.

Here L is the diffusion path length (m), CA(z=L,t) is the concentration of the diffusing species (mol/m3) in the bulk solvent, and CA(z=0,t) is the concentration of the saturated solution at the base of the apparatus. A mass balance on species A in the bulk solvent region gives ⎡ πd 2 ⎤ ⎤ dCA(z = L , t ) ⎡ πd 2 ⎢ t ( H − L )⎥ = |NA(t )|⎢ b ⎥ dt ⎣ 4 ⎦ ⎦ ⎣ 4

(3)

where H is the height of the bulk solvent region above the bung (m), dt is the inside diameter of the test tube, and db is the diameter of the hole in the silicone bung (m). Combining eqs 2 and 3 and integrating from time t = 0 to time t yields an expression for DAB

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THEORETICAL BASIS Estimation of DAB (m2/s) usually relies on the measurement of the concentration of the diffusing species (A) as a function of time and position within a volume of solvent (B). The equation12 describing the total flux of A relative to a fixed reference frame due to convection and diffusion described by Fick’s first law is given by C NA = A (NA + NB) − DAB∇CA (1) C where NA and NB are the molar fluxes of species A and B, respectively (mol/m2s), CA is the concentration of species A, and C is the total system concentration (mol/m3). Recorded values for concentration and time can then be used to calculate a value for DAB according to eq 1.

⎛ CA(z = L , t ) − CA(z = 0, t ) ⎞ db2 ⎟⎟ = D t ln⎜⎜ 2 AB ⎝ CA(z = L , t = 0) − CA(z = 0, t = 0) ⎠ (H − L)Ld t

(4)

Hence, measurements of the concentration of A in the apparatus as a function of elapsed time can be used to calculate a value for the binary diffusion coefficient DAB. The left-hand side of eq 4 is plotted against time, t; then the value for DAB can be calculated from the gradient of a least-squares linear regression. An important assumption in the above analysis is that the concentration of A at z = L is the same as the concentration of A in the bulk solvent region. This is a reasonable assumption if the bulk solvent is regularly mixed, for example, using a wire that can be moved gently up and down as shown in Figure 1. Measurements of the concentrated solution as a function of time are not necessary if, over the total duration of the experiment, CA(z=0,t) ≪ CA(z=0,t).This is easy to arrange in practice by making the solution as concentrated as possible, or perhaps saturating it with excess A present as solid particles in the bottom of the test tube.

Experiment 1

In the apparatus shown in Figure 1, diffusion is essentially onedimensional. After an initial period, during which the linear

Experiment 2

A second method of estimating DAB using the apparatus is to take measurements of non-steady state diffusion within the bulk solvent region. Figure 2 shows a suitable arrangement in which the silicone bung has been replaced by one with a larger hole. The flux of A into the bulk solvent region is now correspondingly

Figure 1. Diffusion cell for experiment 1 showing the pseudo-steadystate concentration profile of A.

concentration gradient is established across the silicone bung, a pseudo-steady-state condition can be assumed. Since the concentration of the diffusing species is low everywhere compared to that of the solvent (CA ≪ C), eq 1 can be simplified and integrated to give an expression for the flux of A across the bung as a function of time: D NA(t ) = − AB (CA(z = L , t ) − CA(z = 0, t )) (2) L

Figure 2. Diffusion cell for experiment 2 showing the time-dependent concentration profile of A. B

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Figure 3. Apparatus for the measurement of diffusion coefficients: (A) cell phone with a camera, (B) cardboard test tube holder, (C) laptop computer displaying white screen, (D) test tubes with standard reference solutions and diffusion cell. Photographs depicting the front and rear of the test tube holder are also shown: (E) denotes the standard solutions of known concentration, (F) is the diffusion cell, and (G) is the semiopaque light diffuser. A scale bar is shown along the top of the center image.

Therefore, it is reasonable to assign pseudo-Fickian diffusivity to the compound CuSO4. This assumption fails when more than one positive or negative ion are present in solution.

much higher than that in experiment 1, and no stirring is used. The data obtained are used in conjunction with Fick’s second law:13 ∂CA ∂ 2C = DAB 2A ∂t ∂z

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EXPERIMENTAL METHOD Concentrations of dilute colored solutions may be determined by measurements of their optical absorbance, which follows the Beer−Lambert law.14

(5)

with appropriate boundary conditions as follows: at t = 0 at z = 0 at z → ∞ (i.e., at z = H)

CA(z,t=0) = constant CA(z=0,t) = constant = CA(z=0) CA(H,t) = constant = CA(z,t=0)

⎛I⎞ A = −log10⎜ ⎟ = εlCA ⎝ I0 ⎠

for all z > 0 for all t > 0 for all t > 0

Here A is the absorbance, (I/I0) is the ratio of the intensity of transmitted light to incident light, ε is the molar absorptivity characteristic of a substance (m2/mol), l is the absorption path length (m), and CA is the concentration of the colored species (mol/m3). Glass test tubes (16 mm o.d. × 150 mm in length) were used for the diffusion cell and for the standard solutions required for calibration of the apparatus. The test tubes were placed in a cardboard holder as shown in Figure 3. An opaque plastic sheet was fitted to the back of the holder, to act as a light diffuser and to minimize reflections from the surrounding laboratory. A survey of all 135 third-year chemical engineering students at the University of Cape Town revealed that all had access to a cell phone with a camera and to a laptop computer. Therefore, in these experiments, a laptop screen was used as a uniform light source (by displaying a blank white MS PowerPoint slide) and a cell phone camera was used to measure the absorbance of the solutions. Blank MS PowerPoint slides were chosen over conventional lighting sources such as incandescent or fluorescent lights because they give rise to minimal stroboscopic interference patterns and have no intensity gradients across the screen. The distance from the laptop screen at which photos were taken was found to have little to no effect on the color information in the picture. However, photos taken from too close cause internal reflections in the test tubes to appear in the photo, and photos taken from too far have a fewer number of pixels in the region of analysis. It is therefore recommended to find the optimal distance for the particular setup used. The resulting digital photos were processed using ImageJ, an open-source image analysis program. On the basis of its availability, cost, and low toxicity, potassium permanganate was chosen as a suitable diffusing species for the experiment. It can be purchased in crystalline form from most pharmacies. Alternatively, food colorants could be used instead of potassium permanganate. These have a very intense color and will need to be diluted sufficiently. When using species where the

for which the solution is CA(z , t ) − CA(z , t = 0) CA(z = 0) − CA(z , t = 0)

⎛ z ⎞ ⎟⎟ = 1 − erf⎜⎜ ⎝ 2 DABt ⎠

(7)

(6)

Hence, plots of concentration of A through the bulk solvent region at a number of elapsed times can be used to calculate a value of DAB. This model approaches the initial concentration of the system asymptotically as z approaches infinity. Due to unavoidable mixing when setting up the experiment, the initial concentration of the bulk solvent region is often non-zero. This value is then determined and used in the model. As the diffusion is assumed to happen from the hole in the silicone bung into an infinite medium, the thickness of the bung and the height of water H are in fact not important in this experiment. The height H of the container used should however be sufficiently long to ensure the assumption of an infinite medium is still valid. Applicability of Fick’s Laws in Electrolytic Systems

The application of Fick’s laws to an electrolytic system is a significant assumption. Ionic diffusion is a complex phenomenon involving a concentration driving force, typical of Fickian diffusion, and the forces associated with maintenance of electric neutrality at any point within a solution. With copper sulfate used as an example, this compound dissolves into Cu2+ ions (ionic radius ∼0.087 nm) and SO42‑ ions (ionic radius ∼0.149 nm) which would, due to their differing size, be expected to have different Fickian diffusion properties in a given solvent. Thus, it would be possible to assign individual diffusion coefficients to ions. However, the fundamental prerequisite for electric neutrality in a solution dictates that each Cu2+ that diffuses at a given rate must be accompanied by a SO42‑ at the same rate. C

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Figure 4. The diffusion cell used in experiment 2 is on the far right. The four solutions on the left of this photo are known potassium permanganate concentrations, with their corresponding absorbance values plotted on the adjacent graph. Error bars are shown at the 95% confidence interval. The experiment was conducted in the test tube on the right; the setup is shown in detail in Figure 2. There is in good agreement with the Beer−Lambert law.

channel. A typical calibration curve, using solutions of known concentration, is shown in Figure 4. The value of DAB in experiment 2 can be found simply by calculating the cumulative squared error between the model profile and raw data at each discrete point. Microsoft Excel’s Solver add-in is then used to minimize this error by adjusting the value of DAB in a designated cell.

concentration cannot be determined easily, concentrations relative to a set of standard dilutions can be used. Calibration curves were developed from four solutions of known concentrations, ranging from pure water to the predicted end-point concentration of the diffusion experiment. The standard solution concentrations of potassium permanganate were chosen as 0.6, 0.3, 0.1, and 0 g/L (pure water). Silicone bungs were prepared by casting silicone into a mold with the outside diameter of the test tube (13.4 mm) and an inside diameter of 2 mm. Alternatively, test tube stoppers (rubber or cork) cut to the correct shape and size or washers of the correct dimensions can be used. For experiment 1, 3 mL of the concentrated colored solution (10 g/L of potassium permanganate crystals in water) was put into the bottom of the test tube using a pipet. The silicone bung was then pushed down into the test tube until it touched the surface of the liquid. The tip of a sharpened pencil was placed into the hole in the silicone bung, allowing the pencil shaft to rest against the rim of the test tube. Water was added with a pipet, so that it ran gently down the shaft of the pencil. Once the test tube was filled, the pencil was slowly withdrawn, taking care not to disturb the concentrated solution below the silicone bung. The liquid level was topped up with water, and the apparatus was left for 30 min to allow it to reach thermal equilibrium with its surroundings, and to establish the linear concentration gradient across the silicone bung. The time between readings is arbitrary as eq 4 shows a linear dependence of the ordinate value with time. In the experiments reported here, readings were taken approximately 15 h apart for a total time of 70 h (experiment 1) and 20 h (experiment 2). Immediately before each reading, the wire was used to mix the dilute solution above the silicone bung, and then five photographs of the apparatus were taken from a distance of 40 cm, with a white LCD screen for backlighting. Experiment 2 was conducted in the same way as experiment 1, except that the mixing wire was removed from the apparatus and the upper dilute solution was not mixed during the experiment. To measure intensities using the ImageJ software, a selection box was created around one test tube at a time. The mean color intensity values for the red, green, and blue color channels in the selected region were obtained using the Measure RGB tool in ImageJ. The resulting values were exported to Microsoft Excel for calculation of the corresponding absorbance values. For experiment 2, the Color Profiler tool in ImageJ was used within the same selection box. It was found that, for potassium permanganate, the best results were obtained with the blue

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RESULTS AND DISCUSSION For experiment 1, concentration-time data were obtained and plotted according to eq 4, the results of which are shown in Figure 5. The data suggest a linear relationship (consistent with

Figure 5. For experiment 1, the data plotted using equation 4 for potassium permanganate diffusing through water at ambient conditions. Error bars are shown at the 95% confidence interval. Due to the size of the error bars relative to the data markers, some error bars are hidden by the data markers.

Fick’s first law). The resulting calculated value for DAB lies within the range reported by Taylor15 and agrees to within 5% with the value reported by Anderson.16 The most significant sources of experimental error are thought to be the imperfect mixing technique used prior to measurement, and the assumption of pseudo steady state when in fact the concentration at the end of the diffusion path is changing during the interval between readings. In experiment 2, the transient concentration profiles of potassium permanganate in the bulk region of the test tube were measured. Equation 6 was then used to obtain a value for DAB (Figure 6). The constant values recorded and used in eq 6 are provided in Table 1 for clarity. D

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CONCLUSIONS In this work, we have demonstrated that basic cell phone cameras can be used by undergraduate students to measure binary diffusivity coefficients. The experimental technique presented here eliminates the need for expensive laboratory equipment, and thus allows even very large undergraduate classes to be offered practical experience of molecular diffusion.

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ASSOCIATED CONTENT

S Supporting Information *

A detailed outline of the method used on ImageJ to extract the color information from the photographs. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

Figure 6. Transient concentration profiles for experiment 2, using potassium permanganate diffusing through water.

*E-mail: [email protected] Present Address §

Table 1. Values of Concentration Used in Equation (6) as Constant for the Determination of the Transient Profiles Shown in Figure 6 Profile Time (h)

CA(z,t=0) (mol/m3)

CA(z=0) (mol/m3)

4.9 18.7

0.1354 0.1291

3.420 4.441

Department of Chemical Engineering & Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge, CB2 3RA, United Kingdom. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors wish to thank Jenni Case and Duncan Fraser for their advice on the pedagogical aspects of the work, as well as advice on publishing. We also thank Duncan Fraser for his input on the use of the technique in his first-year class.

Table 2. Diffusivity Values Obtained for Potassium Permanganate in Experiments 1 and 2 Compared to Literature values

a

Description

Measured Diffusion Coefficienta (m2/s) × 109

Temperature (°C)

Experiment 1 Experiment 2 Literature Value 115 Literature Value 216 Correlation Value17

1.30 ± 9.6% 1.05 ± 12.3% 0.5 < DAB < 1.5 1.04 ± 18% 1.78

23.2 23.3 25.0 23.2 25.0

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REFERENCES

(1) Westbrook, S.; Marek, E. A. A Cross-Age Study of Student Understanding of the Concept of Diffusion. J. Res. Sci. Teach. 1991, 28, 649−660. (2) Meir, E.; Perry, J.; Stal, D.; Maruca, S.; Klopfer, E. How Effective Are Simulated Molecular-Level Experiments for Teaching Diffusion and Osmosis? Cell. Biol. Educ. 2005, 4, 235−248. (3) Marek, E. A.; Cowan, C. C.; Cavallo, A. Students’ Misconceptions about Diffusion: How Can They Be Eliminated? Am. Biol. Teach. 1994, 56, 74−77. (4) Kraus, G.; Tye, R. An Apparatus for the Study of Diffusion in Liquids. J. Chem. Educ. 1949, 26, 489. (5) Nishijima, Y.; Oster, G. Diffusion under the Microscope. J. Chem. Educ. 1961, 38, 114−117. (6) Watts, H. Diffusion in Liquids: A Class Experiment. J. Chem. Educ. 1962, 39, 477. (7) Crooks, J. E. Measurement of Diffusion Coefficients. J. Chem. Educ. 1989, 66, 614−615. (8) Fate, G.; Lynn, D. G. Molecular Diffusion Coefficients: Experimental Determination and Demonstration. J. Chem. Educ. 1990, 67, 536−538. (9) Linder, P. W.; Nassimbeni, L. R.; Polson, A.; Rodgers, A. L. The Diffusion Coefficient of Sucrose in Water. J. Chem. Educ. 1976, 53, 330− 332. (10) Kohl, S. K.; Landmark, J. D.; Stickle, D. F. Demonstration of Absorbance Using Color Image Analysis and Colored Solutions. J. Chem. Educ. 2006, 83, 644−646. (11) Kehoe, E.; Penn, R. L. Introducing Colorimetric Analysis with Camera Phones and Digital Cameras: An Activity for High School or General Chemistry. J. Chem. Educ. 2013, 90, 1191−1195. (12) Welty, J. R.; Wicks, C. E.; Wilson, R. E.; Rorrer, G. L. Fundamentals of Mass Transfer. In Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed.; John Wiley & Sons: Hoboken, NJ, 2008; pp 398−405.

Errors are reported at the 95% confidence interval.

Good agreement was found between the measured profiles and Fick’s second law. Table 2 summarizes the measured values for DAB, compared to those reported in the literature. The most significant source of experimental error is thought to be the hemispherical diffusion gradient that will form close to the outlet of the silicone bung, which is not accounted for in the simplified 1-D mathematical analysis presented here.

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CLASSROOM EXPERIENCE AND FUTURE WORK Some of the aspects of the work reported here have already been used by undergraduate students at the University of Cape Town, South Africa. In February 2013, a class of 140 first-year undergraduate students in the Department of Chemical Engineering used their cell phone cameras to measure concentrations of a series of aqueous potassium permanganate solutions. The course lecturer noted great enthusiasm from the students for the method, and almost all students were able to make measurements of acceptable accuracy in the time available. It is consequently proposed that the complete method presented here be used in 2014 by third-year undergraduates, as the project component of their 13-week core course on mass transfer. E

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(13) Welty, J. R.; Wicks, C. E.; Wilson, R. E.; Rorrer, G. L. UnsteadyState Molecular Diffusion. In Fundamentals of Momentum, Heat, and Mass Transfer, 5th ed.; John Wiley & Sons: Hoboken, NJ, 2008; pp 496− 509. (14) Swinehart, D. F. The Beer-Lambert Law. J. Chem. Educ. 1962, 39, 333−335. (15) Taylor, G. Conditions under Which Dispersion of a Solute in a Stream of Solvent can be Used to Measure Molecular Diffusion. Proc. R. Soc. London, Ser. A. 1954, 225, 473−477. (16) Anderson, K. University of Cape Town, Cape Town, South Africa. Laboratory course diffusion experiment, 2012. (17) Vanysek, P. Ionic Conductivity and Diffusion at Infinite Dilution. In CRC Handbook of Chemistry and Physics, 92; Haynes, W. M, Ed.; CRC Press/Taylor and Francis: Boca Raton, FL, 2011; 5−78.

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